Distance-Preserving Embeddings in Inhomogeneous Random Graphs

# Beyond Worst-Case Distortions: Distance-Preserving Embeddings in Inhomogeneous Random Graphs
Source: [https://arxiv.org/html/2607.10074](https://arxiv.org/html/2607.10074)
\\nameMy Le\\emailmle19@jh\.edu \\addrDepartment of Applied Mathematics & Statistics Johns Hopkins University Baltimore, MD 21218, USA\\nameLuana Ruiz\\emaillrubini1@jh\.edu \\addrDepartment of Applied Mathematics & Statistics Johns Hopkins University Baltimore, MD 21218, USA\\nameSouvik Dhara\\emailsdhara@gatech\.edu \\addrSchool of Industrial and Systems Engineering Georgia Institute of Technology Atlanta, GA 30332, USA

###### Abstract

Graph machine learning provides powerful tools for understanding complex networks and learning meaningful node representations\. A central challenge, however, is designing embeddings with minimal distortion of*both*local and global functionals, such as shortest path lengths\. Prior distortion guarantees for distance\-preserving embeddings are worst\-case in nature, producing overly pessimistic bounds that fail to capture the structure of*typical*large\-scale networks\. To address this, we analyze shortest\-path approximation via landmark\-based embeddings on inhomogeneous random graphs \(IHGs\), a general model with type\-dependent edge probabilities\. By retaining shortest paths to a small set of reference nodes called landmarks, landmark\-based methods effectively function as virtualgraph spanners, where structural heterogeneity and controlled neighborhood expansion modeled via multi\-type branching processes enable significantly tighter dimension–distortion trade\-offs, i\.e\.Ω​\(n1−ε​log⁡n\)\\Omega\\left\(n^\{1\-\\varepsilon\}\\log n\\right\), than classical worst\-case bounds, i\.e\.Ω​\(n2​\(1−ε\)2−ε​log⁡n\)\\Omega\\left\(n^\{\\frac\{2\(1\-\\varepsilon\)\}\{2\-\\varepsilon\}\}\\log n\\right\)for\(1−ε\)\(1\-\\varepsilon\)\-distortion andΩ​\(n22\+ε​log⁡n\)\\Omega\\left\(n^\{\\frac\{2\}\{2\+\\varepsilon\}\}\\log n\\right\)for\(1\+ε\)\(1\+\\varepsilon\)\-distortion\. We extend these guarantees to global, component\-wide averages and unify the analysis across finite\-type and continuous latent spaces through a novel metric sandwiching framework, establishing universal distortion bounds for generalL2L^\{2\}kernel models, including heavy\-tailed and power\-law networks\. Finally, we introduce a GNN\-augmented variant that replaces rigid, computationally expensive exact shortest\-path queries with flexible, structure\-aware neural surrogates\. By leveraging the inherent alignment between graph neural message\-passing and the dynamic programming principles of shortest\-path algorithms, our approach demonstrates that models trained on small\-scale random graphs learn to extract universal distance\-preserving features, achieving robust generalization to large\-scale, real\-world networks that match or exceed the fidelity of classical, exact landmark\-based embeddings\.

Keywords:shortest path, distance\-preserving embeddings, landmarks, graph spanners, inhomogeneous random graphs, heterogeneity, graph neural networks, transferability

## 1Introduction

A central challenge in graph learning is to represent networked data in a form that faithfully captures its essential structural characteristics, including local connectivity patterns, mesoscopic organization, and global topology\. A common strategy is to map nodes into low\-dimensional metric spaces so that distances between embedded points reflect the graph’s original structure\. These embeddings provide a compact and mathematically tractable representation of the network, enabling both statistical analysis and efficient computation at scale in a wide range of inference tasks, including node classification, link prediction, clustering, and routing\(Hamiltonet al\.[2017b](https://arxiv.org/html/2607.10074#bib.bib239), Grover and Leskovec[2016](https://arxiv.org/html/2607.10074#bib.bib240), Belkin and Niyogi[2003](https://arxiv.org/html/2607.10074#bib.bib244)\)\.

Despite significant progress, node embedding techniques are primarily designed to capture local and mesoscopic\-scale structure, failing to accurately preserve global graph functionals\. In particular, a key global functional that is often neglected are shortest\-path distances, which are central to routing, navigation, and network efficiency\. Shortest path distances are often distorted by low\-dimensional embeddings, especially in large\-scale and structurally heterogeneous graphs\(Goyal and Ferrara[2018](https://arxiv.org/html/2607.10074#bib.bib246), Tsitsulinet al\.[2018](https://arxiv.org/html/2607.10074#bib.bib247), Brunner[2021](https://arxiv.org/html/2607.10074#bib.bib261)\)\. This failure to preserve the metric structure of the graph requires a more principled approach to distance\-aware embeddings\.

*Landmark\-based*distance embeddings represent nodes via their distances to a small set of reference nodes called landmarks, naturally inducing a probabilistic virtualgraph spanner\(Ahmedet al\.[2020](https://arxiv.org/html/2607.10074#bib.bib283)\)\. Rooted in both scalable shortest\-path approximation algorithms\(Sarmaet al\.[2010](https://arxiv.org/html/2607.10074#bib.bib16), Potamiaset al\.[2009](https://arxiv.org/html/2607.10074#bib.bib258), Tretyakovet al\.[2011](https://arxiv.org/html/2607.10074#bib.bib259)\)and the mathematical theory of metric embeddings into Hilbert spaces\(Bourgain[1985](https://arxiv.org/html/2607.10074#bib.bib19), Matoušek[1996](https://arxiv.org/html/2607.10074#bib.bib250), Linialet al\.[1995](https://arxiv.org/html/2607.10074#bib.bib249)\), these methods are widely used in practice, offering a coordinate\-based alternative to traditional edge\-deletion spanners\(Peleg and Schäffer[1989](https://arxiv.org/html/2607.10074#bib.bib284), Althöferet al\.[1993](https://arxiv.org/html/2607.10074#bib.bib285)\)while enjoying provable\(1±ε\)\(1\\pm\\varepsilon\)\-distortion guarantees\(Bourgain[1985](https://arxiv.org/html/2607.10074#bib.bib19), Sarmaet al\.[2010](https://arxiv.org/html/2607.10074#bib.bib16)\)\.

Despite their empirical success, existing theoretical guarantees require embedding dimension ofΩ​\(n2​\(1−ε\)2−ε​log⁡n\)\\Omega\\left\(n^\{\\frac\{2\(1\-\\varepsilon\)\}\{2\-\\varepsilon\}\}\\log n\\right\)for\(1−ε\)\(1\-\\varepsilon\)\-distortion andΩ​\(n22\+ε​log⁡n\)\\Omega\\left\(n^\{\\frac\{2\}\{2\+\\varepsilon\}\}\\log n\\right\)for\(1\+ε\)\(1\+\\varepsilon\)\-distortion, which grows polynomially innnand becomes prohibitive at scale\(Matoušek[1996](https://arxiv.org/html/2607.10074#bib.bib250), Sarmaet al\.[2012](https://arxiv.org/html/2607.10074#bib.bib17), Loukas[2020](https://arxiv.org/html/2607.10074#bib.bib15)\)\. These bounds hold for arbitrary graphs and are tight in the worst case, but may be overly pessimistic for the structured, heterogeneous networks that arise in practice\. This raises the question of whether sharper guarantees are achievable under more realistic assumptions on the graph structure\.

In this paper, we analyze landmark\-based distance embeddings on*inhomogeneous random graphs*\(IHGs\), a flexible model that subsumes variants of the stochastic block model and Chung–Lu\-type models\(Chung and Lu[2002](https://arxiv.org/html/2607.10074#bib.bib268), Bollobáset al\.[2007](https://arxiv.org/html/2607.10074#bib.bib264),\\swapHofstadvan der[2024a](https://arxiv.org/html/2607.10074#bib.bib270)\), and captures structural heterogeneity such as community organization and degree variability observed in real\-world networks\. Rather than seeking worst\-case guarantees that hold for arbitrary graphs, we adopt an “average\-case” perspective grounded in random graph theory, analyzing how type\-dependent connectivity governs neighborhood expansion and, in turn, the geometry of landmark\-based distance\-preserving embeddings\.

#### Theoretical Contributions\.

Our main contribution is to demonstrate that, for a broad class of IHGs with either discrete or continuous kernels, landmark\-based embeddings achieve a polynomial improvement in the distortion–dimension trade\-off compared to classical worst\-case guarantees\. We establish these results by characterizing the fine\-grained expansion of local neighborhoods through the lens of multi\-type branching processes\(Athreya and Ney[2012](https://arxiv.org/html/2607.10074#bib.bib279),\\swapHofstadvan der[2024a](https://arxiv.org/html/2607.10074#bib.bib270)\)\. This probabilistic framework bridges local, heterogeneous node connectivity with the global operator\-theoretic properties of the graph, enabling us to map precisely how structural variance inside an IHG controls distance propagation\. To this end, our theoretical framework delivers a three\-fold contribution:

- •Sharp Point\-Wise Guarantees \(Theorems[4\.1](https://arxiv.org/html/2607.10074#S4.ThmtheoremEnv1)and[4\.2](https://arxiv.org/html/2607.10074#S4.ThmtheoremEnv2)\):Within the supercritical regime of IHGs with finitely many types where a unique giant component emerges with high probability \(w\.h\.p\., i\.e\., with probability approaching 1 asn→∞n\\to\\infty\), we obtain embedding dimensionΩ​\(n1−ε​log⁡n\)\\Omega\\left\(n^\{1\-\\varepsilon\}\\log n\\right\)for both\(1±ε\)\(1\\pm\\varepsilon\)\-distortions, which is smaller than the worst\-case bounds\. Since the gain in the embedding dimension isnO​\(ε\)n^\{O\(\\varepsilon\)\}, it diminishes asε→0\\varepsilon\\to 0\. This behavior is consistent with the classical work on Lipschitz embeddings byBourgain \([1985](https://arxiv.org/html/2607.10074#bib.bib19)\), which shows that dense random graphs achieve nearly worst\-case distortion bounds\.
- •Global Average\-Case Stability \(Theorem[4\.5](https://arxiv.org/html/2607.10074#S4.ThmtheoremEnv5)\):We extend our point\-wise guarantees to the entire topology by showing that the empirical spatial average of the metric distortion—normalized over all valid, connected node pairsU=\{\(u1,u2\):u1↔u2,u1≠u2\}U=\\\{\(u\_\{1\},u\_\{2\}\):u\_\{1\}\\leftrightarrow u\_\{2\},u\_\{1\}\\neq u\_\{2\}\\\}—concentrates tightly within a\(1±ε\)\(1\\pm\\varepsilon\)window w\.h\.p\. By decoupling the spatial summation, we prove that the mass of pathologically behaving configurations is asymptotically negligible, ensuring that anomalous structural bottlenecks cannot corrupt global downstream empirical risk optimization\.
- •Universal Continuum Extension \(Theorems[5\.1](https://arxiv.org/html/2607.10074#S5.ThmtheoremEnv1)and[5\.2](https://arxiv.org/html/2607.10074#S5.ThmtheoremEnv2)\):We generalize our finite\-type results to continuous latent spaces via a rigorous probability space coupling that sandwiches an arbitraryL2L^\{2\}kernelκ\\kappabetween two finite step\-function kernelsκδ±\\kappa\_\{\\delta\}^\{\\pm\}\. Under supercriticality, the global metric structure of the continuous network behaves as a stable functional limit of these finite approximations as the approximation resolutionδ→0\\delta\\to 0\. This sandwiching argument allows us to transfer our polynomial dimension\-distortion trade\-offs to non\-parametric, heavy\-tailed network architectures, including unbounded power\-law Chung–Lu configurations\.

Methodological Contributions\.Building on the theoretical framework, we propose a GNN\-augmented variant of landmark\-based embeddings that learns to approximate distances to landmarks directly from graph structure, replacing exact shortest\-path computations\. GNNs are well\-suited for this task due to their alignment with dynamic programming, which underpins shortest\-path algorithms\(Xuet al\.[2019b](https://arxiv.org/html/2607.10074#bib.bib237), Dudzik and Veličković[2022](https://arxiv.org/html/2607.10074#bib.bib238)\)\. Our experiments demonstrate that GNN\-based embeddings match or improve upon exact landmark embeddings, particularly in the strongly supercritical regime where neighborhood expansion grows exponentially at a rate determined by the spectral radius of the graph\. More strikingly, GNNs trained on small graphs generalize effectively to much larger graphs and to real\-world networks, highlighting the practical value of the IHG framework as a training ground for scalable, distance\-aware graph representations\.

### 1\.1Related Work

The fundamental limits of distance\-preserving graph embeddings have been studied extensively\. \([1985](https://arxiv.org/html/2607.10074#bib.bib19)\) established that preserving all pairwise distances up to a factor of\(1±ε\)\(1\\pm\\varepsilon\)requires dimension at leastkε=Ω​\(\(log⁡n\)2/\(log⁡log⁡n\)2\)k\_\{\\varepsilon\}=\\Omega\\big\(\{\(\\log n\)^\{2\}\}/\{\(\\log\\log n\)^\{2\}\}\\big\)for worst\-case graphs\. Subsequent works sharpened these bounds\.[Linialet al\.](https://arxiv.org/html/2607.10074#bib.bib249)\([1995](https://arxiv.org/html/2607.10074#bib.bib249)\) showedkε=Ω​\(\(log⁡n\)2\)k\_\{\\varepsilon\}=\\Omega\(\(\\log n\)^\{2\}\), and[Matoušek](https://arxiv.org/html/2607.10074#bib.bib250)\([1996](https://arxiv.org/html/2607.10074#bib.bib250)\) refined it tokε=Ω​\(nc/\(1\+ε\)\)k\_\{\\varepsilon\}=\\Omega\\left\(n^\{c/\(1\+\\varepsilon\)\}\\right\)for certain graph families\. More recently,[Naor](https://arxiv.org/html/2607.10074#bib.bib251)\([2016](https://arxiv.org/html/2607.10074#bib.bib251)\) and[Naor](https://arxiv.org/html/2607.10074#bib.bib252)\([2021](https://arxiv.org/html/2607.10074#bib.bib252)\) demonstrated that graphs with strong expansion properties require polynomial\-dimensional embeddings, underscoring the difficulty of distance preservation in the worst case\.

On the algorithmic side, landmark\-based methods have been extensively studied as practical alternatives\(Goldberg and Harrelson[2005](https://arxiv.org/html/2607.10074#bib.bib257), Sarmaet al\.[2010](https://arxiv.org/html/2607.10074#bib.bib16), Potamiaset al\.[2009](https://arxiv.org/html/2607.10074#bib.bib258), Tretyakovet al\.[2011](https://arxiv.org/html/2607.10074#bib.bib259), Akibaet al\.[2013](https://arxiv.org/html/2607.10074#bib.bib233), Riziet al\.[2018](https://arxiv.org/html/2607.10074#bib.bib263), Qiet al\.[2020](https://arxiv.org/html/2607.10074#bib.bib262)\), with[Sommer](https://arxiv.org/html/2607.10074#bib.bib42)\([2014](https://arxiv.org/html/2607.10074#bib.bib42)\) providing a comprehensive survey\. These methods select a small subset of reference nodes and approximate pairwise distances via the triangle inequality, trading exactness for scalability\. While empirically effective, their theoretical guarantees inherit the pessimism of worst\-case analyses, motivating the average\-case perspective taken in this paper\.

A broader line of work in graph representation learning studies how structural properties such as local connectivity, higher\-order proximities, and spectral geometry govern the quality of learned embeddings\. Random\-walk\-based methods such as DeepWalk\(Perozziet al\.[2014](https://arxiv.org/html/2607.10074#bib.bib241)\)and Node2Vec\(Grover and Leskovec[2016](https://arxiv.org/html/2607.10074#bib.bib240)\)capture local structure, while GraRep\(Caoet al\.[2015](https://arxiv.org/html/2607.10074#bib.bib242)\), PRONE\(Zhanget al\.[2021](https://arxiv.org/html/2607.10074#bib.bib243)\), and adjacency search embeddings\(Chaitanyaet al\.[2025](https://arxiv.org/html/2607.10074#bib.bib267)\)extend this to higher\-order proximities\. Spectral approaches including Laplacian Eigenmaps\(Belkin and Niyogi[2003](https://arxiv.org/html/2607.10074#bib.bib244)\)capture coarse global structure but can be sensitive to degree variability and edge heterogeneity\(Chung[1997](https://arxiv.org/html/2607.10074#bib.bib272), Von Luxburg[2007](https://arxiv.org/html/2607.10074#bib.bib273)\)\. Cauchy embeddings\(Tanget al\.[2019](https://arxiv.org/html/2607.10074#bib.bib245)\)address this via heavy\-tailed similarity measures, improving robustness in heterogeneous networks\. None of these methods are designed to preserve the metric structure of the graph explicitly, which is the focus of this work\.

Our work is intimately related to the extensive literature on*graph spanners*in theoretical computer science\(Ahmedet al\.[2020](https://arxiv.org/html/2607.10074#bib.bib283)\)\. Introduced to compress network topologies while bounding distance distortion, a classicaltt\-spanner selects a sparse subgraph wherein the shortest path distance between any pair of nodes is at mosttttimes their true distance\(Peleg and Schäffer[1989](https://arxiv.org/html/2607.10074#bib.bib284), Althöferet al\.[1993](https://arxiv.org/html/2607.10074#bib.bib285)\)\. While traditional spanner construction algorithms are primarily algorithmic and worst\-case deterministic, our work provides an average\-case, probabilistic counterpart\. Rather than explicitly computing a sparse physical subgraph, our multi\-scale landmark framework acts as a virtual distance spanner, leveraging the underlying spectral expansion of inhomogeneous random graphs to guarantee a\(1±ε\)\(1\\pm\\varepsilon\)metric stretch w\.h\.p\. using sub\-linear storage overhead and query time\.

Parallel to these representations is the literature on dense and sparse graph limits \(graphons and kernels\), which model large networks as continuous latent spaces\(Bollobáset al\.[2007](https://arxiv.org/html/2607.10074#bib.bib264), Lovász[2012](https://arxiv.org/html/2607.10074#bib.bib123)\)\. While graph limit theory provides strong structural and phase\-transition guarantees, its explicit geometric implications for shortest\-path metric spaces have remained largely unmapped\. None of the aforementioned representation frameworks are designed to explicitly preserve the metric structure of the graph across continuous limits, which is a focus of this work\.

## 2Landmark\-Based Distance\-Preserving Embeddings

In this section, we introduce shortest\-path distances and describe landmark\-based embeddings for distance\-preserving node representation, along with their worst\-case distortion guarantees\. Classical algorithms for exact shortest\-path computation are reviewed briefly to motivate the need for approximate, embedding\-based approaches at scale\.

Throughout, we consider an undirected and unweighted graphG=\(V,E\)G=\(V,E\)with vertex setVVof sizennand edge setEEof sizemm\. Connected components ofGGare indexed in decreasing order of size, and𝒞\(i\)\\mathcal\{C\}\_\{\(i\)\}denotes theii\-th largest connected component\. For any two verticesu1,u2∈Vu\_\{1\},u\_\{2\}\\in V, the notationu1↔u2u\_\{1\}\\leftrightarrow u\_\{2\}indicates thatuuandvvare connected by a path, i\.e\., they belong to the same connected component\.

### 2\.1Shortest Path Distances

For a graphG=\(V,E\)G=\(V,E\)and verticesu1,u2∈Vu\_\{1\},u\_\{2\}\\in V, the shortest path distance \(or simply shortest distance\)d​\(u1,u2\)d\(u\_\{1\},u\_\{2\}\)is defined as the number of edges in a path with the minimum number of edges among all paths connectinguuandvv\. Computing shortest distances is a fundamental problem in graph theory and combinatorial optimization, and serves as a core primitive in our analysis\.

Classical algorithms for computing shortest paths include Dijkstra’s algorithm and its variants\. When implemented with simple data structures, Dijkstra’s algorithm runs inO​\(n2\)O\(n^\{2\}\)time for a single source andO​\(n3\)O\(n^\{3\}\)time for all node pairs\. More efficient implementations based on priority queues reduce the single\-source complexity toO​\(m​log⁡n\)O\(m\\log n\), or toO​\(m\+n​log⁡n\)O\(m\+n\\log n\)in sparse graphs\(Schrijver[2012](https://arxiv.org/html/2607.10074#bib.bib23)\)\.

A number of refinements have been proposed for the single\-source setting, building on early work byDijkstra \([1959](https://arxiv.org/html/2607.10074#bib.bib49)\)andMoore \([1959](https://arxiv.org/html/2607.10074#bib.bib50)\)\. These include bucket\-based methods such as S\-Dial, which achieves a running time ofO​\(m\+n​ℓmax\)O\(m\+n\\ell\_\{\\max\}\)whereℓmax\\ell\_\{\\max\}is the maximum edge length, as well as heap\-based approaches with complexityO​\(m​log⁡n\)O\(m\\log n\)orO​\(n​log⁡n\)O\(n\\log n\)in sparse regimes\(Gallo and Pallottino[1988](https://arxiv.org/html/2607.10074#bib.bib51)\)\. Further improvements were introduced by Fredman and Willard, yielding anO​\(m\+n​log⁡n/log⁡log⁡n\)O\(m\+n\\log n/\\log\\log n\)time algorithm\(Fredman and Willard[1990](https://arxiv.org/html/2607.10074#bib.bib53)\)\.

For the all\-pairs shortest path problem, classical methods such as the Floyd–Warshall algorithm requireO​\(n3\)O\(n^\{3\}\)time\(Gallo and Pallottino[1988](https://arxiv.org/html/2607.10074#bib.bib51)\), while the more advanced hidden\-path algorithm achievesO​\(m​n\+n2​log⁡n\)O\(mn\+n^\{2\}\\log n\)complexity\(Kargeret al\.[1993](https://arxiv.org/html/2607.10074#bib.bib52)\)\. Nevertheless, these remain prohibitive for large graphs, motivating approximate methods and in particular embedding\-based approaches for distance estimation\.

### 2\.2Landmark\-Based Embeddings

Although computing exact shortest distances can be prohibitively expensive on large graphs, often one can precompute distances to local landmarks and use these values to approximate shortest path distances efficiently\(Sarmaet al\.[2010](https://arxiv.org/html/2607.10074#bib.bib16)\)\. This is the core idea of landmark\-based distance\-preserving embeddings\. Explicitly, their construction involves two steps:

Local Step\.Sampler\+1r\+1sets of landmark nodesS0,S1,…,Sr⊂VS\_\{0\},S\_\{1\},\\dots,S\_\{r\}\\subset V\. For each vertexu∈Vu\\in Vand each landmark setSjS\_\{j\}, compute

\[𝐱u\]j=mins∈Sj⁡d​\(u,s\),\[𝝈u\]j=arg⁡mins∈Sj⁡d​\(u,s\),\[\\mathbf\{x\}\_\{u\}\]\_\{j\}=\\min\_\{s\\in S\_\{j\}\}d\(u,s\),\\qquad\[\\bm\{\\sigma\}\_\{u\}\]\_\{j\}=\\arg\\min\_\{s\\in S\_\{j\}\}d\(u,s\),whered​\(u,s\)d\(u,s\)is the shortest distance betweenuuandss\. Here,\[𝐱u\]j\[\\mathbf\{x\}\_\{u\}\]\_\{j\}stores the distance fromuuto its closest landmark inSjS\_\{j\}, and\[𝝈u\]j\[\\bm\{\\sigma\}\_\{u\}\]\_\{j\}records the corresponding landmark\. Collecting these for allj=0,…,rj=0,\\dots,rgives the*local embedding*𝐱u\\mathbf\{x\}\_\{u\}and associated landmark indices𝝈u\\bm\{\\sigma\}\_\{u\}for vertexuu\.

Global Step\.Given local embeddings𝐱u1\\mathbf\{x\}\_\{u\_\{1\}\}and𝐱u2\\mathbf\{x\}\_\{u\_\{2\}\}for two verticesu1,u2∈Vu\_\{1\},u\_\{2\}\\in V, a lower bound on the shortest path distance is obtained as

d¯​\(u1,u2\)=‖𝐱u1−𝐱u2‖∞\.\\underline\{d\}\(u\_\{1\},u\_\{2\}\)=\\\|\\mathbf\{x\}\_\{u\_\{1\}\}\-\\mathbf\{x\}\_\{u\_\{2\}\}\\\|\_\{\\infty\}\.An upper bound is obtained by combining distances through common landmarks:

d¯​\(u1,u2\)=min⁡\{\[𝐱u1\]i\+\[𝐱u2\]j:\[𝝈u1​𝟏⊤−𝟏​𝝈u2⊤\]i​j=0\}\.\\bar\{d\}\(u\_\{1\},u\_\{2\}\)=\\min\\\{\[\\mathbf\{x\}\_\{u\_\{1\}\}\]\_\{i\}\+\[\\mathbf\{x\}\_\{u\_\{2\}\}\]\_\{j\}\\ :\\ \[\\bm\{\\sigma\}\_\{u\_\{1\}\}\\mathbf\{1\}^\{\\top\}\-\\mathbf\{1\}\\bm\{\\sigma\}\_\{u\_\{2\}\}^\{\\top\}\]\_\{ij\}=0\\\}\.
These bounds follow directly from the triangle inequality\. The lower boundd¯​\(u1,u2\)\\underline\{d\}\(u\_\{1\},u\_\{2\}\)relies solely on coordinate\-wise differences and thus involves a search overr\+1r\+1coordinates\. In contrast, the upper boundd¯​\(u1,u2\)\\bar\{d\}\(u\_\{1\},u\_\{2\}\)considers all pairs of coordinates\(i,j\)\(i,j\)such that the closest landmarks coincide and requires at least one landmark set to have strictly one node to preventd¯​\(u1,u2\)\\bar\{d\}\(u\_\{1\},u\_\{2\}\)from being undefined\. In the worst case, this requires checking all\(r\+1\)2\(r\+1\)^\{2\}pairs of landmark indices, since each coordinate of𝐱u1\\mathbf\{x\}\_\{u\_\{1\}\}may correspond to a different landmark than that of𝐱u2\\mathbf\{x\}\_\{u\_\{2\}\}\. Hence, the effective search space for computingd¯​\(u1,u2\)\\bar\{d\}\(u\_\{1\},u\_\{2\}\)can scale quadratically with the embedding dimension\.

From a practical standpoint, this two\-step procedure allows the landmark embeddings computed in the local step to be stored and later retrieved to efficiently computed¯​\(u1,u2\)\\underline\{d\}\(u\_\{1\},u\_\{2\}\)andd¯​\(u1,u2\)\\bar\{d\}\(u\_\{1\},u\_\{2\}\)as approximations ofd​\(u1,u2\)d\(u\_\{1\},u\_\{2\}\)on demand\. The key advantage of landmark\-based algorithms is that they avoid recomputing paths for each query, substantially reducing computational and memory costs while still preserving the graph’s structural information\.

Such lower and upper bounds have been shown to provide reliable estimates of exact distances\(Bourgain[1985](https://arxiv.org/html/2607.10074#bib.bib19), Matoušek[1996](https://arxiv.org/html/2607.10074#bib.bib250), Sarmaet al\.[2010](https://arxiv.org/html/2607.10074#bib.bib16), Gubichevet al\.[2010](https://arxiv.org/html/2607.10074#bib.bib45), Sommer[2014](https://arxiv.org/html/2607.10074#bib.bib42), Akibaet al\.[2014](https://arxiv.org/html/2607.10074#bib.bib46), Menget al\.[2015](https://arxiv.org/html/2607.10074#bib.bib43), Jianget al\.[2021](https://arxiv.org/html/2607.10074#bib.bib47), Awasthiet al\.[2022](https://arxiv.org/html/2607.10074#bib.bib18)\), enabling rapid online queries for individual node pairs as well as efficient all\-pairs distance approximation\. The quality of these approximations is characterized by their distortion relative to the true shortest\-path distances\. Provable guarantees on this distortion holding for any graph are discussed next\.

### 2\.3Distortion on Worst\-Case Graphs

The distortion guarantee for the shortest distance lower bound follows from[Matoušek](https://arxiv.org/html/2607.10074#bib.bib250)\([1996](https://arxiv.org/html/2607.10074#bib.bib250)\), building on Bourgain’s embedding theorem \([1985](https://arxiv.org/html/2607.10074#bib.bib19)\)\.

###### Theorem 2\.1\(Worst\-Case Lower\-Bound Distortion\)

LetG=\(V,E\)G=\(V,E\)be a graph withn≥3n\\geq 3nodes and letu1,u2∈Vu\_\{1\},u\_\{2\}\\in V\. For anyc\>1c\>1, there exist embeddings𝐱u1∗,𝐱u2∗∈ℝD\{\\mathbf\{x\}\}^\{\*\}\_\{u\_\{1\}\},\{\\mathbf\{x\}\}^\{\*\}\_\{u\_\{2\}\}\\in\{\\mathbb\{R\}\}^\{D\}withD=Ω​\(n1/c​log⁡n\)D=\\Omega\(n^\{1/c\}\\log n\)such that the lower\-bound estimatord¯​\(u1,u2\)\\underline\{d\}\(u\_\{1\},u\_\{2\}\)satisfies

12​c−1​d​\(u1,u2\)≤d¯​\(u1,u2\)≤d​\(u1,u2\)\.\\frac\{1\}\{2c\-1\}d\(u\_\{1\},u\_\{2\}\)\\leq\\underline\{d\}\(u\_\{1\},u\_\{2\}\)\\leq d\(u\_\{1\},u\_\{2\}\)\.

The distortion guarantee for the upper bound was derived by[Sarmaet al\.](https://arxiv.org/html/2607.10074#bib.bib16)\([2010](https://arxiv.org/html/2607.10074#bib.bib16)\)\.

###### Theorem 2\.2\(Worst\-Case Upper\-Bound Distortion\)

LetG=\(V,E\)G=\(V,E\)be a graph withn≥3n\\geq 3nodes and letu1,u2∈Vu\_\{1\},u\_\{2\}\\in V\. For anyc\>1c\>1, there exist embeddings𝐱u1∗,𝐱u2∗∈ℝD\{\\mathbf\{x\}\}^\{\*\}\_\{u\_\{1\}\},\{\\mathbf\{x\}\}^\{\*\}\_\{u\_\{2\}\}\\in\{\\mathbb\{R\}\}^\{D\}withD=Ω​\(n1/c​log⁡n\)D=\\Omega\(n^\{1/c\}\\log n\)such that the upper\-bound estimatord¯​\(u1,u2\)\\bar\{d\}\(u\_\{1\},u\_\{2\}\)satisfies

d​\(u1,u2\)≤d¯​\(u1,u2\)≤\(2​c−1\)​d​\(u1,u2\)\.d\(u\_\{1\},u\_\{2\}\)\\leq\\bar\{d\}\(u\_\{1\},u\_\{2\}\)\\leq\(2c\-1\)d\(u\_\{1\},u\_\{2\}\)\.

The distortion bounds in Theorems[2\.1](https://arxiv.org/html/2607.10074#S2.ThmtheoremEnv1)and[2\.2](https://arxiv.org/html/2607.10074#S2.ThmtheoremEnv2)rely on embeddings that are optimal in an information\-theoretic sense, i\.e\., in the sense that they match known lower bounds on the minimum dimension required to preserve all pairwise distances \(within a prescribed distortion\) in worst\-case metrics\(Bourgain[1985](https://arxiv.org/html/2607.10074#bib.bib19), Matoušek[1996](https://arxiv.org/html/2607.10074#bib.bib250), Sarmaet al\.[2010](https://arxiv.org/html/2607.10074#bib.bib16)\)\. More precisely, these embeddings achieve the best possible trade\-off between dimension and distortion up to constant factors, independent of computational considerations\. However, Theorems[2\.1](https://arxiv.org/html/2607.10074#S2.ThmtheoremEnv1)and[2\.2](https://arxiv.org/html/2607.10074#S2.ThmtheoremEnv2)only guarantee the existence of embeddings achieving the stated dimension–distortion trade\-offs; they do not ensure that the landmark construction described in Section[2\.2](https://arxiv.org/html/2607.10074#S2.SS2)realizes them\.

Indeed, the choice of landmark sampling is critical\.[Sarmaet al\.](https://arxiv.org/html/2607.10074#bib.bib16)\([2010](https://arxiv.org/html/2607.10074#bib.bib16)\) advocate a multiscale sampling scheme in which landmark setsSiS\_\{i\}are drawn at exponentially increasing scales, with\|Si\|=2i\|S\_\{i\}\|=2^\{i\}fori=0,…,ri=0,\\ldots,r\. This multiresolution design serves complementary roles in controlling the two estimatorsd¯\\underline\{d\}andd¯\\bar\{d\}\. For the lower bound, small\-scale landmark sets are required to place a landmark within theσ1​d​\(u1,u2\)\\sigma\_\{1\}d\(u\_\{1\},u\_\{2\}\)\-hop neighborhood ofu1u\_\{1\}while avoiding the largerσ2​d​\(u1,u2\)\\sigma\_\{2\}d\(u\_\{1\},u\_\{2\}\)\-hop neighborhood ofu2u\_\{2\}, with0<σ1<σ20<\\sigma\_\{1\}<\\sigma\_\{2\}satisfyingσ1\+σ2<1\\sigma\_\{1\}\+\\sigma\_\{2\}<1\(see Figure[1](https://arxiv.org/html/2607.10074#S4.F1)\)\. Conversely, for the upper bound, larger landmark sets allow w\.h\.p\. the presence of at least one landmark in the overlap of the⌈d​\(u1,u2\)/2⌉\\lceil d\(u\_\{1\},u\_\{2\}\)/2\\rceil\-hop neighborhoods ofu1u\_\{1\}andu2u\_\{2\}\. We adopt this sampling scheme in our analysis, generating sets of size\|S0\|=M0\|S\_\{0\}\|=M^\{0\},\|S1\|=M1\|S\_\{1\}\|=M^\{1\}, …,\|Sr\|=Mr\|S\_\{r\}\|=M^\{r\}for some integerM\>1M\>1\.

Another consideration is mitigating the effect of poor quality sample sets\. To address it, we sampleRRsets of each size\. WhenR=Ω​\(n1/c\)R=\\Omega\(n^\{1/c\}\)and the landmark set sizes grow exponentially withr=O​\(log⁡n\)r=O\(\\log n\), the total embedding dimension scales asΘ​\(n1/c​log⁡n\)\\Theta\(n^\{1/c\}\\log n\)\. Under this regime, the resulting distance estimates satisfy the guarantees of Theorems[2\.1](https://arxiv.org/html/2607.10074#S2.ThmtheoremEnv1)and[2\.2](https://arxiv.org/html/2607.10074#S2.ThmtheoremEnv2)simultaneously for all node pairs on arbitrary graphs w\.h\.p\.

While these guarantees hold for any graph, they may be overly pessimistic for structured networks\. In the following sections, we show that for inhomogeneous random graphs—a broad class of models capturing sparsity, community structure, and heterogeneous connectivity, significantly tighter distortion–dimension trade\-offs are achievable\.

## 3Inhomogeneous Random Graphs \(IHGs\)

In the following we formally introduce the inhomogeneous random graph model\. This model captures structural heterogeneity through type\-dependent connectivity and provides a tractable probabilistic framework for studying neighborhood expansion and distance behavior\. We then formalize a set of assumptions under which the graph exhibits controlled growth and concentration properties, enabling a precise characterization of neighborhood dynamics \(Section[4\.3](https://arxiv.org/html/2607.10074#S4.SS3)\) that will be used in our main results \(Section[4\.1](https://arxiv.org/html/2607.10074#S4.SS1)\)\.

### 3\.1Graph Model

The inhomogeneous graph model withnnnodes andTTdistinct node types will be denotedI​H​G​\(n→,D\)IHG\(\\vec\{n\},D\), wheren→=\[n1,…,nT\]∈ℕT\\vec\{n\}=\[n\_\{1\},\\dots,n\_\{T\}\]\\in\\mathbb\{N\}^\{T\}is the*type size vector*andD∈ℝT×TD\\in\\mathbb\{R\}^\{\{T\}\\times\{T\}\}is the*affinity*\(or*connectivity*\) matrix\. There arentn\_\{t\}nodes of each typet∈\[T\]=\{1,…,T\}t\\in\[T\]=\\\{1,\\dots,\{T\}\\\}, and each node belongs to exactly one type\. The probability of an edge between a node of typepp\(parent\) and a node of typecc\(child\) isPp​c=Dp​c/ncP\_\{pc\}=D\_\{pc\}/n\_\{c\}with0≤Dp​c≤nc0\\leq D\_\{pc\}\\leq n\_\{c\}\(Chung and Lu[2002](https://arxiv.org/html/2607.10074#bib.bib268), Bollobáset al\.[2007](https://arxiv.org/html/2607.10074#bib.bib264)\)\.

The matrixDDis non\-negative and has a symmetric support, meaningDi​j\>0D\_\{ij\}\>0if and only ifDj​i\>0D\_\{ji\}\>0\. However,DDis not necessarily symmetric: even in undirected graphs wherePp​c=Pc​pP\_\{pc\}=P\_\{cp\}, differing type sizes withnc≠npn\_\{c\}\\neq n\_\{p\}result inDp​c≠Dc​pD\_\{pc\}\\neq D\_\{cp\}\. Consequently, the spectral radiusρ​\(D\)\\rho\(D\)is at most its spectral norm‖D‖2\\\|D\\\|\_\{2\}but not necessarily equal, and the largest magnitude eigenvalueλ1\\lambda\_\{1\}has value at most‖D‖2\\\|D\\\|\_\{2\}\.

This model captures heterogeneity in network structure by allowing edge probabilities to depend on the types of the endnodes, rather than being uniform across all nodes\. At the same time, it generalizes the classical Erdős–Rényi random graph since, ifT=1T=1, then the model reduces to an Erdős–Rényi graph with connection probabilityp=D11/n1p=D\_\{11\}/\{n\_\{1\}\}\. The type\-dependent connectivity matrixDDcontrols the expected number of neighbors of each type and introduces structured variability in local neighborhoods, which can better reflect realistic networks where nodes exhibit group\-specific interaction patterns\.

### 3\.2Model Assumptions

The structure of the affinity matrixDD—whether it is reducible or irreducible, periodic or aperiodic—determines the asymptotic growth and support of its powersDkD^\{k\}\. In particular, the entry\[Dk\]i​j\[D^\{k\}\]\_\{ij\}admits a natural combinatorial interpretation, corresponding to the total weighted number of length\-kktype sequences through which interactions originating from typejjcan influence typeii\. When\[Dk\]i​j=ℓ\>0\[D^\{k\}\]\_\{ij\}=\\ell\>0, this indicates that there existℓ\\elldistinct admissible walks of lengthkkin the type\-interaction graph from typejjto typeii, counted with multiplicity according to the weights ofDD\. Consequently, nodes of typeiireceive contributions from nodes of typejjthroughℓ\\ellpropagation channels afterkksteps\.

###### Assumption 3\.1

The affinity matrixDDis primitive\.

We begin with the case whereDDis*primitive*, i\.e\., irreducible and aperiodic, and subsequently extend the results to the imprimitive \(irreducible but periodic\) and reducible settings\. IfDDis irreducible, the Perron–Frobenius theorem ensures that its spectral radiusρ​\(D\)\\rho\(D\)is a positive and simple eigenvalueλ1\\lambda\_\{1\}associated with strictly positive left and right eigenvectors\. If, in addition,DDis primitive, then the convergence of powers ofDDtoward the Perron eigendirection is uniform and non\-oscillatory\. In particular,\[Dk\]i​j=Θ​\(λ1k\)\[D^\{k\}\]\_\{ij\}=\\Theta\(\\lambda\_\{1\}^\{k\}\)\(Horn and Johnson[2012](https://arxiv.org/html/2607.10074#bib.bib274), Theorem 8\.5\.1\)for sufficiently largekkfor all type pairs\(i,j\)\(i,j\)\.

###### Assumption 3\.2

The affinity matrixDDis uniformly supercritical; that is, there exists a constantϵ\>0\\epsilon\>0such thatλ1​\(D\)≥1\+ϵ\\lambda\_\{1\}\(D\)\\geq 1\+\\epsilonfor allnn, whereλ1​\(D\)\\lambda\_\{1\}\(D\)denotes the Perron\-Frobenius eigenvalue ofDD\.

In this regime, whereλ1\>1\\lambda\_\{1\}\>1is strictly bounded away from the critical threshold, we ensure the existence of a unique giant component with high probability\. This condition guarantees the exponential growth of the entries of the matrix power\[Dk\]i​j\[D^\{k\}\]\_\{ij\}\. Specifically, sinceλ1≥1\+ϵ\\lambda\_\{1\}\\geq 1\+\\epsilon, the expected number of walks of lengthkkbetween any pair of types grows asΘ​\(λ1k\)\\Theta\(\\lambda\_\{1\}^\{k\}\), which remains non\-vanishing fork=Θ​\(log⁡n\)k=\\Theta\(\\log n\)\. Conversely, ifλ1≤1\\lambda\_\{1\}\\leq 1, the expected number of walks decays or grows at most sub\-exponentially, precluding macroscopic connectivity\. This aligns with the phase transition thresholds established in inhomogeneous random graph theoryBollobáset al\.\([2007](https://arxiv.org/html/2607.10074#bib.bib264)\)\(Theorems 3\.1 and 3\.12\), where the emergence of a giant component is guaranteed in the strictly supercritical regimeρ​\(D\)\>1\\rho\(D\)\>1\.

###### Assumption 3\.3

There exist constantsαt\>0\\alpha\_\{t\}\>0such that∑t=1Tαt=1\\sum\_\{t=1\}^\{T\}\\alpha\_\{t\}=1and the number of nodes of typett, denotedntn\_\{t\}, satisfiesnt/n→αt\{n\_\{t\}\}/\{n\}\\to\\alpha\_\{t\}\.

Assumption[3\.3](https://arxiv.org/html/2607.10074#S3.ThmtheoremEnv3)ensures that each type represents a non\-vanishing fraction of the total node population\. Because each type contains a linear number of nodes, edge counts concentrate sharply around their expectations, with the number of potential edges between any two typesppandccscaling asΘ​\(n2\)\\Theta\(n^\{2\}\)\. Consequently, the expected degree contributionsdp​cd\_\{pc\}serve as consistent, first\-order descriptors of pairwise connectivity\. This structure guarantees sufficient regularity for the multi\-type branching process approximation, as local exploration processes are not distorted by vanishingly small or volatile type populations\.

Moreover, because the number of typesTTis fixed, the empirical type distributionνn=\(n1/n,…,nT/n\)\\nu\_\{n\}=\(n\_\{1\}/n,\\dots,n\_\{T\}/n\)converges to a stable limiting distributionν\\nuasn→∞n\\to\\infty\. This stability ensures that the affinity matrixDDmaintains a constant, finite dimension, allowing the macroscopic interaction structure of the graph to remain well\-defined and analytically tractable in the asymptotic limit\.

## 4Embedding Distortion in IHGs with Primitive Affinity Matrices

LetG∼I​H​G​\(n→,D\)G\\sim IHG\(\\vec\{n\},D\)\. Our main results establish\(1±ε\)\(1\\pm\\varepsilon\)\-distortion guarantees for landmark\-based distance\-preserving embeddings ofGG, under suitable scaling of the number of landmark sets and their sizes\. The supporting neighborhood growth and intersection lemmas are developed in Section[4\.3](https://arxiv.org/html/2607.10074#S4.SS3)\.

### 4\.1\(1±ε\)\(1\\pm\\varepsilon\)\-Approximation Guarantees

For integersM,r\>1M,r\>1, we sampleRRlandmark sets of sizesM0,M1,…,MrM^\{0\},M^\{1\},\\ldots,M^\{r\}\. Our main results demonstrate that by appropriately scalingRRandrras functions of the graph sizennand the error toleranceε\\varepsilon, the lower bound estimatord¯​\(u1,u2\)\\underline\{d\}\(u\_\{1\},u\_\{2\}\)\(Theorem[4\.1](https://arxiv.org/html/2607.10074#S4.ThmtheoremEnv1)\) and the upper bound estimatord¯​\(u1,u2\)\\bar\{d\}\(u\_\{1\},u\_\{2\}\)\(Theorem[4\.2](https://arxiv.org/html/2607.10074#S4.ThmtheoremEnv2)\) simultaneously achieve\(1±ε\)\(1\\pm\\varepsilon\)multiplicative accuracy w\.h\.p\. This multi\-scale sampling scheme guarantees a strictly tighter embedding dimension requirement than classical worst\-case bounds\.

###### Theorem 4\.1\(Lower\-Bound Distortion\)

Letu1u\_\{1\}andu2u\_\{2\}be any two nodes chosen uniformly at random from the graphG∼I​H​G​\(n→,D\)G\\sim IHG\(\\vec\{n\},D\)that satisfies Assumptions[3\.1](https://arxiv.org/html/2607.10074#S3.ThmtheoremEnv1)\-[3\.3](https://arxiv.org/html/2607.10074#S3.ThmtheoremEnv3)\. Letd¯​\(u1,u2\)\\underline\{d\}\(u\_\{1\},u\_\{2\}\)be the lower bound on the shortest distanced​\(u1,u2\)d\(u\_\{1\},u\_\{2\}\)as defined in Section[2\.2](https://arxiv.org/html/2607.10074#S2.SS2)\. Letε∈\(0,1\)\\varepsilon\\in\(0,1\)andς\>0\\varsigma\>0be arbitrarily small\. Define

\(1\)​θ∈\(0,ε\),\\displaystyle\(1\)\\;\\theta\\in\(0,\\varepsilon\),\(2\)​r=⌊θlog⁡M​log⁡n⌋,\\displaystyle\(2\)\\;r=\\left\\lfloor\\frac\{\\theta\}\{\\log M\}\\log n\\right\\rfloor,\(3\)​R=Ω​\(n1−θ−min⁡\{ε2,ε−θ\}\+ς\)\.\\displaystyle\(3\)\\;R=\\Omega\\left\(n^\{1\-\\theta\-\\min\\left\\\{\\frac\{\\varepsilon\}\{2\},\\varepsilon\-\\theta\\right\\\}\+\\varsigma\}\\right\)\.Then w\.h\.p\.,

\(1−ε\)​d​\(u1,u2\)≤d¯​\(u1,u2\)≤d​\(u1,u2\),\(1\-\\varepsilon\)d\(u\_\{1\},u\_\{2\}\)\\leq\\underline\{d\}\(u\_\{1\},u\_\{2\}\)\\leq d\(u\_\{1\},u\_\{2\}\),i\.e\.d¯​\(u1,u2\)\\underline\{d\}\(u\_\{1\},u\_\{2\}\)provides a\(1−ε\)\(1\-\\varepsilon\)\-approximation ofd​\(u1,u2\)d\(u\_\{1\},u\_\{2\}\)\.

###### Theorem 4\.2\(Upper\-Bound Distortion\)

Letu1u\_\{1\}andu2u\_\{2\}be any two nodes chosen uniformly at random from the graphG∼I​H​G​\(n→,D\)G\\sim IHG\(\\vec\{n\},D\)that satisfies Assumptions[3\.1](https://arxiv.org/html/2607.10074#S3.ThmtheoremEnv1)\-[3\.3](https://arxiv.org/html/2607.10074#S3.ThmtheoremEnv3)\. Letd¯​\(u1,u2\)\\bar\{d\}\(u\_\{1\},u\_\{2\}\)be the upper bound on the shortest distanced​\(u1,u2\)d\(u\_\{1\},u\_\{2\}\)as defined in Section[2\.2](https://arxiv.org/html/2607.10074#S2.SS2)\. Letε∈\(0,1\)\\varepsilon\\in\(0,1\)andς\>0\\varsigma\>0be arbitrarily small\. Define

\(1\)​θ∈\(0,1−ε2\),\\displaystyle\(1\)\\;\\theta\\in\\left\(0,\\frac\{1\-\\varepsilon\}\{2\}\\right\),\(2\)​r=⌊θlog⁡M​log⁡n⌋,\\displaystyle\(2\)\\;r=\\left\\lfloor\\frac\{\\theta\}\{\\log M\}\\log n\\right\\rfloor,\(3\)​R=Ω​\(n1−ε\+ς\)\.\\displaystyle\(3\)\\;R=\\Omega\\left\(n^\{1\-\\varepsilon\+\\varsigma\}\\right\)\.Then w\.h\.p\.,

d​\(u1,u2\)≤d¯​\(u1,u2\)≤\(1\+ε\)​d​\(u1,u2\),d\(u\_\{1\},u\_\{2\}\)\\leq\\bar\{d\}\(u\_\{1\},u\_\{2\}\)\\leq\(1\+\\varepsilon\)d\(u\_\{1\},u\_\{2\}\),i\.e\.d¯​\(u1,u2\)\\bar\{d\}\(u\_\{1\},u\_\{2\}\)provides a\(1\+ε\)\(1\+\\varepsilon\)\-approximation ofd​\(u1,u2\)d\(u\_\{1\},u\_\{2\}\)\.

In the supercritical regime whereλ1\>1\\lambda\_\{1\}\>1, the graphGGcontains a giant connected component, and two nodesu1u\_\{1\}andu2u\_\{2\}chosen independently and uniformly at random lie in the same connected component with probability bounded away from zero, and the probability ofu1u\_\{1\}andu2u\_\{2\}not in the giant component is negligible as Theorems 3\.1 and 3\.12 fromBollobáset al\.\([2007](https://arxiv.org/html/2607.10074#bib.bib264)\)imply

ℙ\(u1↔u2butu1,u2∉𝒞\(1\)∣G\)=1n2∑i≥2\|𝒞\(i\)\|2≤\|𝒞\(2\)\|n→ℙ0,\\mathbb\{P\}\(u\_\{1\}\\leftrightarrow u\_\{2\}\\text\{ but \}u\_\{1\},u\_\{2\}\\notin\\mathcal\{C\}\_\{\\scriptscriptstyle\(1\)\}\\mid G\)=\\frac\{1\}\{n^\{2\}\}\\sum\_\{i\\geq 2\}\|\\mathcal\{C\}\_\{\\scriptscriptstyle\(i\)\}\|^\{2\}\\leq\\frac\{\|\\mathcal\{C\}\_\{\\scriptscriptstyle\(2\)\}\|\}\{n\}\\xrightarrow\{\\scriptscriptstyle\\mathbb\{P\}\}0,where𝒞\(i\)\\mathcal\{C\}\_\{\\scriptscriptstyle\(i\)\}denotes theii\-th largest connected component\. The key to both theorems is understanding how the neighborhoods ofu1u\_\{1\}andu2u\_\{2\}, conditionally on being in the giant component, grow and interact as a function of spectral properties ofDD\. Here, we provide a proof sketch and defer the complete proofs to Sections[7\.6](https://arxiv.org/html/2607.10074#S7.SS6)and[7\.7](https://arxiv.org/html/2607.10074#S7.SS7)\.

#### Proof Sketch\.

For any nodeuuand positive integerkk, letNk​\(u\)N\_\{k\}\(u\)denote the set of nodes at graph distance at mostkkfromuuand∂Nk​\(u\)\\partial N\_\{k\}\(u\)denote the set of nodes at distance exactlykk\. LetVtV\_\{t\}denote nodes of typettand sont=\|Vt\|n\_\{t\}=\|V\_\{t\}\|\. We further write∂Nk​\(u\)t=∂Nk​\(u\)∩Vt\\partial N\_\{k\}\(u\)\_\{t\}=\\partial N\_\{k\}\(u\)\\cap V\_\{t\}andNk​\(u\)t=Nk​\(u\)∩VtN\_\{k\}\(u\)\_\{t\}=N\_\{k\}\(u\)\\cap V\_\{t\}\. We also denote\|∂Nk​\(u\)\|→=\(\|∂Nk​\(u\)1\|,\|∂Nk​\(u\)2\|,…,\|∂Nk​\(u\)T\|\)⊤\\overrightarrow\{\|\\partial N\_\{k\}\(u\)\|\}=\\left\(\|\\partial N\_\{k\}\(u\)\_\{1\}\|,\|\\partial N\_\{k\}\(u\)\_\{2\}\|,\\dots,\|\\partial N\_\{k\}\(u\)\_\{T\}\|\\right\)^\{\\top\}as the vector of type\-specific boundary sizes andt​\(u\)t\(u\)as the type of nodeuu\. Letete\_\{t\}be the standard basis vector inℝT\\mathbb\{R\}^\{T\}\.

The following result shows that conditionally onu1u\_\{1\}andu2u\_\{2\}being in the giant component, neighborhoods expand exponentially at rateλ1\\lambda\_\{1\}at radiusL=Θ​\(log⁡n\)L=\\Theta\(\\log n\)and continue to grow exponentially for an additionalk=O​\(log⁡n\)k=O\(\\log n\)generations w\.h\.p\. Specifically, lettingℰn,k\\mathcal\{E\}\_\{n,k\}denote the event that the neighborhood sizes\|∂NL\+k​\(ui\)\|→\\overrightarrow\{\|\\partial N\_\{L\+k\}\(u\_\{i\}\)\|\}lie within a\(1±ε\)\(1\\pm\\varepsilon\)multiplicative window around the deterministic predictionet​\(ui\)⊤​DL\+ke\_\{t\(u\_\{i\}\)\}^\{\\top\}D^\{L\+k\}fori∈\{1,2\}i\\in\\\{1,2\\\}, we have the following:

###### Proposition 4\.3

Letu1u\_\{1\}andu2u\_\{2\}be any two nodes in the giant component ofG∼I​H​G​\(n→,D\)G\\sim IHG\(\\vec\{n\},D\)that satisfies Assumptions[3\.1](https://arxiv.org/html/2607.10074#S3.ThmtheoremEnv1)\-[3\.3](https://arxiv.org/html/2607.10074#S3.ThmtheoremEnv3)\. Then∩l=0kℰn,l\\cap\_\{l=0\}^\{k\}\\mathcal\{E\}\_\{n,l\}occurs w\.h\.p\. for anyL\+k<logλ1⁡nL\+k<\\log\_\{\\lambda\_\{1\}\}n\.

ProofSee Section[7\.4](https://arxiv.org/html/2607.10074#S7.SS4)\.

![Refer to caption](https://arxiv.org/html/2607.10074v1/x1.png)Figure 1:Schematic depicting the computation of the lower boundd¯​\(u1,u2\)\\underline\{d\}\(u\_\{1\},u\_\{2\}\), wherek2−k1≥\(1−ε\)​d​\(u1,u2\)k\_\{2\}\-k\_\{1\}\\geq\(1\-\\varepsilon\)d\(u\_\{1\},u\_\{2\}\)\. Blue nodes are the sourceu1u\_\{1\}and targetu2u\_\{2\}, orange nodes are landmarks in setSS, and gray nodes are arbitrary nodes\.SinceDDis primitive,\[Dk\]i​j=Θ​\(λ1k\)\[D^\{k\}\]\_\{ij\}=\\Theta\(\\lambda\_\{1\}^\{k\}\)for any type pair\(i,j\)\(i,j\), so Proposition[4\.3](https://arxiv.org/html/2607.10074#S4.ThmtheoremEnv3)implies that neighborhoods grow exponentially at rateλ1k\\lambda\_\{1\}^\{k\}for1≪k≤logλ1⁡n1\\ll k\\leq\\log\_\{\\lambda\_\{1\}\}n\. Hence, the local step of the landmark\-based approximation, w\.h\.p\., selects a landmark setSSthat intersectsNk1​\(u1\)N\_\{k\_\{1\}\}\(u\_\{1\}\)but not the disjointNk2​\(u2\)N\_\{k\_\{2\}\}\(u\_\{2\}\), wherek2−k1≥\(1−ε\)​d​\(u1,u2\)k\_\{2\}\-k\_\{1\}\\geq\(1\-\\varepsilon\)d\(u\_\{1\},u\_\{2\}\)\. This yields the lower bound on distortion in Theorem[4\.1](https://arxiv.org/html/2607.10074#S4.ThmtheoremEnv1)as

d¯​\(u1,u2\)=d​\(u2,S\)−d​\(u1,S\)≥k2−k1≥\(1−ε\)​d​\(u1,u2\)w\.h\.p\.\\underline\{d\}\(u\_\{1\},u\_\{2\}\)=d\(u\_\{2\},S\)\-d\(u\_\{1\},S\)\\geq k\_\{2\}\-k\_\{1\}\\geq\(1\-\\varepsilon\)d\(u\_\{1\},u\_\{2\}\)\\quad\\text\{w\.h\.p\.\}See Figure[1](https://arxiv.org/html/2607.10074#S4.F1)for an illustration\.

For Theorem[4\.2](https://arxiv.org/html/2607.10074#S4.ThmtheoremEnv2), controlling the upper bound requires a landmark to fall in the*intersection*Nk​\(u1\)∩Nk​\(u2\)N\_\{k\}\(u\_\{1\}\)\\cap N\_\{k\}\(u\_\{2\}\)fork≤1\+ε2​d​\(u1,u2\)k\\leq\\frac\{1\+\\varepsilon\}\{2\}d\(u\_\{1\},u\_\{2\}\)\. This requires a finer analysis of how the two neighborhoods overlap, given by the following result:

###### Proposition 4\.4

Letu1u\_\{1\}andu2u\_\{2\}be any two nodes in the giant component ofG∼I​H​G​\(n→,D\)G\\sim IHG\(\\vec\{n\},D\)that satisfies Assumptions[3\.1](https://arxiv.org/html/2607.10074#S3.ThmtheoremEnv1)\-[3\.3](https://arxiv.org/html/2607.10074#S3.ThmtheoremEnv3)\. Letε∈\(0,1\)\\varepsilon\\in\(0,1\),κ0∈\(0,1\)\\kappa\_\{0\}\\in\(0,1\),κ∈\(0,1−κ0\)\\kappa\\in\(0,1\-\\kappa\_\{0\}\),L=κ0​logλ1⁡nL=\\kappa\_\{0\}\\log\_\{\\lambda\_\{1\}\}n, and arbitrarily smallρ\>0\\rho\>0\. Then for anyt∈\[T\]t\\in\[T\]andL<k1,k2≤\(κ0\+κ\)​logλ1⁡nL<k\_\{1\},k\_\{2\}\\leq\(\\kappa\_\{0\}\+\\kappa\)\\log\_\{\\lambda\_\{1\}\}nsatisfyingk1\+k2≥\(1\+ρ\)​logλ1⁡ntk\_\{1\}\+k\_\{2\}\\geq\(1\+\\rho\)\\log\_\{\\lambda\_\{1\}\}n\_\{t\},

\|∂Nk1​\(u1\)t∩∂Nk2​\(u2\)t\|∈\[\(1−ε\)3​c2​λ1k1\+k22​nt,\(1\+ε\)3​C2​λ1k1\+k2nt\]w\.h\.p\.\\displaystyle\|\\partial N\_\{k\_\{1\}\}\(u\_\{1\}\)\_\{t\}\\cap\\partial N\_\{k\_\{2\}\}\(u\_\{2\}\)\_\{t\}\|\\in\\left\[\\frac\{\(1\-\\varepsilon\)^\{3\}c^\{2\}\\lambda\_\{1\}^\{k\_\{1\}\+k\_\{2\}\}\}\{2n\_\{t\}\},\\frac\{\(1\+\\varepsilon\)^\{3\}C^\{2\}\\lambda\_\{1\}^\{k\_\{1\}\+k\_\{2\}\}\}\{n\_\{t\}\}\\right\]\\quad\\text\{w\.h\.p\.\}for some constantsc,C\>0c,C\>0\.

ProofSee Section[7\.5](https://arxiv.org/html/2607.10074#S7.SS5)\.

![Refer to caption](https://arxiv.org/html/2607.10074v1/x2.png)Figure 2:Schematic depicting the computation of the upper boundd¯​\(u1,u2\)\\bar\{d\}\(u\_\{1\},u\_\{2\}\), wherek1=k2=1\+ε2​d​\(u1,u2\)k\_\{1\}=k\_\{2\}=\\tfrac\{1\+\\varepsilon\}\{2\}d\(u\_\{1\},u\_\{2\}\)\. Blue nodes are the sourceu1u\_\{1\}and targetu2u\_\{2\}, orange nodes are landmarks in setSS, and gray nodes are arbitrary nodes\.Proposition[4\.4](https://arxiv.org/html/2607.10074#S4.ThmtheoremEnv4)shows that oncek1\+k2k\_\{1\}\+k\_\{2\}exceedslogλ1⁡nt\\log\_\{\\lambda\_\{1\}\}n\_\{t\}, the intersection∂Nk1​\(u1\)t∩∂Nk2​\(u2\)t\\partial N\_\{k\_\{1\}\}\(u\_\{1\}\)\_\{t\}\\cap\\partial N\_\{k\_\{2\}\}\(u\_\{2\}\)\_\{t\}is non\-trivial and grows asλ1k1\+k2/nt\{\\lambda\_\{1\}^\{k\_\{1\}\+k\_\{2\}\}\}/\{n\_\{t\}\}w\.h\.p\. In other words, once neighborhoods grow sufficiently large and exceed a logarithmic threshold, overlaps between different neighborhoods become predictable and scale proportionally to the product of their exponential growth factors, normalized by the type size\. This ensures that, w\.h\.p\., the local step selects a landmark setSSthat intersectsNk​\(u1\)∩Nk​\(u2\)N\_\{k\}\(u\_\{1\}\)\\cap N\_\{k\}\(u\_\{2\}\)but not the exclusive parts of their neighborhoods fork≤1\+ε2​d​\(u1,u2\)k\\leq\\tfrac\{1\+\\varepsilon\}\{2\}d\(u\_\{1\},u\_\{2\}\), yielding the upper bound on distortion:

d¯​\(u1,u2\)=d​\(u1,S\)\+d​\(u2,S\)≤k\+k≤\(1\+ε\)​d​\(u1,u2\)w\.h\.p\.\\overline\{d\}\(u\_\{1\},u\_\{2\}\)=d\(u\_\{1\},S\)\+d\(u\_\{2\},S\)\\leq k\+k\\leq\(1\+\\varepsilon\)\\,d\(u\_\{1\},u\_\{2\}\)\\quad\\text\{w\.h\.p\.\}See Figure[2](https://arxiv.org/html/2607.10074#S4.F2)for an illustration\.

The structural regularity of IHGs allows for a strictly lower embedding dimension than what is required under classical, worst\-case metric spaces\. We provide a comparative analysis of the embedding dimension requirements in Remark[1](https://arxiv.org/html/2607.10074#Thmtheorem1):

### 4\.2Main Results: Average\-Case Metric Distortions

The performance bounds established in Theorems[4\.1](https://arxiv.org/html/2607.10074#S4.ThmtheoremEnv1)and[4\.2](https://arxiv.org/html/2607.10074#S4.ThmtheoremEnv2)provide strong, point\-wise guarantees for pairs of nodes selected uniformly at random across the graph space\. However, in many practical networking and machine learning applications, such as matrix factorization or graph neural network embedding calibrations, one is primarily concerned with the aggregate behavior of the metric estimator across all viable pairs simultaneously\. To address this, we extend our analysis from point\-wise high\-probability bounds to global, component\-wide distortion averages\. The following theorem demonstrates that the landmark embedding framework successfully stabilizes metric distortion across the entire topology, ensuring that anomalous structural bottlenecks do not corrupt the global average accuracy\.

###### Theorem 4\.5\(Average\-Case Distortions\)

Letε\>0\\varepsilon\>0andU=\{\(u1,u2\):u1↔u2\}U=\\\{\(u\_\{1\},u\_\{2\}\):u\_\{1\}\\leftrightarrow u\_\{2\}\\\}denote the set of ordered node pairs belonging to the same connected component\.

1. 1\.Under the conditions as in Theorem[4\.1](https://arxiv.org/html/2607.10074#S4.ThmtheoremEnv1), the average lower\-bound distortion satisfies: 1\|U\|​∑u1↔u2d¯​\(u1,u2\)d​\(u1,u2\)≥1−εw\.h\.p\.\\frac\{1\}\{\|U\|\}\\sum\_\{u\_\{1\}\\leftrightarrow u\_\{2\}\}\\frac\{\\underline\{d\}\(u\_\{1\},u\_\{2\}\)\}\{d\(u\_\{1\},u\_\{2\}\)\}\\geq 1\-\\varepsilon\\quad\\text\{w\.h\.p\.\}
2. 2\.Under the conditions as in Theorem[4\.2](https://arxiv.org/html/2607.10074#S4.ThmtheoremEnv2), the average upper\-bound distortion satisfies: 1\|U\|​∑u1↔u2d¯​\(u1,u2\)d​\(u1,u2\)≤1\+εw\.h\.p\.\\frac\{1\}\{\|U\|\}\\sum\_\{u\_\{1\}\\leftrightarrow u\_\{2\}\}\\frac\{\\bar\{d\}\(u\_\{1\},u\_\{2\}\)\}\{d\(u\_\{1\},u\_\{2\}\)\}\\leq 1\+\\varepsilon\\quad\\text\{w\.h\.p\.\}

ProofSee Section[7\.8](https://arxiv.org/html/2607.10074#S7.SS8)\.

Theorem[4\.5](https://arxiv.org/html/2607.10074#S4.ThmtheoremEnv5)highlights the global metric regularity of IHGs under our multi\-scale landmark framework\. While worst\-case metric spaces are prone to localized geometric pathologies—where small, isolated clusters can arbitrarily inflate average distortion metrics—the branching process dynamics driving the IHG topology fundamentally bound these distortions\. By decoupling the summation over valid pairs into successful approximation sets and highly localized failure configurations, the proof establishes that any pathologically behaving node pairs are statistically negligible\.

Furthermore, this global convergence behavior carries significant implications for the implementation of downstream graph learning algorithms\. Because the average distortion remains tightly controlled within a\(1±ε\)\(1\\pm\\varepsilon\)window w\.h\.p\., any empirical risks or loss functions that rely on pairwise distance approximations will remain stable\. This guarantees that metric embeddings optimized using this hierarchical landmark strategy will preserve global structural properties\. Crucially, it achieves this without requiring dense landmark sampling, avoiding the computational overhead typically required to patch localized estimation errors in more adversarial graph geometries\.

### 4\.3Supporting Lemmas on Neighborhood Growth

The proofs of Theorems[4\.1](https://arxiv.org/html/2607.10074#S4.ThmtheoremEnv1)and[4\.2](https://arxiv.org/html/2607.10074#S4.ThmtheoremEnv2)rely on a precise characterization of how node neighborhoods expand inG∼I​H​G​\(n→,D\)G\\sim IHG\(\\vec\{n\},D\)\. This subsection develops supporting lemmas establishing that neighborhoods grow exponentially at rateλ1\\lambda\_\{1\}\. The first such result is an upper bound on the expected neighborhood growth\.

###### Lemma 4\.6

For every nodeuuinGGand every typet∈\[T\]t\\in\[T\],

𝔼​\(\|∂Nk​\(u\)t\|\)≤‖D‖2kand𝔼​\(\|Nk​\(u\)t\|\)=∑l=0k‖D‖2l\.\\mathbb\{E\}\(\|\\partial N\_\{k\}\(u\)\_\{t\}\|\)\\leq\\\|D\\\|\_\{2\}^\{k\}\\quad\\text\{and\}\\quad\\mathbb\{E\}\(\|N\_\{k\}\(u\)\_\{t\}\|\)=\\sum\_\{l=0\}^\{k\}\\\|D\\\|\_\{2\}^\{l\}\.Furthermore, under Assumptions[3\.1](https://arxiv.org/html/2607.10074#S3.ThmtheoremEnv1)and[3\.2](https://arxiv.org/html/2607.10074#S3.ThmtheoremEnv2)fork=Θ​\(log⁡n\)k=\\Theta\(\\log n\),

𝔼​\(\|∂Nk​\(u\)t\|\)=O​\(λ1k\)and𝔼​\(\|Nk​\(u\)t\|\)=O​\(λ1k\)\.\\mathbb\{E\}\(\|\\partial N\_\{k\}\(u\)\_\{t\}\|\)=O\(\\lambda\_\{1\}^\{k\}\)\\quad\\text\{and\}\\quad\\mathbb\{E\}\(\|N\_\{k\}\(u\)\_\{t\}\|\)=O\(\\lambda\_\{1\}^\{k\}\)\.

ProofSee Section[7\.1](https://arxiv.org/html/2607.10074#S7.SS1)\.

Lemma[4\.6](https://arxiv.org/html/2607.10074#S4.ThmtheoremEnv6)states that neighborhood growth inI​H​G​\(n→,D\)IHG\(\\vec\{n\},D\)is controlled by the spectral norm of the affinity matrixDD\. In expectation, the breadth\-first exploration process behaves like a multi\-type branching process with mean offspring matrixDD, and therefore the size of thekk\-th layer grows at most exponentially at rate‖D‖2\\\|D\\\|\_\{2\}\. Consequently, the expected size of thekk\-hop neighborhood is bounded by a geometric series whose growth rate is determined entirely by the operator norm ofDD\. In particular, when‖D‖2<1\\\|D\\\|\_\{2\}<1, the model stays in a subcritical regime where neighborhoods remain small and exploration dies out quickly\. Under Assumption[3\.2](https://arxiv.org/html/2607.10074#S3.ThmtheoremEnv2)\(‖D‖2≥λ1\>1\\\|D\\\|\_\{2\}\\geq\\lambda\_\{1\}\>1\), the model enters a supercritical regime characterized by exponential expansion\. This entails that a positive fraction of nodes belong to a giant connected component, while the remaining components are typically small \([Bollobáset al\.](https://arxiv.org/html/2607.10074#bib.bib264),[2007](https://arxiv.org/html/2607.10074#bib.bib264), Theorems 3\.1 and 3\.12\)\.

Furthermore, under Assumption[3\.1](https://arxiv.org/html/2607.10074#S3.ThmtheoremEnv1)whereDDis primitive,\[Dk\]i​j=Θ​\(λ1k\)\[D^\{k\}\]\_\{ij\}=\\Theta\(\\lambda\_\{1\}^\{k\}\)for any type pair\(i,j\)\(i,j\)\. As a result, we achieve a tighter bound on the growth rate with𝔼​\(\|∂Nk​\(u\)t\|\)=Θ​\(λ1k\)\\mathbb\{E\}\(\|\\partial N\_\{k\}\(u\)\_\{t\}\|\)=\\Theta\(\\lambda\_\{1\}^\{k\}\)and𝔼​\(\|Nk​\(u\)t\|\)=O​\(λ1k\)\\mathbb\{E\}\(\|N\_\{k\}\(u\)\_\{t\}\|\)=O\(\\lambda\_\{1\}^\{k\}\)\. Building on this exponential growth result, we establish exponential growth of neighborhoods from fixed starting nodes, conditionally on node being in the giant component\. The following lemma shows that this conditioning event is asymptotically equivalent to the local event that both nodes’ neighborhoods exhibit the exponential growth up to generationL=Θ​\(log⁡n\)L=\\Theta\(\\log n\)\. This equivalence is what allows us to work conditionally on connectivity throughout the analysis\.

###### Lemma 4\.7

Letu1u\_\{1\}andu2u\_\{2\}be any two nodes inG∼I​H​G​\(n→,D\)G\\sim IHG\(\\vec\{n\},D\)that satisfies Assumptions[3\.2](https://arxiv.org/html/2607.10074#S3.ThmtheoremEnv2)\-[3\.3](https://arxiv.org/html/2607.10074#S3.ThmtheoremEnv3)\. Letε∈\(0,1\)\\varepsilon\\in\(0,1\),κ0∈\(0,1\)\\kappa\_\{0\}\\in\(0,1\),L=κ0​logλ1⁡nL=\\kappa\_\{0\}\\log\_\{\\lambda\_\{1\}\}n, andt​\(u\)t\(u\)denote the type of nodeuu\. Define the events

An\\displaystyle A\_\{n\}=\{\|∂NL\(ui\)t\|∈\[\(1−ε\)\[DL\]t​\(ui\)​t,\(1\+ε\)\[DL\]t​\(ui\)​t\]:i=1,2;t∈\[T\]\},\\displaystyle=\\\{\|\\partial N\_\{L\}\(u\_\{i\}\)\_\{t\}\|\\in\\left\[\(1\-\\varepsilon\)\[D^\{L\}\]\_\{t\(u\_\{i\}\)t\},\(1\+\\varepsilon\)\[D^\{L\}\]\_\{t\(u\_\{i\}\)t\}\\right\]\\\>:\\\>i=1,2\\\>;\\\>t\\in\[T\]\\\},Bn\\displaystyle B\_\{n\}=\{u1andu2are in the giant component\}\.\\displaystyle=\\\{\\text\{$u\_\{1\}$ and $u\_\{2\}$ are in the giant component\}\\\}\.Thenℙ​\(An∖Bn\)→0\\mathbb\{P\}\(A\_\{n\}\\setminus B\_\{n\}\)\\to 0andℙ​\(Bn∖An\)→0\\mathbb\{P\}\(B\_\{n\}\\setminus A\_\{n\}\)\\to 0asn→∞n\\to\\infty\.

ProofSee Section[7\.2](https://arxiv.org/html/2607.10074#S7.SS2)\.

Intuitively, since the local exploration is well approximated by a branching process, ifu1u\_\{1\}andu2u\_\{2\}lie in the giant component, then the corresponding branching processes survive\. As branching processes either die out or grow exponentially, the neighborhoods grow exponentially conditioned on survival\. Thus, local exponential expansion and global connectivity occur simultaneously in the large\-nnlimit\. Once the neighborhood size at generationLLis well\-approximated by branching process, subsequent generations continue to grow in a stable and multiplicative manner, as described by the following lemma:

###### Lemma 4\.8

Letu1u\_\{1\}andu2u\_\{2\}be any two nodes in the giant component ofG∼I​H​G​\(n→,D\)G\\sim IHG\(\\vec\{n\},D\)that satisfies Assumptions[3\.1](https://arxiv.org/html/2607.10074#S3.ThmtheoremEnv1)\-[3\.3](https://arxiv.org/html/2607.10074#S3.ThmtheoremEnv3)\. Letε∈\(0,1\)\\varepsilon\\in\(0,1\),κ0∈\(0,1\)\\kappa\_\{0\}\\in\(0,1\),κ∈\(0,1−κ0\)\\kappa\\in\(0,1\-\\kappa\_\{0\}\), andL=κ0​logλ1⁡nL=\\kappa\_\{0\}\\log\_\{\\lambda\_\{1\}\}n\. Define

Abm,bM\\displaystyle A\_\{b^\{m\},b^\{M\}\}=\{\|∂NL​\(ui\)\|→∈\[bim,biM\]:i=1,2\},\\displaystyle=\\left\\\{\\overrightarrow\{\|\\partial N\_\{L\}\(u\_\{i\}\)\|\}\\in\[b\_\{i\}^\{m\},b\_\{i\}^\{M\}\]:i=1,2\\right\\\},ℰn,k\\displaystyle\\mathcal\{E\}\_\{n,k\}=\{\|∂NL\+k​\(ui\)\|→∈\[bim​Dk,biM​Dk\]:i=1,2\},\\displaystyle=\\left\\\{\\overrightarrow\{\|\\partial N\_\{L\+k\}\(u\_\{i\}\)\|\}\\in\[b\_\{i\}^\{m\}D^\{k\},b\_\{i\}^\{M\}D^\{k\}\]:i=1,2\\right\\\},wherebim=\(1−ε\)​et​\(ui\)⊤​DLb\_\{i\}^\{m\}=\(1\-\\varepsilon\)e\_\{t\(u\_\{i\}\)\}^\{\\top\}D^\{L\}andbiM=\(1\+ε\)​et​\(ui\)⊤​DLb\_\{i\}^\{M\}=\(1\+\\varepsilon\)e\_\{t\(u\_\{i\}\)\}^\{\\top\}D^\{L\}\. Then there existsδ\>0\\delta\>0such that

ℙ\(⋂l=0kℰn,l\|Abm,bM\)≥\(1−\(2T\+1\)n−δ\)k\\mathbb\{P\}\\\!\\left\(\\bigcap\_\{l=0\}^\{k\}\\mathcal\{E\}\_\{n,l\}\\middle\|A\_\{b^\{m\},b^\{M\}\}\\right\)\\geq\\left\(1\-\(2T\+1\)n^\{\-\\delta\}\\right\)^\{k\}for anyk≤κ​logλ1⁡nk\\leq\\kappa\\log\_\{\\lambda\_\{1\}\}n, for all sufficiently largenn\.

ProofSee Section[7\.3](https://arxiv.org/html/2607.10074#S7.SS3)\.

Conditional on the eventAbm,bMA\_\{b^\{m\},b^\{M\}\}, which ensures that theLL\-th layer lies within a controlled multiplicative window aroundDLD^\{L\}, Lemma[4\.8](https://arxiv.org/html/2607.10074#S4.ThmtheoremEnv8)establishes that the neighborhood sizes remain close to the deterministic trajectoryDkD^\{k\}for an additionalk=O​\(log⁡n\)k=O\(\\log n\)generations w\.h\.p\. This result shows that, in the supercritical regime, neighborhood expansion not only initiates exponentially but persists in a stable, predictable fashion over logarithmic scales that closely tracks the multi\-type branching process with mean matrixDD, reinforcing the connection between local expansion dynamics and the global structure of the graph\.

## 5Extension to Continuous Type Spaces \(Kernel Model\)

In the finite model, the geometry of the graph is governed by the spectral structure of the affinity matrixDD, which controls the rate of neighborhood expansion and the probability that independently growing neighborhoods intersect\. These mechanisms are precisely what drive the\(1±ε\)\(1\\pm\\varepsilon\)\-distortion guarantees established in Section[4\.1](https://arxiv.org/html/2607.10074#S4.SS1)\. Many real\-world networks, however, are more accurately described by continuously varying node attributes rather than a finite collection of discrete classes, and the analysis developed for finitely many node types extends naturally to a continuous latent\-space setting\.

To capture this setting, we consider an inhomogeneous random graph model\(Bollobáset al\.[2007](https://arxiv.org/html/2607.10074#bib.bib264)\)in which each nodeiiis assigned a latent positionxi∈𝒳,x\_\{i\}\\in\\mathcal\{X\},sampled independently from a probability space\(𝒳,μ\)\(\\mathcal\{X\},\\mu\)\. Connectivity is determined by a measurable kernelκ∈L2​\(𝒳×𝒳,μ×μ\)\\kappa\\in L^\{2\}\(\\mathcal\{X\}\\times\\mathcal\{X\},\\mu\\times\\mu\), whereκ​\(x,y\)\\kappa\(x,y\)quantifies the affinity between positionsxxandyy\. Conditional on the latent positions, edges are generated independently with probabilityPi​j=min⁡\{κ​\(xi,xj\)/n,1\}\.P\_\{ij\}=\\min\\left\\\{\{\\kappa\(x\_\{i\},x\_\{j\}\)\}/\{n\},1\\right\\\}\.

This formulation generalizes the finite\-type model: if𝒳\\mathcal\{X\}is partitioned into finitely many regions andκ\\kappais constant on each block, then the model reduces to the previously studied discrete affinity\-matrix framework\. By working within theL2L^\{2\}space that allows the kernel to possess localized singularities, our framework serves as a unified continuum limit for a vast taxonomy of network architectures—encompassing heavy\-tailed scale\-free networks via unbounded Chung\-Lu kernels\(Chung and Lu[2002](https://arxiv.org/html/2607.10074#bib.bib268)\)and continuous variants of the stochastic block model with fluid community boundaries\(Hollandet al\.[1983](https://arxiv.org/html/2607.10074#bib.bib281)\)\.

### 5\.1The Integral Operator

In the finite\-type setting, the affinity matrixDDacts on vectors whose coordinates encode type\-specific neighborhood densities\. IteratingDDdescribes how neighborhoods expand through successive graph distances, while the leading eigenvalue determines the asymptotic growth rate of this exploration process\.

In the kernel setting, the matrixDDis replaced by the integral operator

\(𝒯κ​f\)​\(x\)=∫𝒳κ​\(x,y\)​f​\(y\)​𝑑μ​\(y\),\(\\mathcal\{T\}\_\{\\kappa\}f\)\(x\)=\\int\_\{\\mathcal\{X\}\}\\kappa\(x,y\)f\(y\)\\,d\\mu\(y\),whereμ\\muis the probability measure describing the distribution of latent positions in𝒳\\mathcal\{X\}, andf∈L2​\(𝒳,μ\)f\\in L^\{2\}\(\\mathcal\{X\},\\mu\)is a square\-integrable function representing the density of reachable mass or influence across the latent space\. Thus,𝒯κ\\mathcal\{T\}\_\{\\kappa\}plays exactly the same role as matrix multiplication in the finite\-dimensional setting: repeated application of the operator models the propagation and expansion of neighborhoods across the graph, effectively describing a multitype branching process in a continuous state space\.

The spectral radiusρ​\(𝒯κ\)\\rho\(\\mathcal\{T\}\_\{\\kappa\}\), equivalently the principal eigenvalueλ1\\lambda\_\{1\}, governs the large\-scale geometry of the graph\. Whenλ1\>1\\lambda\_\{1\}\>1, the graph is supercritical and contains a giant connected component w\.h\.p\.\(Bollobáset al\.[2007](https://arxiv.org/html/2607.10074#bib.bib264), Theorem 3\.1\)\. Moreover, neighborhood volumes grow asymptotically at exponential rateλ1\\lambda\_\{1\}, implying that typical graph distances scale aslogλ1⁡n\\log\_\{\\lambda\_\{1\}\}n\(Bollobáset al\.[2007](https://arxiv.org/html/2607.10074#bib.bib264), Theorem 3\.14\)\. Consequently, the same spectral mechanism underlying the finite\-type distortion bounds continues to govern metric behavior in the continuous model\.

### 5\.2The Sandwiching Argument

A key technical step in extending these results is the approximation of the continuous kernel by finite\-dimensional block models\. For everyδ\>0\\delta\>0, there exists a measurable partition𝒳=⨆t=1T𝒳t\\mathcal\{X\}=\\bigsqcup\_\{t=1\}^\{T\}\\mathcal\{X\}\_\{t\}such thatκ\\kappaadmits a step\-function approximationκδ\\kappa\_\{\\delta\}satisfying‖κ−κδ‖∞<δ\.\\\|\\kappa\-\\kappa\_\{\\delta\}\\\|\_\{\\infty\}<\\delta\.This approximating kernelκδ\\kappa\_\{\\delta\}is constant on each rectangle𝒳i×𝒳j\\mathcal\{X\}\_\{i\}\\times\\mathcal\{X\}\_\{j\}, effectively inducing a finite\-type random graph model by leveraging the fact that step functions are dense in the space of kernels under the relevant metric\(Lovász[2012](https://arxiv.org/html/2607.10074#bib.bib123)\)\.

As the partition is refined, the associated operator𝒯κδ\\mathcal\{T\}\_\{\\kappa\_\{\\delta\}\}converges to𝒯κ\\mathcal\{T\}\_\{\\kappa\}in operator norm\. Because the spectral radius is a continuous functional of the operator for compact kernels\(Dunfordet al\.[1963](https://arxiv.org/html/2607.10074#bib.bib275)\), this convergence implies that the leading eigenvalues and the resulting exponential neighborhood growth rates of the approximation approach those of the continuous kernel\. By “sandwiching” the true kernel between two such step functions,κδ−\\kappa\_\{\\delta\}^\{\-\}andκδ\+\\kappa\_\{\\delta\}^\{\+\}, we can bound the true shortest\-path distancesdκd\_\{\\kappa\}between those of the finite\-type models\. The lower approximationκδ−\\kappa\_\{\\delta\}^\{\-\}generates a sparser graph, so graph distances in the corresponding model are typically larger \(dκδ−​\(u1,u2\)≥dκ​\(u1,u2\)d\_\{\\kappa\_\{\\delta\}^\{\-\}\}\(u\_\{1\},u\_\{2\}\)\\geq d\_\{\\kappa\}\(u\_\{1\},u\_\{2\}\)\)\. Conversely, the upper approximationκδ\+\\kappa\_\{\\delta\}^\{\+\}produces a denser graph with shorter distances \(dκδ\+​\(u1,u2\)≤dκ​\(u1,u2\)d\_\{\\kappa\_\{\\delta\}^\{\+\}\}\(u\_\{1\},u\_\{2\}\)\\leq d\_\{\\kappa\}\(u\_\{1\},u\_\{2\}\)\)\. These approximations create a geometric “sandwich” around the true metric structure\.

Since both bounding models are finite\-type graphs, we will show that the distortion results from Section[4\.1](https://arxiv.org/html/2607.10074#S4.SS1)apply directly to them\. As the partition becomes finer \(i\.e\. the number of typesT→∞T\\to\\infty\),‖𝒯κδ±−𝒯κ‖→0\\\|\\mathcal\{T\}\_\{\\kappa\_\{\\delta\}^\{\\pm\}\}\-\\mathcal\{T\}\_\{\\kappa\}\\\|\\to 0\(Krasnosel’skiiet al\.[1976](https://arxiv.org/html/2607.10074#bib.bib276), Chapter 2\), and thereforeλ1​\(𝒯κδ±\)→λ1​\(𝒯κ\)\\lambda\_\{1\}\(\\mathcal\{T\}\_\{\\kappa\_\{\\delta\}^\{\\pm\}\}\)\\to\\lambda\_\{1\}\(\\mathcal\{T\}\_\{\\kappa\}\)\(Kato and Katåo[1966](https://arxiv.org/html/2607.10074#bib.bib277), Theorem 3\.16\)\. Because shortest\-path distances scale logarithmically with the effective branching factord​\(u1,u2\)/log⁡n​⟶𝑝​1/log⁡λ1\{d\(u\_\{1\},u\_\{2\}\)\}/\{\\log n\}\\overset\{p\}\{\\longrightarrow\}\{1\}/\{\\log\\lambda\_\{1\}\}\(\\swapHofstadvan der[2024b](https://arxiv.org/html/2607.10074#bib.bib10), Theorem 6\.2\), the geometric behavior of the finite approximations converges to that of the original kernel model\. We formalize a rigorous probability space coupling and spectral perturbation framework for arbitrary bounded kernels in the following theorem\.

###### Theorem 5\.1\(Metric Sandwiching for Kernel Models\)

Let\(𝒳,μ\)\(\\mathcal\{X\},\\mu\)be a probability space andκ:𝒳×𝒳→\[0,∞\)\\kappa:\\mathcal\{X\}\\times\\mathcal\{X\}\\to\[0,\\infty\)be a bounded, primitive kernel with integral operator spectral radiusλ1​\(𝒯κ\)\>1\\lambda\_\{1\}\(\\mathcal\{T\}\_\{\\kappa\}\)\>1\. For anyε∈\(0,1\)\\varepsilon\\in\(0,1\), there exists aδ\>0\\delta\>0and a finite measurable partition𝒫δ=\{𝒳1,…,𝒳T\}\\mathcal\{P\}\_\{\\delta\}=\\\{\\mathcal\{X\}\_\{1\},\\dots,\\mathcal\{X\}\_\{T\}\\\}of𝒳\\mathcal\{X\}defining bounded, primitive step\-function kernelsκδ−\\kappa\_\{\\delta\}^\{\-\}andκδ\+\\kappa\_\{\\delta\}^\{\+\}such that‖κ−κδ±‖∞≤δ\\\|\\kappa\-\\kappa\_\{\\delta\}^\{\\pm\}\\\|\_\{\\infty\}\\leq\\delta\.

Let𝒢n=\(Gκδ−,Gκ,Gκδ\+\)\\mathcal\{G\}\_\{n\}=\(G\_\{\\kappa\_\{\\delta\}^\{\-\}\},G\_\{\\kappa\},G\_\{\\kappa\_\{\\delta\}^\{\+\}\}\)be a joint coupling of random graphs constructed on a shared sequence of latent positionsx1,…,xn∼μx\_\{1\},\\dots,x\_\{n\}\\sim\\muand independent edge variablesUi​j∼Uniform​\(0,1\)U\_\{ij\}\\sim\\mathrm\{Uniform\}\(0,1\), where an edge exists inGfG\_\{f\}ifUi​j≤f​\(xi,xj\)/nU\_\{ij\}\\leq f\(x\_\{i\},x\_\{j\}\)/n\. Then the following properties hold simultaneously:

1. 1\.Edge\-Set Inclusion:The respective edge sets are monotonically nested on the same probability space, satisfying: E​\(Gκδ−\)⊆E​\(Gκ\)⊆E​\(Gκδ\+\)w\.h\.p\.E\(G\_\{\\kappa\_\{\\delta\}^\{\-\}\}\)\\subseteq E\(G\_\{\\kappa\}\)\\subseteq E\(G\_\{\\kappa\_\{\\delta\}^\{\+\}\}\)\\quad\\text\{w\.h\.p\.\}
2. 2\.Spectral Radius Perturbation:The spectral radii of the corresponding step\-kernel operatorsλ1​\(𝒯κδ−\)\\lambda\_\{1\}\(\\mathcal\{T\}\_\{\\kappa\_\{\\delta\}^\{\-\}\}\)andλ1​\(𝒯κδ\+\)\\lambda\_\{1\}\(\\mathcal\{T\}\_\{\\kappa\_\{\\delta\}^\{\+\}\}\)satisfy a linear perturbation bound around the true kernel spectral radius: 1<λ1​\(𝒯κδ−\)≤min⁡\{λ1​\(𝒯κδ\+\),λ1​\(𝒯κ\)\}withλ1​\(𝒯κδ±\)=λ1​\(𝒯κ\)±O​\(δ\)\.1<\\lambda\_\{1\}\(\\mathcal\{T\}\_\{\\kappa\_\{\\delta\}^\{\-\}\}\)\\leq\\min\\\{\\lambda\_\{1\}\(\\mathcal\{T\}\_\{\\kappa\_\{\\delta\}^\{\+\}\}\),\\lambda\_\{1\}\(\\mathcal\{T\}\_\{\\kappa\}\)\\\}\\quad\\text\{with\}\\quad\\lambda\_\{1\}\(\\mathcal\{T\}\_\{\\kappa\_\{\\delta\}^\{\\pm\}\}\)=\\lambda\_\{1\}\(\\mathcal\{T\}\_\{\\kappa\}\)\\pm O\(\\delta\)\.

ProofSee Section[7\.9](https://arxiv.org/html/2607.10074#S7.SS9)\.

This framework serves as a rigorous “bridge” between finite\-type affinity matrices and continuous latent\-space kernels\. The result shows that, for supercritical graphs withλ1\>1\\lambda\_\{1\}\>1, the asymptotic structural behavior is a continuous functional of the underlying kernel\. Although individual edge modifications are discrete and potentially unstable, the large\-nngeometric behavior is governed by the integral operator𝒯κ\\mathcal\{T\}\_\{\\kappa\}\. Consequently, continuous graphons can be treated as limits of finite block models while preserving the underlying random graph mechanics\. More generally, the theorem shows that the structural framework of an inhomogeneous random graph is spectrally robust as small perturbations ofκ\\kappain theL2L^\{2\}norm induce only controllable, continuous variations in the network’s spectral radius\.

From a proof\-theoretical standpoint, this theorem provides a rigorous reduction argument\. By “sandwiching” the true kernel between two finite step\-function approximations, we transfer the analysis of infinite\-dimensional operators to the more tractable linear algebra of finite affinity matrices\. Because the exact edge inclusions and spectral bounds hold for the bounding modelsκδ±\\kappa\_\{\\delta\}^\{\\pm\}, and since these coupled systems can be made arbitrarily tight relative to the true kernel operator𝒯κ\\mathcal\{T\}\_\{\\kappa\}, the fundamental structural and topological properties of the continuous model are fully constrained\.

### 5\.3Main Result: Universal Kernel Distortion

By combining the finite\-type distortion guarantees with the metric sandwiching framework, we can now lift our analysis from discrete blocks to continuous latent spaces\. The following theorem formalizes this transition, proving that the\(1±ε\)\(1\\pm\\varepsilon\)\-distortion bounds established in Theorems[4\.1](https://arxiv.org/html/2607.10074#S4.ThmtheoremEnv1)and[4\.2](https://arxiv.org/html/2607.10074#S4.ThmtheoremEnv2)generalize universally to graphs generated by arbitrary square\-integrable kernels, accommodating both bounded topologies and heavy\-tailed, unbounded architectures\.

###### Theorem 5\.2\(Universal Kernel Distortion\)

LetGκG\_\{\\kappa\}be an inhomogeneous random graph generated by a primitive kernelκ∈L2​\(𝒳×𝒳,μ×μ\)\\kappa\\in L^\{2\}\(\\mathcal\{X\}\\times\\mathcal\{X\},\\mu\\times\\mu\)with spectral radiusλ1\>1\\lambda\_\{1\}\>1\. Letε∈\(0,1\)\\varepsilon\\in\(0,1\)\. Suppose the landmark\-based estimatorsd¯\\underline\{d\}andd¯\\overline\{d\}are constructed as defined in Section[2\.2](https://arxiv.org/html/2607.10074#S2.SS2)\. Then for any two verticesu1,u2u\_\{1\},u\_\{2\}chosen uniformly at random from the graphGκG\_\{\\kappa\}, we have the following:

1. 1\.If parameters\(θ,r,R\)\(\\theta,r,R\)satisfy the conditions in Theorem[4\.1](https://arxiv.org/html/2607.10074#S4.ThmtheoremEnv1), \(1−ε\)​dκ​\(u1,u2\)≤d¯​\(u1,u2\)≤dκ​\(u1,u2\)w\.h\.p\.\(1\-\\varepsilon\)\\,d\_\{\\kappa\}\(u\_\{1\},u\_\{2\}\)\\leq\\underline\{d\}\(u\_\{1\},u\_\{2\}\)\\leq d\_\{\\kappa\}\(u\_\{1\},u\_\{2\}\)\\quad\\text\{w\.h\.p\.\}
2. 2\.If parameters\(θ,r,R\)\(\\theta,r,R\)satisfy the conditions in Theorem[4\.2](https://arxiv.org/html/2607.10074#S4.ThmtheoremEnv2), dκ​\(u1,u2\)≤d¯​\(u1,u2\)≤\(1\+ε\)​dκ​\(u1,u2\)w\.h\.p\.d\_\{\\kappa\}\(u\_\{1\},u\_\{2\}\)\\leq\\overline\{d\}\(u\_\{1\},u\_\{2\}\)\\leq\(1\+\\varepsilon\)\\,d\_\{\\kappa\}\(u\_\{1\},u\_\{2\}\)\\quad\\text\{w\.h\.p\.\}

Consequently, the landmark\-based embedding provides a\(1±ε\)\(1\\pm\\varepsilon\)\-approximation of the true shortest\-path distance for arbitraryL2L^\{2\}kernels\.

ProofSee Section[7\.10](https://arxiv.org/html/2607.10074#S7.SS10)\.

Theorem[5\.2](https://arxiv.org/html/2607.10074#S5.ThmtheoremEnv2)establishes that the tight\(1±ε\)\(1\\pm\\varepsilon\)\-distortion guarantees derived for finite\-type affinity matrices are not structural artifacts of discrete block models, but instead reflect a fundamental metric property of general inhomogeneous random graphs\. By abstracting node attributes into a continuous latent space\(𝒳,μ\)\(\\mathcal\{X\},\\mu\), the theorem extends these guarantees to real\-world networks in which connectivity is driven by smooth, continuous traits \(such as geographical positions, semantic text embeddings, or dense social proximity vectors\) as well as heavy\-tailed degree distributions governed by unbounded power\-law configurations, rather than rigid category assignments\.

Through a metric sandwiching argument, we show that when the underlying kernelκ\\kappais square\-integrable and primitive, the continuous graph metric behaves as a stable limit of finite\-dimensional constructions governed strictly by the structural parameters\(θ,r,R\)\(\\theta,r,R\)\. This ensures that the computational benefits of landmark\-based approaches \(namely, avoiding explicit all\-pairs shortest path calculations\) extend naturally to non\-parametric settings\.

Algorithmically, this highlights the robustness of our multiscale sampling scheme\. Even though continuous neighborhoods expand along smoothly varying frontiers rather than uniform block boundaries, the contraction of the propagation operator𝒯κ\\mathcal\{T\}\_\{\\kappa\}toward its principal eigendirection preserves the geometric intersection behavior seen in the discrete model\. Ultimately, because kernel perturbations and boundary fluctuations induce only controlled, continuous distortions in the operator spectrum, Theorem[5\.2](https://arxiv.org/html/2607.10074#S5.ThmtheoremEnv2)validates distance\-preserving landmark embeddings as a robust and scalable framework for structural representation across a broad class of network models\.

## 6GNN\-based Landmark Embeddings and Experimental Results

Sarmaet al\.\([2010](https://arxiv.org/html/2607.10074#bib.bib16)\)proposed using one BFS to calculate the shortest path distance between every node and each landmark set in the local step given in Section[2\.2](https://arxiv.org/html/2607.10074#S2.SS2), requiringDDBFS runs to generate landmark embeddings inℝD\{\\mathbb\{R\}\}^\{D\}\. This is prohibitive for large graphs, with complexityO​\(n\+m\)O\(n\+m\)per source andO​\(n​\(n\+m\)\)O\(n\(n\+m\)\)for all pairs\(Cormenet al\.[2009](https://arxiv.org/html/2607.10074#bib.bib48)\)\. We propose replacing BFS in the local step with a GNN, which approximates landmark distances from graph structure\. Once trained, GNN inference is computationally efficient, and crucially GNNs trained on small graphs can be transferred to larger ones, leveraging the transferability properties of graph neural networks on convergent graph sequences\(Ruizet al\.[2020](https://arxiv.org/html/2607.10074#bib.bib66),[2023](https://arxiv.org/html/2607.10074#bib.bib61)\)\. This last property is directly motivated by the IHG framework; since our theoretical results characterize distance geometry in terms of the spectral properties ofDD, graphs from the same IHG model share structural regularities that a GNN can learn and exploit across sizes\.

Formally, GNNs are deep convolutional architectures tailored to graph data\(Scarselliet al\.[2008](https://arxiv.org/html/2607.10074#bib.bib29), Kipf and Welling[2017](https://arxiv.org/html/2607.10074#bib.bib88), Defferrardet al\.[2016](https://arxiv.org/html/2607.10074#bib.bib87), Ruizet al\.[2021](https://arxiv.org/html/2607.10074#bib.bib57)\)\. Focusing on node\-level data represented as𝐗∈ℝn×d\{\\mathbf\{X\}\}\\in\{\\mathbb\{R\}\}^\{n\\times d\}, each GNN layer applies a graph convolution followed by a pointwise nonlinearity,

𝐗ℓ=σ​\(∑k=0K−1𝐀k​𝐗ℓ−1​𝐖ℓ,k\),\{\\mathbf\{X\}\}\_\{\\ell\}=\\sigma\\left\(\\sum\_\{k=0\}^\{K\-1\}\{\\mathbf\{A\}\}^\{k\}\{\\mathbf\{X\}\}\_\{\\ell\-1\}\{\\mathbf\{W\}\}\_\{\\ell,k\}\\right\),where𝐀∈ℝn×n\{\\mathbf\{A\}\}\\in\{\\mathbb\{R\}\}^\{n\\times n\}is the graph adjacency,𝐖ℓ,k∈ℝdℓ−1×dℓ\{\\mathbf\{W\}\}\_\{\\ell,k\}\\in\{\\mathbb\{R\}\}^\{d\_\{\\ell\-1\}\\times d\_\{\\ell\}\}are learnable parameters, andσ\\sigmais a pointwise nonlinearity\. The full GNN is written compactly as𝐘=Φ​\(𝐗,𝐀;𝒲\)\{\\mathbf\{Y\}\}=\\Phi\(\{\\mathbf\{X\}\},\{\\mathbf\{A\}\};\{\\mathcal\{W\}\}\)\. A key property inherited from graph convolutions is locality: each layer exchanges information only within one\-hop neighborhoods, soLL\-layer GNNs aggregate information withinLL\-hop neighborhoods\. This aligns naturally with the landmark\-based embedding task, where the relevant quantity—the distance from a node to a landmark—is determined by local neighborhood structure up to the appropriate radius\.

### Experimental Setup

We train GNNs to approximate landmark distances in sparse, undirected, unweighted random graphs\. As the canonicalT=1T=1special case of the IHG model, we use Erdős–Rényi graphsERn​\(λ/n\)\\text\{ER\}\_\{n\}\(\\lambda/n\), where each pair of nodes is connected independently with probabilityλ/n\\lambda/n\. Setting1<λ≪n1<\\lambda\\ll nensures sparsity and the existence of a giant component w\.h\.p\., placing the model firmly in the supercritical regime of our theory\. We considerλ∈\{3,4,5,6\}\\lambda\\in\\\{3,4,5,6\\\}andn∈\{25,50,100,200,400,800,1600,3200\}n\\in\\\{25,50,100,200,400,800,1600,3200\\\}\.

We evaluate four standard GNN architectures—GCN\(Kipf and Welling[2017](https://arxiv.org/html/2607.10074#bib.bib88)\), GraphSAGE\(Hamiltonet al\.[2017a](https://arxiv.org/html/2607.10074#bib.bib38)\), GAT\(Veličkovićet al\.[2018](https://arxiv.org/html/2607.10074#bib.bib190)\), and GIN\(Xuet al\.[2019a](https://arxiv.org/html/2607.10074#bib.bib177)\)—all using sum aggregation, dropout, and ReLU activations\. For each architecture, we evaluate nine models with⌊n⌋\\lfloor\\sqrt\{n\}\\rfloornodes in the first and last layers and varying hidden\-layer depth and width\.

Each graph is treated as a batch of nodes with a 200\-50\-50 train\-validation\-test split\. Input signals𝐗∈ℝn×r\{\\mathbf\{X\}\}\\in\{\\mathbb\{R\}\}^\{n\\times r\}one\-hot encode landmark nodes, and outputs𝐘∈ℝn×r\{\\mathbf\{Y\}\}\\in\{\\mathbb\{R\}\}^\{n\\times r\}represent shortest path distances\[𝐘\]u​s=d​\(u,s\)\[\{\\mathbf\{Y\}\}\]\_\{us\}=d\(u,s\)\. Training runs for 1000 epochs with early stopping \(100 epochs\), MSE loss, Adam optimizer \(lr=0\.01, weight decay=0\.0001\), and a cyclic\-cosine learning rate schedule \(0\.001–0\.1 for 10 cycles\)\.

### 6\.1Experiment 1: Learning the GNNs

In the first experiment, we evaluate the ability of trained GNNs to compute end\-to\-end shortest paths\. We considern=50n=50and set the GNN depth to be larger than⌈logλ⁡n⌉\\lceil\\log\_\{\\lambda\}n\\rceil\. Figure[3](https://arxiv.org/html/2607.10074#S6.F3)plots the actual shortest path distances versus those predicted by our selected GNN architectures\. Predictions for distances beyond the GNN depth saturate, indicating that GNNs cannot capture longer distances even with depth exceeding the expected path length\. As expected, GNNs are not suitable for computing end\-to\-end shortest path distances, especially on sparser graphs withλ∈\{3,4\}\\lambda\\in\\\{3,4\\\}, which tend to exhibit longer paths\.

![Refer to caption](https://arxiv.org/html/2607.10074v1/paper/figures/lambda3-raw.png)

![Refer to caption](https://arxiv.org/html/2607.10074v1/paper/figures/lambda4-raw.png)

![Refer to caption](https://arxiv.org/html/2607.10074v1/paper/figures/lambda5-raw.png)

![Refer to caption](https://arxiv.org/html/2607.10074v1/paper/figures/lambda6-raw.png)

Figure 3:End\-to\-end shortest path distance predictions from⌊n⌋​\-64\-32\-16\-​⌊n⌋\\lfloor\\sqrt\{n\}\\rfloor\\text\{\-64\-32\-16\-\}\\lfloor\\sqrt\{n\}\\rfloorGNNs trained on graphs generated byERn​\(λ/n\)\\mathrm\{ER\}\_\{n\}\(\\lambda/n\)\. The evaluation data consists of graphs from the same model\.
### 6\.2Experiment 2: Comparing BFS\-Based and GNN\-Based Landmark Embeddings

In this experiment, we compare the lower bounds resulting from BFS\-based and GNN\-based landmark embeddings against the actual shortest path distances\. We focus on lower bounds since they depend only on coordinate\-wise differences and are thus a clean measure of embedding quality independent of the landmark index matching required for upper bounds\.

To construct the landmark embeddings, we sampler\+1r\+1landmark setsS0,S1,…,SrS\_\{0\},S\_\{1\},\\dots,S\_\{r\}of cardinalities20,21,…,2r2^\{0\},2^\{1\},\\dots,2^\{r\}withr=⌊log⁡n⌋r=\\lfloor\\log n\\rfloorforRRrepetitions\. In Figure[4](https://arxiv.org/html/2607.10074#S6.F4)\(a\-d\), GNN\-based lower bounds underperform the vanilla lower bounds for smallerλ∈\{3,4\}\\lambda\\in\\\{3,4\\\}, but yield substantial improvements for largerλ∈\{5,6\}\\lambda\\in\\\{5,6\\\}across all three tested values ofRR\. Although bothλ\\lambdavalues are in the supercritical regime \(λ\>1\\lambda\>1\), several factors explain this difference\. As shown in Figure[3](https://arxiv.org/html/2607.10074#S6.F3), the GNN learns poorer landmark embeddings forλ∈\{3,4\}\\lambda\\in\\\{3,4\\\}, even on small 50\-node graphs\. Additionally, for largenn, graphs are almost surely connected whenλ∈\{5,6\}\\lambda\\in\\\{5,6\\\}but not whenλ∈\{3,4\}\\lambda\\in\\\{3,4\\\}\. Finally, Figure[4](https://arxiv.org/html/2607.10074#S6.F4)\(e\) illustrates that GNN\-based embeddings can be generated faster than BFS\-based embeddings, particularly on large graphs as exact local embedding computations via BFS scale poorly with graph size\.

![Refer to caption](https://arxiv.org/html/2607.10074v1/paper/figures/exp2.png)Figure 4:\(a\)\-\(d\) Error rates of BFS\-based and GNN\-based lower bounds on graphs generated byERn​\(λ/n\)\\mathrm\{ER\}\_\{n\}\(\\lambda/n\), with the GNNs trained on graphs from the same model\. \(e\) Time required to generate all node\-to\-landmark distances innn\-node ER graphs by NetworkX’s highly optimized BFS compared to our widest and deepest GNNs\. All GCN, GraphSage, GAT, and GIN models are represented by the same color and solid lines for the sameRR, and the deviations between them are insignificant\.
### 6\.3Experiment 3: Transferability

In our last experiment, we investigate whether GNNs trained on small graphs can be transferred to compute landmark embeddings on larger networks for downstream shortest path approximation via LBs\. This is motivated by[Ruizet al\.](https://arxiv.org/html/2607.10074#bib.bib66)\([2020](https://arxiv.org/html/2607.10074#bib.bib66)\) and[Ruizet al\.](https://arxiv.org/html/2607.10074#bib.bib61)\([2023](https://arxiv.org/html/2607.10074#bib.bib61)\), which show that GNNs are transferable as their outputs converge on convergent graph sequences\. This, in turn, allows models trained on smaller graphs to generalize to similar larger graphs\.

Here, we focus onλ∈\{5,6\}\\lambda\\in\\\{5,6\\\}and train a sequence of eight GNNs on ER graphs ranging fromn=25n=25ton=3200n=3200nodes\. These GNNs are then used to generate local node embeddings on graphs from the ER model with the sameλ\\lambdaandn′=12800n^\{\\prime\}=12800nodes\. Figure[5](https://arxiv.org/html/2607.10074#S6.F5)\(a,d\) shows the MSE for each instance as the training graph size increases, with flat dashed lines indicating the MSE of BFS\-based LBs on then′n^\{\\prime\}\-node graph\. We observe a steady decrease in MSE asnngrows, with GNN\-based embeddings matching BFS\-based performance when GNNs are trained on graphs ofn=100n=100, which is 128 times smaller than the target graph\.

![Refer to caption](https://arxiv.org/html/2607.10074v1/paper/figures/exp3.png)Figure 5:Error rates of BFS\-based and GNN\-based lower bounds on \(a,d\) test Erdős–Rényi graphs generated byERn′​\(λ/n′\)\\text\{ER\}\_\{n^\{\\prime\}\}\(\\lambda/n^\{\\prime\}\), \(b,e\) Arxiv COND\-MAT collaboration network with 21,364 nodes, and \(c,f\) GEMSEC company network with 14,113 nodes, with the GNNs trained on graphs fromERn​\(λ/n\)\\mathrm\{ER\}\_\{n\}\(\\lambda/n\)\. Legend is the same as in Figure[4](https://arxiv.org/html/2607.10074#S6.F4)\.Table 1:Details on the largest connected component of selected benchmark networks\.![Refer to caption](https://arxiv.org/html/2607.10074v1/paper/figures/graph2.png)

![Refer to caption](https://arxiv.org/html/2607.10074v1/paper/figures/graph3.png)

![Refer to caption](https://arxiv.org/html/2607.10074v1/paper/figures/graph4.png)

![Refer to caption](https://arxiv.org/html/2607.10074v1/paper/figures/graph5.png)

![Refer to caption](https://arxiv.org/html/2607.10074v1/paper/figures/graph6.png)

![Refer to caption](https://arxiv.org/html/2607.10074v1/paper/figures/graph7.png)

![Refer to caption](https://arxiv.org/html/2607.10074v1/paper/figures/graph8.png)

![Refer to caption](https://arxiv.org/html/2607.10074v1/paper/figures/graph9.png)

![Refer to caption](https://arxiv.org/html/2607.10074v1/paper/figures/graph11.png)

![Refer to caption](https://arxiv.org/html/2607.10074v1/paper/figures/graph12.png)

![Refer to caption](https://arxiv.org/html/2607.10074v1/paper/figures/graph13.png)

![Refer to caption](https://arxiv.org/html/2607.10074v1/paper/figures/graph14.png)

![Refer to caption](https://arxiv.org/html/2607.10074v1/paper/figures/graph15.png)

![Refer to caption](https://arxiv.org/html/2607.10074v1/paper/figures/graph16.png)

Figure 6:Additional transferability results on real networks, with the GNNs trained on graphs fromERn​\(λ/n\)\\mathrm\{ER\}\_\{n\}\(\\lambda/n\)\. Legend is the same as in Figure[4](https://arxiv.org/html/2607.10074#S6.F4)\.When examining the transferability of the same set of GNNs on sixteen real\-world networks listed in Table[1](https://arxiv.org/html/2607.10074#S6.T1), we again observe that MSE improves with training graph size and that GNN\-based lower bounds outperform BFS\-based lower bounds, even though the landmark embeddings are learned on much smaller graphs \(see Figures[5](https://arxiv.org/html/2607.10074#S6.F5)and[6](https://arxiv.org/html/2607.10074#S6.F6)\)\. This can be explained as random graphs can model real\-world networks in certain scenarios, and networks with similar sparsity likely exhibit similar local structures which local message\-passing in GNNs can learn with sufficient training\.

All experiments use PyTorch Geometric\(Fey and Lenssen[2019](https://arxiv.org/html/2607.10074#bib.bib41)\)on a Lambda Vector 1 machine \(AMD Ryzen Threadripper PRO 5955WX CPU, 16 cores, 128 GB RAM, 2× NVIDIA RTX 4090 GPUs, no parallel training\)\. Code is available at[https://github\.com/ruiz\-lab/shortest\-path](https://github.com/ruiz-lab/shortest-path)\.

## 7Proofs

This section contains the proofs of all theoretical results in the paper\.

### 7\.1Proof of Lemma[4\.6](https://arxiv.org/html/2607.10074#S4.ThmtheoremEnv6)

Let

D=\[d11d12⋯d1​Td21d22⋮d2​T⋮⋮⋱⋮dT​1dT​2⋯dT​T\]=\[v1v2⋮vT\]=\[v1′v2′⋯vT′\]\.D=\\begin\{bmatrix\}d\_\{11\}&d\_\{12\}&\\cdots&d\_\{1T\}\\\\ d\_\{21\}&d\_\{22\}&\\vdots&d\_\{2T\}\\\\ \\vdots&\\vdots&\\ddots&\\vdots\\\\ d\_\{T1\}&d\_\{T2\}&\\cdots&d\_\{TT\}\\end\{bmatrix\}=\\begin\{bmatrix\}v\_\{1\}\\\\ v\_\{2\}\\\\ \\vdots\\\\ v\_\{T\}\\end\{bmatrix\}=\\begin\{bmatrix\}v^\{\\prime\}\_\{1\}&v^\{\\prime\}\_\{2\}&\\cdots&v^\{\\prime\}\_\{T\}\\end\{bmatrix\}\.By construction ofGG, we have∂Nk​\(u\)=∪t=1T∂Nk​\(u\)t\\partial N\_\{k\}\(u\)=\\cup\_\{t=1\}^\{T\}\\partial N\_\{k\}\(u\)\_\{t\}and∂Nk​\(u\)t∩∂Nk′​\(u\)t′=∅\\partial N\_\{k\}\(u\)\_\{t\}\\cap\\partial N\_\{k^\{\\prime\}\}\(u\)\_\{t^\{\\prime\}\}=\\emptysetfor any distancesk≠k′k\\neq k^\{\\prime\}or any typest≠t′t\\neq t^\{\\prime\}\. LetIx​yI\_\{xy\}be the indicator random variable for the edge\{x,y\}\\\{x,y\\\}being present\. Then fork≥1k\\geq 1,

𝔼​\(\|∂Nk​\(u\)t\|∣Nk−1​\(u\)\)\\displaystyle\\mathbb\{E\}\(\|\\partial N\_\{k\}\(u\)\_\{t\}\|\\mid N\_\{k\-1\}\(u\)\)≤𝔼​\(∑t′=1T∑x∈∂Nk−1​\(u\)t′∑y​is of type​ty∉Nk−1​\(u\)Ix​y∣Nk−1​\(u\)\)\\displaystyle\\leq\\mathbb\{E\}\\left\(\\sum\_\{t^\{\\prime\}=1\}^\{T\}\\sum\_\{x\\in\\partial N\_\{k\-1\}\(u\)\_\{t^\{\\prime\}\}\}\\sum\_\{\\begin\{subarray\}\{c\}y\\text\{ is of type \}t\\\\ y\\notin N\_\{k\-1\}\(u\)\\end\{subarray\}\}I\_\{xy\}\\mid N\_\{k\-1\}\(u\)\\right\)=∑t′=1T\|∂Nk−1​\(u\)t′\|​\(nt−∑l=0k−1\|∂Nl​\(u\)t\|\)​dt′​tnt\.\\displaystyle=\\sum\_\{t^\{\\prime\}=1\}^\{T\}\|\\partial N\_\{k\-1\}\(u\)\_\{t^\{\\prime\}\}\|\\left\(n\_\{t\}\-\\sum\_\{l=0\}^\{k\-1\}\|\\partial N\_\{l\}\(u\)\_\{t\}\|\\right\)\\frac\{d\_\{t^\{\\prime\}t\}\}\{n\_\{t\}\}\.It follows that

𝔼​\(\|∂Nk​\(u\)tk\|\)\\displaystyle\\mathbb\{E\}\(\|\\partial N\_\{k\}\(u\)\_\{t\_\{k\}\}\|\)=𝔼​\(𝔼​\(\|∂Nk​\(u\)tk\|∣Nk−1​\(u\)\)\)\\displaystyle=\\mathbb\{E\}\(\\mathbb\{E\}\(\|\\partial N\_\{k\}\(u\)\_\{t\_\{k\}\}\|\\mid N\_\{k\-1\}\(u\)\)\)≤𝔼​\(∑tk−1=1T\|∂Nk−1​\(u\)tk−1\|​\(ntk−∑l=0k−1\|∂Nl​\(u\)tk\|\)​dtk−1​tkntk\)\\displaystyle\\leq\\mathbb\{E\}\\left\(\\sum\_\{t\_\{k\-1\}=1\}^\{T\}\|\\partial N\_\{k\-1\}\(u\)\_\{t\_\{k\-1\}\}\|\\left\(n\_\{t\_\{k\}\}\-\\sum\_\{l=0\}^\{k\-1\}\|\\partial N\_\{l\}\(u\)\_\{t\_\{k\}\}\|\\right\)\\frac\{d\_\{t\_\{k\-1\}t\_\{k\}\}\}\{n\_\{t\_\{k\}\}\}\\right\)≤∑tk−1=1Tdtk−1​tk​𝔼​\(\|∂Nk−1​\(u\)tk−1\|\)\\displaystyle\\leq\\sum\_\{t\_\{k\-1\}=1\}^\{T\}d\_\{t\_\{k\-1\}t\_\{k\}\}\\mathbb\{E\}\(\|\\partial N\_\{k\-1\}\(u\)\_\{t\_\{k\-1\}\}\|\)≤∑tk−1=1Tdtk−1​tk​∑tk−2=1Tdtk−2​tk−1​𝔼​\(\|∂Nk−2​\(u\)tk−2\|\)\\displaystyle\\leq\\sum\_\{t\_\{k\-1\}=1\}^\{T\}d\_\{t\_\{k\-1\}t\_\{k\}\}\\sum\_\{t\_\{k\-2\}=1\}^\{T\}d\_\{t\_\{k\-2\}t\_\{k\-1\}\}\\mathbb\{E\}\(\|\\partial N\_\{k\-2\}\(u\)\_\{t\_\{k\-2\}\}\|\)≤⋯≤∑tk−1=1Tdtk−1​tk​∑tk−2=1Tdtk−2​tk−1​⋯​∑t1=1Tdt1​t2​𝔼​\(\|∂N1​\(u\)t1\|\)\.\\displaystyle\\leq\\cdots\\leq\\sum\_\{t\_\{k\-1\}=1\}^\{T\}d\_\{t\_\{k\-1\}t\_\{k\}\}\\sum\_\{t\_\{k\-2\}=1\}^\{T\}d\_\{t\_\{k\-2\}t\_\{k\-1\}\}\\cdots\\sum\_\{t\_\{1\}=1\}^\{T\}d\_\{t\_\{1\}t\_\{2\}\}\\mathbb\{E\}\(\|\\partial N\_\{1\}\(u\)\_\{t\_\{1\}\}\|\)\.Since

𝔼​\(\|∂N1​\(u\)t1\|\)=\{1⋅nt1​dt0​t1nt1if​t1≠t01⋅\(nt1−1\)​dt0​t1nt1otherwise≤dt0​t1,\\displaystyle\\mathbb\{E\}\(\|\\partial N\_\{1\}\(u\)\_\{t\_\{1\}\}\|\)=\\begin\{cases\}1\\cdot n\_\{t\_\{1\}\}\\frac\{d\_\{t\_\{0\}t\_\{1\}\}\}\{n\_\{t\_\{1\}\}\}&\\text\{if \}t\_\{1\}\\neq t\_\{0\}\\\\ 1\\cdot\(n\_\{t\_\{1\}\}\-1\)\\frac\{d\_\{t\_\{0\}t\_\{1\}\}\}\{n\_\{t\_\{1\}\}\}&\\text\{otherwise\}\\end\{cases\}\\leq d\_\{t\_\{0\}t\_\{1\}\},we have for allk≥2k\\geq 2that

𝔼​\(\|∂Nk​\(u\)tk\|\)\\displaystyle\\mathbb\{E\}\(\|\\partial N\_\{k\}\(u\)\_\{t\_\{k\}\}\|\)≤∑tk−1=1Tdtk−1​tk​∑tk−2=1Tdtk−2​tk−1​⋯​∑t1=1Tdt1​t2​dt0​t1\\displaystyle\\leq\\sum\_\{t\_\{k\-1\}=1\}^\{T\}d\_\{t\_\{k\-1\}t\_\{k\}\}\\sum\_\{t\_\{k\-2\}=1\}^\{T\}d\_\{t\_\{k\-2\}t\_\{k\-1\}\}\\cdots\\sum\_\{t\_\{1\}=1\}^\{T\}d\_\{t\_\{1\}t\_\{2\}\}d\_\{t\_\{0\}t\_\{1\}\}=∑tk−1=1Tdtk−1​tk​∑tk−2=1Tdtk−2​tk−1​⋯​∑t2=1Tdt2​t3​⟨vt0,vt2′⟩\\displaystyle=\\sum\_\{t\_\{k\-1\}=1\}^\{T\}d\_\{t\_\{k\-1\}t\_\{k\}\}\\sum\_\{t\_\{k\-2\}=1\}^\{T\}d\_\{t\_\{k\-2\}t\_\{k\-1\}\}\\cdots\\sum\_\{t\_\{2\}=1\}^\{T\}d\_\{t\_\{2\}t\_\{3\}\}\\left<v\_\{t\_\{0\}\},v^\{\\prime\}\_\{t\_\{2\}\}\\right\>=∑tk−1=1Tdtk−1​tk​∑tk−2=1Tdtk−2​tk−1​⋯​∑t3=1Tdt3​t4​⟨vt0,∑t2=1Tdt2​t3​vt2′⟩\\displaystyle=\\sum\_\{t\_\{k\-1\}=1\}^\{T\}d\_\{t\_\{k\-1\}t\_\{k\}\}\\sum\_\{t\_\{k\-2\}=1\}^\{T\}d\_\{t\_\{k\-2\}t\_\{k\-1\}\}\\cdots\\sum\_\{t\_\{3\}=1\}^\{T\}d\_\{t\_\{3\}t\_\{4\}\}\\left<v\_\{t\_\{0\}\},\\sum\_\{t\_\{2\}=1\}^\{T\}d\_\{t\_\{2\}t\_\{3\}\}v^\{\\prime\}\_\{t\_\{2\}\}\\right\>=∑tk−1=1Tdtk−1​tk​∑tk−2=1Tdtk−2​tk−1​⋯​∑t3=1Tdt3​t4​⟨vt0,D​vt3′⟩\\displaystyle=\\sum\_\{t\_\{k\-1\}=1\}^\{T\}d\_\{t\_\{k\-1\}t\_\{k\}\}\\sum\_\{t\_\{k\-2\}=1\}^\{T\}d\_\{t\_\{k\-2\}t\_\{k\-1\}\}\\cdots\\sum\_\{t\_\{3\}=1\}^\{T\}d\_\{t\_\{3\}t\_\{4\}\}\\left<v\_\{t\_\{0\}\},Dv^\{\\prime\}\_\{t\_\{3\}\}\\right\>=∑tk−1=1Tdtk−1​tk​∑tk−2=1Tdtk−2​tk−1​⋯​∑t3=1Tdt3​t4​⟨D⊤​vt0,vt3′⟩\\displaystyle=\\sum\_\{t\_\{k\-1\}=1\}^\{T\}d\_\{t\_\{k\-1\}t\_\{k\}\}\\sum\_\{t\_\{k\-2\}=1\}^\{T\}d\_\{t\_\{k\-2\}t\_\{k\-1\}\}\\cdots\\sum\_\{t\_\{3\}=1\}^\{T\}d\_\{t\_\{3\}t\_\{4\}\}\\left<D^\{\\top\}v\_\{t\_\{0\}\},v^\{\\prime\}\_\{t\_\{3\}\}\\right\>=∑tk−1=1Tdtk−1​tk​∑tk−2=1Tdtk−2​tk−1​⋯​∑t4=1Tdt4​t5​⟨\(D⊤\)2​vt0,vt4′⟩\\displaystyle=\\sum\_\{t\_\{k\-1\}=1\}^\{T\}d\_\{t\_\{k\-1\}t\_\{k\}\}\\sum\_\{t\_\{k\-2\}=1\}^\{T\}d\_\{t\_\{k\-2\}t\_\{k\-1\}\}\\cdots\\sum\_\{t\_\{4\}=1\}^\{T\}d\_\{t\_\{4\}t\_\{5\}\}\\left<\(D^\{\\top\}\)^\{2\}v\_\{t\_\{0\}\},v^\{\\prime\}\_\{t\_\{4\}\}\\right\>=⋯=⟨\(D⊤\)k−2​vt0,vtk′⟩=⟨\(D⊤\)k−2​D⊤​et0,D​etk⟩=\[Dk\]t0​tk\\displaystyle=\\cdots=\\left<\(D^\{\\top\}\)^\{k\-2\}v\_\{t\_\{0\}\},v^\{\\prime\}\_\{t\_\{k\}\}\\right\>=\\left<\(D^\{\\top\}\)^\{k\-2\}D^\{\\top\}e\_\{t\_\{0\}\},De\_\{t\_\{k\}\}\\right\>=\[D^\{k\}\]\_\{t\_\{0\}t\_\{k\}\}and

𝔼\(\|Nk\(\\displaystyle\\mathbb\{E\}\(\|N\_\{k\}\(u\)t\|\)=𝔼\(\|∂N0\(u\)t\|\)\+𝔼\(\|∂N1\(u\)t\|\)\+∑l=2k𝔼\(\|∂Nl\(u\)t\|\)\\displaystyle u\)\_\{t\}\|\)=\\mathbb\{E\}\(\|\\partial N\_\{0\}\(u\)\_\{t\}\|\)\+\\mathbb\{E\}\(\|\\partial N\_\{1\}\(u\)\_\{t\}\|\)\+\\sum\_\{l=2\}^\{k\}\\mathbb\{E\}\(\|\\partial N\_\{l\}\(u\)\_\{t\}\|\)≤1\+dt0​t\+∑l=2k‖D‖2l≤1\+‖D‖2\+∑l=2k‖D‖2l\.\\displaystyle\\leq 1\+d\_\{t\_\{0\}t\}\+\\sum\_\{l=2\}^\{k\}\\\|D\\\|\_\{2\}^\{l\}\\leq 1\+\\\|D\\\|\_\{2\}\+\\sum\_\{l=2\}^\{k\}\\\|D\\\|\_\{2\}^\{l\}\.
Under Assumptions[3\.1](https://arxiv.org/html/2607.10074#S3.ThmtheoremEnv1)and[3\.2](https://arxiv.org/html/2607.10074#S3.ThmtheoremEnv2)withk=Θ​\(log⁡n\)k=\\Theta\(\\log n\), there existsC\>0C\>0such that for allk\>Kk\>K,

𝔼​\(\|∂Nk​\(u\)tk\|\)\\displaystyle\\mathbb\{E\}\(\|\\partial N\_\{k\}\(u\)\_\{t\_\{k\}\}\|\)≤⟨et0,Dk​etk⟩=\[Dk\]t0​tk≤C​λ1k\\displaystyle\\leq\\left<e\_\{t\_\{0\}\},D^\{k\}e\_\{t\_\{k\}\}\\right\>=\[D^\{k\}\]\_\{t\_\{0\}t\_\{k\}\}\\leq C\\lambda\_\{1\}^\{k\}and

𝔼\(\|Nk\(u\)t\\displaystyle\\mathbb\{E\}\(\|N\_\{k\}\(u\)\_\{t\}\|\)=𝔼\(\|∂N0\(u\)t\|\)\+𝔼\(\|∂N1\(u\)t\|\)\+∑l=2k𝔼\(\|∂Nl\(u\)t\|\)\\displaystyle\|\)=\\mathbb\{E\}\(\|\\partial N\_\{0\}\(u\)\_\{t\}\|\)\+\\mathbb\{E\}\(\|\\partial N\_\{1\}\(u\)\_\{t\}\|\)\+\\sum\_\{l=2\}^\{k\}\\mathbb\{E\}\(\|\\partial N\_\{l\}\(u\)\_\{t\}\|\)≤1\+dt0​t\+∑l=2K−1‖D‖2l\+∑l=KkC​λ1l=O​\(λ1k\)\.\\displaystyle\\leq 1\+d\_\{t\_\{0\}t\}\+\\sum\_\{l=2\}^\{K\-1\}\\\|D\\\|\_\{2\}^\{l\}\+\\sum\_\{l=K\}^\{k\}C\\lambda\_\{1\}^\{l\}=O\(\\lambda\_\{1\}^\{k\}\)\.

### 7\.2Proof of Lemma[4\.7](https://arxiv.org/html/2607.10074#S4.ThmtheoremEnv7)

Recall inhomogeneous random graph theory:

1. \(1\)Sinceλ1\>1\\lambda\_\{1\}\>1, the largest component satisfies\|𝒞\(1\)\|=Θ​\(n\)\|\\mathcal\{C\}\_\{\(1\)\}\|=\\Theta\(n\)and all other components satisfy\|𝒞\(i\)\|=O​\(log⁡n\)\|\\mathcal\{C\}\_\{\(i\)\}\|=O\(\\log n\)fori\>1i\>1w\.h\.p\.\(Bollobáset al\.[2007](https://arxiv.org/html/2607.10074#bib.bib264), Theorems 3\.1 and 3\.12\)\.
2. \(2\)Local neighborhood growth \(starting from a fixed node\) can be approximated by a multitype Poisson branching process with mean matrixDD\(Bollobáset al\.[2007](https://arxiv.org/html/2607.10074#bib.bib264), Sections 4\.1–4\.2\)\.
3. \(3\)The local neighborhood explorations of two randomly chosen nodes can, w\.h\.p\., be coupled to two independent copies of the multitype Poisson branching process up to a slowly growing logarithmic depthk≤κ​logλ1⁡nk\\leq\\kappa\\log\_\{\\lambda\_\{1\}\}nwithκ∈\(0,1\)\\kappa\\in\(0,1\)\(Bollobáset al\.[2007](https://arxiv.org/html/2607.10074#bib.bib264), Lemma 11\.5\)\.
4. \(4\)Let\(Zki\)k≥0\(Z^\{i\}\_\{k\}\)\_\{k\\geq 0\}be the branching process starting with one node of typeii, and letZki​\(j\)Z^\{i\}\_\{k\}\(j\)denote the number of nodes of typejjat generationkk\. A Kesten–Stigum type theorem for a supercritical multitype branching process \([Gramaet al\.](https://arxiv.org/html/2607.10074#bib.bib266),[2023](https://arxiv.org/html/2607.10074#bib.bib266), Theorem 2\.7, Remark 2\.8, and Corollary 2\.9\) states that - •On the event of survival/explosion,Zki​\(j\)Z^\{i\}\_\{k\}\(j\)grows asymptotically like\[Dk\]i​j\[D^\{k\}\]\_\{ij\}\([Gramaet al\.](https://arxiv.org/html/2607.10074#bib.bib266),[2023](https://arxiv.org/html/2607.10074#bib.bib266), Equations 1\.2, 1\.7, and 2\.21\)\. - •On the event of extinction,Zki​\(j\)→0Z^\{i\}\_\{k\}\(j\)\\to 0w\.h\.p\. \([Gramaet al\.](https://arxiv.org/html/2607.10074#bib.bib266),[2023](https://arxiv.org/html/2607.10074#bib.bib266), Equations 2\.19 and 2\.20\)\.

IfBnB\_\{n\}does not occur, then at least one ofu1u\_\{1\}oru2u\_\{2\}must lie in𝒞\(i\)\\mathcal\{C\}\_\{\(i\)\}for somei\>1i\>1\. Since\|𝒞\(i\)\|=O​\(log⁡n\)\|\\mathcal\{C\}\_\{\(i\)\}\|=O\(\\log n\)for alli\>1i\>1w\.h\.p\. according to \(1\),AnA\_\{n\}cannot occur as it explicitly requires both components containingu1u\_\{1\}andu2u\_\{2\}to have size at leastnΩ​\(1\)n^\{\\Omega\(1\)\}w\.h\.p\. SinceBnc⊆AncB\_\{n\}^\{c\}\\subseteq A\_\{n\}^\{c\}, we immediately haveℙ​\(An∖Bn\)→0\\mathbb\{P\}\(A\_\{n\}\\setminus B\_\{n\}\)\\to 0asn→∞n\\to\\infty\.

Conversely, to establish thatℙ​\(Bn∖An\)→0\\mathbb\{P\}\(B\_\{n\}\\setminus A\_\{n\}\)\\to 0, we look at the neighborhood exploration ofu1u\_\{1\}andu2u\_\{2\}given thatBnB\_\{n\}occurs\. By \(2\) and \(3\), the local explorations can be jointly coupled to two branching processes until generationk≤κ​logλ1⁡nk\\leq\\kappa\\log\_\{\\lambda\_\{1\}\}n\. Under this coupling, a node belongs to the giant component𝒞\(1\)\\mathcal\{C\}\_\{\(1\)\}if and only if its locally coupled branching process avoids early extinction and enters the survival/explosion regime\. BecauseBnB\_\{n\}impliesu1,u2∈𝒞\(1\)u\_\{1\},u\_\{2\}\\in\\mathcal\{C\}\_\{\(1\)\}, it follows that both independent branching processes𝒵1\\mathcal\{Z\}\_\{1\}and𝒵2\\mathcal\{Z\}\_\{2\}survive w\.h\.p\. By the Kesten–Stigum theorem \(4\), conditional on survival, the generation sizes of both processes grow exponentially at rateλ1k\\lambda\_\{1\}^\{k\}, tracking the matrix elements\[Dk\]i​j\[D^\{k\}\]\_\{ij\}\. This exponential growth directly fulfills the conditions defining eventAnA\_\{n\}\. Consequently, conditional onBnB\_\{n\}, eventAnA\_\{n\}occurs w\.h\.p\., meaningℙ​\(An∣Bn\)→1\\mathbb\{P\}\(A\_\{n\}\\mid B\_\{n\}\)\\to 1, which provesℙ​\(Bn∖An\)→0\\mathbb\{P\}\(B\_\{n\}\\setminus A\_\{n\}\)\\to 0\.

### 7\.3Proof of Lemma[4\.8](https://arxiv.org/html/2607.10074#S4.ThmtheoremEnv8)

From Markov’s inequality and Lemma[4\.6](https://arxiv.org/html/2607.10074#S4.ThmtheoremEnv6), there existδ\>0\\delta\>0for anyγ∈\(κ0\+κ,1\)\\gamma\\in\(\\kappa\_\{0\}\+\\kappa,1\)such that

ℙ​\(\|Nk​\(ui\)t\|≥nγ\)≤O​\(λ1k\)nγ≤O​\(nκ0\+κ\)nγ≤n−δ\\mathbb\{P\}\(\|N\_\{k\}\(u\_\{i\}\)\_\{t\}\|\\geq n^\{\\gamma\}\)\\leq\\frac\{O\(\\lambda\_\{1\}^\{k\}\)\}\{n^\{\\gamma\}\}\\leq\\frac\{O\\left\(n^\{\\kappa\_\{0\}\+\\kappa\}\\right\)\}\{n^\{\\gamma\}\}\\leq n^\{\-\\delta\}fori=1,2i=1,2andt∈\[T\]t\\in\[T\]withk≤\(κ0\+κ\)​logλ1⁡nk\\leq\(\\kappa\_\{0\}\+\\kappa\)\\log\_\{\\lambda\_\{1\}\}nwith sufficiently largenn\. Then for each fixedk≤\(κ0\+κ\)​logλ1⁡nk\\leq\(\\kappa\_\{0\}\+\\kappa\)\\log\_\{\\lambda\_\{1\}\}n,

ℙ\(\|Nk\(ui\)t\|\\displaystyle\\mathbb\{P\}\(\|N\_\{k\}\(u\_\{i\}\)\_\{t\}\|≤nγfori=1,2andt∈\[T\]\)\\displaystyle\\leq n^\{\\gamma\}\\text\{ for \}i=1,2\\text\{ and \}t\\in\[T\]\)=1−ℙ\(∃i∈\{1,2\},t∈\[T\]s\.t\.\|Nk\(ui\)t\|≥nγ\)\\displaystyle=1\-\\mathbb\{P\}\(\\exists i\\in\\\{1,2\\\},t\\in\[T\]s\.t\.\\left\|N\_\{k\}\(u\_\{i\}\)\_\{t\}\\right\|\\geq n^\{\\gamma\}\)≥1−∑i=1,2∑t=1Tℙ​\(\|Nk​\(ui\)t\|≥nγ\)≥1−2​T​n−δ\.\\displaystyle\\geq 1\-\\sum\_\{i=1,2\}\\sum\_\{t=1\}^\{T\}\\mathbb\{P\}\(\|N\_\{k\}\(u\_\{i\}\)\_\{t\}\|\\geq n^\{\\gamma\}\)\\geq 1\-2Tn^\{\-\\delta\}\.\(1\)
Letδn=n−β\\delta\_\{n\}=n^\{\-\\beta\}with0<β<κ020<\\beta<\\frac\{\\kappa\_\{0\}\}\{2\}\. The upper bound onβ\\betais chosen so that the upper bound in \([4](https://arxiv.org/html/2607.10074#S7.E4)\) vanishes faster than2​T​n−δ2Tn^\{\-\\delta\}asnκ0−2​β→∞n^\{\\kappa\_\{0\}\-2\\beta\}\\to\\infty, which explains the failure probability in \([5](https://arxiv.org/html/2607.10074#S7.E5)\)\. Sincent/n≥α\{n\_\{t\}\}/\{n\}\\geq\\alphafor allt∈\[T\]t\\in\[T\]for some constantα\>0\\alpha\>0by Assumption[3\.3](https://arxiv.org/html/2607.10074#S3.ThmtheoremEnv3)\. Define

ℰn,k′=\{\|∂NL\+k​\(ui\)\|→∈\[b′im​\(1−δn\)k​\(1−α−1​nγ−1\)k​Dk,b′iM​\(1\+δn\)k​Dk\]:i=1,2\}\\mathcal\{E\}^\{\\prime\}\_\{n,k\}=\\\{\\overrightarrow\{\|\\partial N\_\{L\+k\}\(u\_\{i\}\)\|\}\\in\[\{b^\{\\prime\}\}\_\{i\}^\{m\}\(1\-\\delta\_\{n\}\)^\{k\}\(1\-\\alpha^\{\-1\}n^\{\\gamma\-1\}\)^\{k\}D^\{k\},\{b^\{\\prime\}\}\_\{i\}^\{M\}\(1\+\\delta\_\{n\}\)^\{k\}D^\{k\}\]:i=1,2\\\}whereb′im=\(1−ε′\)​et​\(ui\)⊤​DL\{b^\{\\prime\}\}\_\{i\}^\{m\}=\(1\-\\varepsilon^\{\\prime\}\)e\_\{t\(u\_\{i\}\)\}^\{\\top\}D^\{L\}andb′iM=\(1\+ε′\)​et​\(ui\)⊤​DL\{b^\{\\prime\}\}\_\{i\}^\{M\}=\(1\+\\varepsilon^\{\\prime\}\)e\_\{t\(u\_\{i\}\)\}^\{\\top\}D^\{L\}with0<ε′<ε0<\\varepsilon^\{\\prime\}<\\varepsilon, which reflects tighter bounds than inℰn,k\\mathcal\{E\}\_\{n,k\}\. Our next step is to bound𝔼​\(\|∂NL\+k​\(ui\)\|→∣NL\+k−1​\(ui\)\)\\mathbb\{E\}\(\\overrightarrow\{\|\\partial N\_\{L\+k\}\(u\_\{i\}\)\|\}\\mid N\_\{L\+k\-1\}\(u\_\{i\}\)\)and then use the Chernoff\-Hoeffding bound \([Dubhashi and Panconesi](https://arxiv.org/html/2607.10074#bib.bib265),[2009](https://arxiv.org/html/2607.10074#bib.bib265), Theorem 1\.1\) to prove thatℰn,k′\\mathcal\{E\}^\{\\prime\}\_\{n,k\}occurs w\.h\.p\. given∩l=0k−1ℰn,l′\\cap\_\{l=0\}^\{k\-1\}\\mathcal\{E\}^\{\\prime\}\_\{n,l\}andAbm,bMA\_\{b^\{m\},b^\{M\}\}; that is, local expansions from two uniformly random nodes continue to grow exponentially up to the neighborhood radius\(κ0\+κ\)​logλ1⁡n\(\\kappa\_\{0\}\+\\kappa\)\\log\_\{\\lambda\_\{1\}\}n, provided that the previous layers exhibit exponential growth\.

LetIx​yI\_\{xy\}be the indicator random variable for the edge\{x,y\}\\\{x,y\\\}being present\. Let\[∂Nk​\(u\)\]t′​t\[\\partial N\_\{k\}\(u\)\]\_\{t^\{\\prime\}t\}be nodes of typettnot inNk−1​\(u\)N\_\{k\-1\}\(u\)that have an edge with a node of typet′t^\{\\prime\}in∂Nk−1​\(u\)\\partial N\_\{k\-1\}\(u\)\. Conditionally onAbm,bM=\{\|∂NL​\(ui\)\|→∈\[bim,biM\]:i=1,2\}A\_\{b^\{m\},b^\{M\}\}=\\left\\\{\\overrightarrow\{\|\\partial N\_\{L\}\(u\_\{i\}\)\|\}\\in\[b\_\{i\}^\{m\},b\_\{i\}^\{M\}\]:i=1,2\\right\\\},

𝔼\(\|∂\\displaystyle\\mathbb\{E\}\(\|\\partialNL\+k\(ui\)t\|∣NL\+k−1\(ui\)\)\\displaystyle N\_\{L\+k\}\(u\_\{i\}\)\_\{t\}\|\\mid N\_\{L\+k\-1\}\(u\_\{i\}\)\)=𝔼​\(∑x∉NL\+k−1​\(ui\)t​\(x\)=t𝟙\{∃t′∈\[T\]​∃y∈∂NL\+k−1​\(ui\)t′:Ix​y=1\}∣NL\+k−1​\(ui\)\)\\displaystyle=\\mathbb\{E\}\\left\(\\sum\_\{\\begin\{subarray\}\{c\}x\\notin N\_\{L\+k\-1\}\(u\_\{i\}\)\\\\ t\(x\)=t\\end\{subarray\}\}\\mathbbm\{1\}\_\{\\\{\\exists t^\{\\prime\}\\in\[T\]\\\>\\exists y\\in\\partial N\_\{L\+k\-1\}\(u\_\{i\}\)\_\{t^\{\\prime\}\}\\\>:\\\>I\_\{xy\}=1\\\}\}\\mid N\_\{L\+k\-1\}\(u\_\{i\}\)\\right\)=\(nt−\|NL\+k−1​\(ui\)t\|\)​\(1−∏t′=1T\(1−dt′​tnt\)\|∂NL\+k−1​\(ui\)t′\|\)\\displaystyle=\(n\_\{t\}\-\|N\_\{L\+k\-1\}\(u\_\{i\}\)\_\{t\}\|\)\\left\(1\-\\prod\_\{t^\{\\prime\}=1\}^\{T\}\\left\(1\-\\frac\{d\_\{t^\{\\prime\}t\}\}\{n\_\{t\}\}\\right\)^\{\|\\partial N\_\{L\+k\-1\}\(u\_\{i\}\)\_\{t^\{\\prime\}\}\|\}\\right\)Conditioning onℰn,k−1′\\mathcal\{E\}^\{\\prime\}\_\{n,k\-1\}withλ1\>1\\lambda\_\{1\}\>1,dt′​tnt∈\[0,1\]\\frac\{d\_\{t^\{\\prime\}t\}\}\{n\_\{t\}\}\\in\[0,1\], andDDbeing primitive, we have that

\|∂NL\+k−1​\(ui\)t′\|​dt′​tnt\\displaystyle\|\\partial N\_\{L\+k\-1\}\(u\_\{i\}\)\_\{t^\{\\prime\}\}\|\\frac\{d\_\{t^\{\\prime\}t\}\}\{n\_\{t\}\}≤\(1\+ε′\)​\(1\+δn\)k​\[DL\+k−1\]t​\(ui\)​t′​dt′​tα​n\\displaystyle\\leq\(1\+\\varepsilon^\{\\prime\}\)\(1\+\\delta\_\{n\}\)^\{k\}\[D^\{L\+k\-1\}\]\_\{t\(u\_\{i\}\)t^\{\\prime\}\}\\frac\{d\_\{t^\{\\prime\}t\}\}\{\\alpha n\}≤\(1\+ε′\)​\(1\+n−β\)k​C​λ1L\+k−1​dt′​tα​n\\displaystyle\\leq\(1\+\\varepsilon^\{\\prime\}\)\(1\+n^\{\-\\beta\}\)^\{k\}C\\lambda\_\{1\}^\{L\+k\-1\}\\frac\{d\_\{t^\{\\prime\}t\}\}\{\\alpha n\}≤\(1\+ε′\)​\(1\+n−β\)k​C​dt′​t​nκ0\+κα​n\\displaystyle\\leq\(1\+\\varepsilon^\{\\prime\}\)\(1\+n^\{\-\\beta\}\)^\{k\}Cd\_\{t^\{\\prime\}t\}\\frac\{n^\{\\kappa\_\{0\}\+\\kappa\}\}\{\\alpha n\}for allt′∈\[T\]t^\{\\prime\}\\in\[T\]and alli=1,2i=1,2for some constantC\>0C\>0\. Since−β<0\-\\beta<0andk≤κ​logλ1⁡nk\\leq\\kappa\\log\_\{\\lambda\_\{1\}\}n,\(1\+n−β\)k→1\(1\+n^\{\-\\beta\}\)^\{k\}\\to 1asn→∞n\\to\\infty\. Sinceκ0\+κ<1\\kappa\_\{0\}\+\\kappa<1,\|∂NL\+k−1​\(ui\)t′\|​dt′​tnt≤\(1\+ε′\)​\(1\+n−β\)k​C​dt′​t​α​nκ0\+κα​n\|\\partial N\_\{L\+k\-1\}\(u\_\{i\}\)\_\{t^\{\\prime\}\}\|\\frac\{d\_\{t^\{\\prime\}t\}\}\{n\_\{t\}\}\\leq\(1\+\\varepsilon^\{\\prime\}\)\(1\+n^\{\-\\beta\}\)^\{k\}Cd\_\{t^\{\\prime\}t\}\\alpha\\frac\{n^\{\\kappa\_\{0\}\+\\kappa\}\}\{\\alpha n\}vanishes, and so

\(1−dt′​tnt\)\|∂NL\+k−1​\(ui\)t′\|=1−\|∂NL\+k−1​\(ui\)t′\|​dt′​tnt​\(1\+o​\(1\)\),\\displaystyle\\left\(1\-\\frac\{d\_\{t^\{\\prime\}t\}\}\{n\_\{t\}\}\\right\)^\{\|\\partial N\_\{L\+k\-1\}\(u\_\{i\}\)\_\{t^\{\\prime\}\}\|\}=1\-\|\\partial N\_\{L\+k\-1\}\(u\_\{i\}\)\_\{t^\{\\prime\}\}\|\\frac\{d\_\{t^\{\\prime\}t\}\}\{n\_\{t\}\}\(1\+o\(1\)\),which implies

𝔼\(\|∂\\displaystyle\\mathbb\{E\}\(\|\\partialNL\+k\(ui\)t\|∣NL\+k−1\(ui\)\)\\displaystyle N\_\{L\+k\}\(u\_\{i\}\)\_\{t\}\|\\mid N\_\{L\+k\-1\}\(u\_\{i\}\)\)=\(nt−\|NL\+k−1​\(ui\)t\|\)​\(1−∏t′=1T\(1−\|∂NL\+k−1​\(ui\)t′\|​dt′​tnt​\(1\+o​\(1\)\)\)\)\.\\displaystyle=\(n\_\{t\}\-\|N\_\{L\+k\-1\}\(u\_\{i\}\)\_\{t\}\|\)\\left\(1\-\\prod\_\{t^\{\\prime\}=1\}^\{T\}\\left\(1\-\|\\partial N\_\{L\+k\-1\}\(u\_\{i\}\)\_\{t^\{\\prime\}\}\|\\frac\{d\_\{t^\{\\prime\}t\}\}\{n\_\{t\}\}\(1\+o\(1\)\)\\right\)\\right\)\.\(2\)SinceTTis finite, we have the identity

∏t′=1T\(1−At′​\(1\+o​\(1\)\)\)=1−\(∑t′=1TAt′\)​\(1\+o​\(1\)\)\\displaystyle\\prod\_\{t^\{\\prime\}=1\}^\{T\}\\left\(1\-A\_\{t^\{\\prime\}\}\(1\+o\(1\)\)\\right\)=1\-\\left\(\\sum\_\{t^\{\\prime\}=1\}^\{T\}A\_\{t^\{\\prime\}\}\\right\)\(1\+o\(1\)\)\(3\)Applying this identity to \([2](https://arxiv.org/html/2607.10074#S7.E2)\), we obtain

𝔼\(\|∂\\displaystyle\\mathbb\{E\}\(\|\\partialNL\+k\(ui\)t\|∣NL\+k−1\(ui\)\)\\displaystyle N\_\{L\+k\}\(u\_\{i\}\)\_\{t\}\|\\mid N\_\{L\+k\-1\}\(u\_\{i\}\)\)=\(nt−\|NL\+k−1​\(ui\)t\|\)​\(∑t′=1T\|∂NL\+k−1​\(ui\)t′\|​dt′​tnt\)​\(1\+o​\(1\)\)\\displaystyle=\(n\_\{t\}\-\|N\_\{L\+k\-1\}\(u\_\{i\}\)\_\{t\}\|\)\\left\(\\sum\_\{t^\{\\prime\}=1\}^\{T\}\|\\partial N\_\{L\+k\-1\}\(u\_\{i\}\)\_\{t^\{\\prime\}\}\|\\frac\{d\_\{t^\{\\prime\}t\}\}\{n\_\{t\}\}\\right\)\(1\+o\(1\)\)=\(1−\|NL\+k−1​\(ui\)t\|nt\)​\(1\+o​\(1\)\)​⟨\|∂NL\+k−1​\(ui\)\|→,D​et⟩\.\\displaystyle=\\left\(1\-\\frac\{\|N\_\{L\+k\-1\}\(u\_\{i\}\)\_\{t\}\|\}\{n\_\{t\}\}\\right\)\(1\+o\(1\)\)\\left\\langle\\overrightarrow\{\|\\partial N\_\{L\+k\-1\}\(u\_\{i\}\)\|\},De\_\{t\}\\right\\rangle\.Conditionally on∩l=0k−1ℰn,l′\\cap\_\{l=0\}^\{k\-1\}\\mathcal\{E\}^\{\\prime\}\_\{n,l\}andAbm,bMA\_\{b^\{m\},b^\{M\}\}, from \([1](https://arxiv.org/html/2607.10074#S7.E1)\) we have

b′im​\(1−δn\)k−1​\(1−α−1​nγ−1\)k​Dk≤𝔼​\(\|∂NL\+k​\(ui\)\|→∣NL\+k−1​\(ui\)\)≤b′iM​\(1\+δn\)k−1​Dk\\displaystyle\{b^\{\\prime\}\}\_\{i\}^\{m\}\(1\-\\delta\_\{n\}\)^\{k\-1\}\(1\-\\alpha^\{\-1\}n^\{\\gamma\-1\}\)^\{k\}D^\{k\}\\leq\\mathbb\{E\}\(\\overrightarrow\{\|\\partial N\_\{L\+k\}\(u\_\{i\}\)\|\}\\mid N\_\{L\+k\-1\}\(u\_\{i\}\)\)\\leq\{b^\{\\prime\}\}\_\{i\}^\{M\}\(1\+\\delta\_\{n\}\)^\{k\-1\}D^\{k\}with probability at least1−2​T​n−δ1\-2Tn^\{\-\\delta\}since1−α−1​nγ−1≤1−\|NL\+k−1​\(ui\)t\|nt≤11\-\\alpha^\{\-1\}n^\{\\gamma\-1\}\\leq 1\-\\frac\{\|N\_\{L\+k\-1\}\(u\_\{i\}\)\_\{t\}\|\}\{n\_\{t\}\}\\leq 1for alli=1,2i=1,2andt∈\[T\]t\\in\[T\]with sufficiently largenn\. Denote this eventRkR\_\{k\}\. Usingℙ​\(A\)≤ℙ​\(A∣B\)\+ℙ​\(Bc\)\\mathbb\{P\}\(A\)\\leq\\mathbb\{P\}\(A\\mid B\)\+\\mathbb\{P\}\(B^\{c\}\), we obtain

ℙ\(ℰ′n,kc\\displaystyle\\mathbb\{P\}\(\{\\mathcal\{E\}^\{\\prime\}\}\_\{n,k\}^\{c\}∣∩l=0k−1ℰn,l′,Abm,bM\)≤ℙ\(ℰ′n,kc∣Rk,∩l=0k−1ℰn,l′,Abm,bM\)\+2Tn−δ\.\\displaystyle\\mid\\cap\_\{l=0\}^\{k\-1\}\\mathcal\{E\}^\{\\prime\}\_\{n,l\},A\_\{b^\{m\},b^\{M\}\}\)\\leq\\mathbb\{P\}\(\{\\mathcal\{E\}^\{\\prime\}\}\_\{n,k\}^\{c\}\\mid R\_\{k\},\\cap\_\{l=0\}^\{k\-1\}\\mathcal\{E\}^\{\\prime\}\_\{n,l\},A\_\{b^\{m\},b^\{M\}\}\)\+2Tn^\{\-\\delta\}\.Then by union bound and Chernoff\-Hoeffding bound \([Dubhashi and Panconesi](https://arxiv.org/html/2607.10074#bib.bib265),[2009](https://arxiv.org/html/2607.10074#bib.bib265), Theorem 1\.1\),

ℙ​\(ℰ′n,kc∣Rk,∩l=0k−1ℰn,l′,Abm,bM\)\\displaystyle\\mathbb\{P\}\(\{\\mathcal\{E\}^\{\\prime\}\}\_\{n,k\}^\{c\}\\mid R\_\{k\},\\cap\_\{l=0\}^\{k\-1\}\\mathcal\{E\}^\{\\prime\}\_\{n,l\},A\_\{b^\{m\},b^\{M\}\}\)≤∑i=1,2∑t=1Tℙ​\(\|\|∂NL\+k​\(ui\)t\|−𝔼​\(\|∂NL\+k​\(ui\)t\|\)\|≥δn​𝔼​\(\|∂NL\+k​\(ui\)t\|\)∣Rk,∩l=0k−1ℰn,l′,Abm,bM\)\\displaystyle\\leq\\sum\_\{i=1,2\}\\sum\_\{t=1\}^\{T\}\\mathbb\{P\}\(\|\|\\partial N\_\{L\+k\}\(u\_\{i\}\)\_\{t\}\|\-\\mathbb\{E\}\(\|\\partial N\_\{L\+k\}\(u\_\{i\}\)\_\{t\}\|\)\|\\geq\\delta\_\{n\}\\mathbb\{E\}\(\|\\partial N\_\{L\+k\}\(u\_\{i\}\)\_\{t\}\|\)\\mid R\_\{k\},\\cap\_\{l=0\}^\{k\-1\}\\mathcal\{E\}^\{\\prime\}\_\{n,l\},A\_\{b^\{m\},b^\{M\}\}\)≤∑i=1,2∑t=1T2​exp⁡\(−δn23​𝔼​\(\|∂NL\+k​\(ui\)t\|∣Rk,∩l=0k−1ℰn,l′,Abm,bM\)\)\\displaystyle\\leq\\sum\_\{i=1,2\}\\sum\_\{t=1\}^\{T\}2\\exp\\left\(\-\\frac\{\\delta\_\{n\}^\{2\}\}\{3\}\\mathbb\{E\}\(\|\\partial N\_\{L\+k\}\(u\_\{i\}\)\_\{t\}\|\\mid R\_\{k\},\\cap\_\{l=0\}^\{k\-1\}\\mathcal\{E\}^\{\\prime\}\_\{n,l\},A\_\{b^\{m\},b^\{M\}\}\)\\right\)≤4​T​exp⁡\(−δn23​b′im​\(1−δn\)k−1​\(1−α−1​nγ−1\)k​Dk​et\)\\displaystyle\\leq 4T\\exp\\left\(\-\\frac\{\\delta\_\{n\}^\{2\}\}\{3\}\{b^\{\\prime\}\}\_\{i\}^\{m\}\(1\-\\delta\_\{n\}\)^\{k\-1\}\(1\-\\alpha^\{\-1\}n^\{\\gamma\-1\}\)^\{k\}D^\{k\}e\_\{t\}\\right\)=4​T​exp⁡\(−δn23​\(1−ε′\)​\(1−n−β\)k−1​\(1−α−1​nγ−1\)k​\[DL\+k\]t​\(ui\)​t\)\.\\displaystyle=4T\\exp\\left\(\-\\frac\{\\delta\_\{n\}^\{2\}\}\{3\}\(1\-\\varepsilon^\{\\prime\}\)\(1\-n^\{\-\\beta\}\)^\{k\-1\}\(1\-\\alpha^\{\-1\}n^\{\\gamma\-1\}\)^\{k\}\[D^\{L\+k\}\]\_\{t\(u\_\{i\}\)t\}\\right\)\.SinceDDis primitive, there existsc\>0c\>0such that\[DL\+k\]i​j≥c​λ1L\+k≥c​nκ0\[D^\{L\+k\}\]\_\{ij\}\\geq c\\lambda\_\{1\}^\{L\+k\}\\geq cn^\{\\kappa\_\{0\}\}for all pairs of types\(i,j\)\(i,j\)and so

ℙ\(ℰ′n,kc\\displaystyle\\mathbb\{P\}\(\{\\mathcal\{E\}^\{\\prime\}\}\_\{n,k\}^\{c\}∣Rk,∩l=0k−1ℰn,l′,Abm,bM\)\\displaystyle\\mid R\_\{k\},\\cap\_\{l=0\}^\{k\-1\}\\mathcal\{E\}^\{\\prime\}\_\{n,l\},A\_\{b^\{m\},b^\{M\}\}\)≤4​T​exp⁡\(−n−2​β3​\(1−ε′\)​\(1−n−β\)k−1​\(1−α−1​nγ−1\)k​c​nκ0\)\.\\displaystyle\\leq 4T\\exp\\left\(\-\\frac\{n^\{\-2\\beta\}\}\{3\}\(1\-\\varepsilon^\{\\prime\}\)\(1\-n^\{\-\\beta\}\)^\{k\-1\}\(1\-\\alpha^\{\-1\}n^\{\\gamma\-1\}\)^\{k\}cn^\{\\kappa\_\{0\}\}\\right\)\.\(4\)Since−β<0\-\\beta<0andγ−1<0\\gamma\-1<0withk≤κ​logλ1⁡nk\\leq\\kappa\\log\_\{\\lambda\_\{1\}\}n,\(1−n−β\)k−1→1\(1\-n^\{\-\\beta\}\)^\{k\-1\}\\to 1and\(1−α−1​nγ−1\)k→1\(1\-\\alpha^\{\-1\}n^\{\\gamma\-1\}\)^\{k\}\\to 1asn→∞n\\to\\infty\. Also since2​β<κ02\\beta<\\kappa\_\{0\},4​T​exp⁡\(−n−2​β3​\(1−ε′\)​\(1−n−β\)k−1​\(1−α−1​nγ−1\)k​c​nκ0\)4T\\exp\\left\(\-\\frac\{n^\{\-2\\beta\}\}\{3\}\(1\-\\varepsilon^\{\\prime\}\)\(1\-n^\{\-\\beta\}\)^\{k\-1\}\(1\-\\alpha^\{\-1\}n^\{\\gamma\-1\}\)^\{k\}cn^\{\\kappa\_\{0\}\}\\right\)vanishes faster than2​T​n−δ2Tn^\{\-\\delta\}\. Thus,

ℙ​\(ℰn,k′∣∩l=0k−1ℰn,l′,Abm,bM\)≥1−\(2​T\+1\)​n−δ\\displaystyle\\mathbb\{P\}\(\\mathcal\{E\}^\{\\prime\}\_\{n,k\}\\mid\\cap\_\{l=0\}^\{k\-1\}\\mathcal\{E\}^\{\\prime\}\_\{n,l\},A\_\{b^\{m\},b^\{M\}\}\)\\geq 1\-\(2T\+1\)n^\{\-\\delta\}\(5\)for sufficiently largenn\. Then by induction,

ℙ\(\\displaystyle\\mathbb\{P\}\(∩l=0kℰn,l′∣Abm,bM\)\\displaystyle\\cap\_\{l=0\}^\{k\}\\mathcal\{E\}^\{\\prime\}\_\{n,l\}\\mid A\_\{b^\{m\},b^\{M\}\}\)=ℙ​\(ℰn,k′∣∩l=0k−1ℰn,l′,Abm,bM\)​ℙ​\(ℰn,k−1′∣∩l=0k−2ℰn,l′,Abm,bM\)​…​ℙ​\(ℰn,0′∣Abm,bM\)\\displaystyle=\\mathbb\{P\}\(\\mathcal\{E\}^\{\\prime\}\_\{n,k\}\\mid\\cap\_\{l=0\}^\{k\-1\}\\mathcal\{E\}^\{\\prime\}\_\{n,l\},A\_\{b^\{m\},b^\{M\}\}\)\\mathbb\{P\}\(\\mathcal\{E\}^\{\\prime\}\_\{n,k\-1\}\\mid\\cap\_\{l=0\}^\{k\-2\}\\mathcal\{E\}^\{\\prime\}\_\{n,l\},A\_\{b^\{m\},b^\{M\}\}\)\\dots\\mathbb\{P\}\(\\mathcal\{E\}^\{\\prime\}\_\{n,0\}\\mid A\_\{b^\{m\},b^\{M\}\}\)≥\(1−\(2​T\+1\)​n−δ\)⋅\(1−\(2​T\+1\)​n−δ\)​…​\(1−\(2​T\+1\)​n−δ\)​ℙ​\(ℰn,0′∣Abm,bM\)\\displaystyle\\geq\(1\-\(2T\+1\)n^\{\-\\delta\}\)\\cdot\(1\-\(2T\+1\)n^\{\-\\delta\}\)\\dots\(1\-\(2T\+1\)n^\{\-\\delta\}\)\\mathbb\{P\}\(\\mathcal\{E\}^\{\\prime\}\_\{n,0\}\\mid A\_\{b^\{m\},b^\{M\}\}\)=\(1−\(2​T\+1\)​n−δ\)k​ℙ​\(ℰn,0′∣Abm,bM\)\.\\displaystyle=\(1\-\(2T\+1\)n^\{\-\\delta\}\)^\{k\}\\mathbb\{P\}\(\\mathcal\{E\}^\{\\prime\}\_\{n,0\}\\mid A\_\{b^\{m\},b^\{M\}\}\)\.Since\(1−ε′\)​\(1−δn\)k​\(1−α−1​nγ−1\)k≥1−ε\(1\-\\varepsilon^\{\\prime\}\)\(1\-\\delta\_\{n\}\)^\{k\}\(1\-\\alpha^\{\-1\}n^\{\\gamma\-1\}\)^\{k\}\\geq 1\-\\varepsilonand\(1\+ε′\)​\(1\+δn\)k≤1\+ε\(1\+\\varepsilon^\{\\prime\}\)\(1\+\\delta\_\{n\}\)^\{k\}\\leq 1\+\\varepsilonfor sufficiently largenn,b′im​\(1−δn\)k​\(1−α−1​nγ−1\)k≥bim\{b^\{\\prime\}\}\_\{i\}^\{m\}\(1\-\\delta\_\{n\}\)^\{k\}\(1\-\\alpha^\{\-1\}n^\{\\gamma\-1\}\)^\{k\}\\geq b\_\{i\}^\{m\}andb′iM​\(1\+δn\)k≤biM\{b^\{\\prime\}\}\_\{i\}^\{M\}\(1\+\\delta\_\{n\}\)^\{k\}\\leq b\_\{i\}^\{M\}\. Therefore,ℰn,0′⊆Abm,bM\\mathcal\{E\}^\{\\prime\}\_\{n,0\}\\subseteq A\_\{b^\{m\},b^\{M\}\}andℰn,k′⊆ℰn,k\\mathcal\{E\}^\{\\prime\}\_\{n,k\}\\subseteq\\mathcal\{E\}\_\{n,k\}for allk≥0k\\geq 0\. Hence,ℙ​\(ℰn,0′∣Abm,bM\)=1\\mathbb\{P\}\(\\mathcal\{E\}^\{\\prime\}\_\{n,0\}\\mid A\_\{b^\{m\},b^\{M\}\}\)=1and so

ℙ​\(∩l=0kℰn,l∣Abm,bM\)≥ℙ​\(∩l=0kℰn,l′∣Abm,bM\)≥\(1−\(2​T\+1\)​n−δ\)k\.\\mathbb\{P\}\(\\cap\_\{l=0\}^\{k\}\\mathcal\{E\}\_\{n,l\}\\mid A\_\{b^\{m\},b^\{M\}\}\)\\geq\\mathbb\{P\}\(\\cap\_\{l=0\}^\{k\}\\mathcal\{E\}^\{\\prime\}\_\{n,l\}\\mid A\_\{b^\{m\},b^\{M\}\}\)\\geq\(1\-\(2T\+1\)n^\{\-\\delta\}\)^\{k\}\.

### 7\.4Proof of Proposition[4\.3](https://arxiv.org/html/2607.10074#S4.ThmtheoremEnv3)

Recall all the notation from Lemma[4\.8](https://arxiv.org/html/2607.10074#S4.ThmtheoremEnv8)and its proof\. By Lemma[4\.8](https://arxiv.org/html/2607.10074#S4.ThmtheoremEnv8), there existsδ\>0\\delta\>0such thatℙ​\(∩l=0kℰn,l∣An\)≥\(1−\(2​T\+1\)​n−δ\)k→1\\mathbb\{P\}\(\\cap\_\{l=0\}^\{k\}\\mathcal\{E\}\_\{n,l\}\\mid A\_\{n\}\)\\geq\(1\-\(2T\+1\)n^\{\-\\delta\}\)^\{k\}\\to 1for anyk≤κ​logλ1⁡nk\\leq\\kappa\\log\_\{\\lambda\_\{1\}\}nfor all sufficiently largenn\. By Lemma[4\.7](https://arxiv.org/html/2607.10074#S4.ThmtheoremEnv7),ℙ​\(An∖Bn\)→0\\mathbb\{P\}\(A\_\{n\}\\setminus B\_\{n\}\)\\to 0andℙ​\(Bn∖An\)→0\\mathbb\{P\}\(B\_\{n\}\\setminus A\_\{n\}\)\\to 0, so any event that holds w\.h\.p\. underAnA\_\{n\}also holds w\.h\.p\. underBnB\_\{n\}\.

### 7\.5Proof of Proposition[4\.4](https://arxiv.org/html/2607.10074#S4.ThmtheoremEnv4)

Similar to the proof of Lemma[4\.8](https://arxiv.org/html/2607.10074#S4.ThmtheoremEnv8), this proof also consists of two main steps: first, we bound𝔼​\(\|∂Nk1​\(u1\)t∩∂Nk2​\(u2\)t\|\|Nk1​\(u1\),Nk2−1​\(u2\)\)\\mathbb\{E\}\\big\(\|\\partial N\_\{k\_\{1\}\}\(u\_\{1\}\)\_\{t\}\\cap\\partial N\_\{k\_\{2\}\}\(u\_\{2\}\)\_\{t\}\|\\big\|N\_\{k\_\{1\}\}\(u\_\{1\}\),N\_\{k\_\{2\}\-1\}\(u\_\{2\}\)\\big\); then we use the Chernoff–Hoeffding bound \([Dubhashi and Panconesi](https://arxiv.org/html/2607.10074#bib.bib265),[2009](https://arxiv.org/html/2607.10074#bib.bib265), Theorem 1\.1\) to show that the intersection grows w\.h\.p\. While the upper bound on𝔼​\(\|∂Nk1​\(u1\)t∩∂Nk2​\(u2\)t\|\|Nk1​\(u1\),Nk2−1​\(u2\)\)\\mathbb\{E\}\\big\(\|\\partial N\_\{k\_\{1\}\}\(u\_\{1\}\)\_\{t\}\\cap\\partial N\_\{k\_\{2\}\}\(u\_\{2\}\)\_\{t\}\|\\big\|N\_\{k\_\{1\}\}\(u\_\{1\}\),N\_\{k\_\{2\}\-1\}\(u\_\{2\}\)\\big\)is straightforward from the neighborhood growth results in Proposition[4\.3](https://arxiv.org/html/2607.10074#S4.ThmtheoremEnv3), the lower bound concerns distinct radius regimesj≤Lj\\leq Landj\>Lj\>L\. The former, where the branching process approximation is valid but does not yield any contribution to the intersection, utilizes Markov’s inequality to prove an upper bound on\|NL​\(u2\)t\|\|N\_\{L\}\(u\_\{2\}\)\_\{t\}\|, while the latter uses Lemma[4\.8](https://arxiv.org/html/2607.10074#S4.ThmtheoremEnv8)together with Theorem 2\.8 and Corollary 2\.4 fromJansonet al\.\([2000](https://arxiv.org/html/2607.10074#bib.bib12)\)to establish an upper bound on\|∂Nk1​\(u1\)t∩∂Nj​\(u2\)t\|\|\\partial N\_\{k\_\{1\}\}\(u\_\{1\}\)\_\{t\}\\cap\\partial N\_\{j\}\(u\_\{2\}\)\_\{t\}\|\.

Recall all the notation from Lemma[4\.8](https://arxiv.org/html/2607.10074#S4.ThmtheoremEnv8)and its proof\. Then for everyL<k1,j≤k=\(κ0\+κ\)​logλ1⁡nL<k\_\{1\},j\\leq k=\(\\kappa\_\{0\}\+\\kappa\)\\log\_\{\\lambda\_\{1\}\}nandt∈\[T\]t\\in\[T\],

𝔼\\displaystyle\\mathbb\{E\}\(\|∂Nk1​\(u1\)t∩∂Nj​\(u2\)t\|∣Nk1​\(u1\),Nj−1​\(u2\)\)\\displaystyle\(\|\\partial N\_\{k\_\{1\}\}\(u\_\{1\}\)\_\{t\}\\cap\\partial N\_\{j\}\(u\_\{2\}\)\_\{t\}\|\\mid N\_\{k\_\{1\}\}\(u\_\{1\}\),N\_\{j\-1\}\(u\_\{2\}\)\)=𝔼​\(∑x∈∂Nk1​\(u1\)t∖Nj−1​\(u2\)t𝟙\{∃t′∈\[T\]​∃y∈∂Nj−1​\(u2\)t′:Ix​y=1\}∣Nk1​\(u1\),Nj−1​\(u2\)\)\\displaystyle=\\mathbb\{E\}\\left\(\\sum\_\{x\\in\\partial N\_\{k\_\{1\}\}\(u\_\{1\}\)\_\{t\}\\setminus N\_\{j\-1\}\(u\_\{2\}\)\_\{t\}\}\\mathbbm\{1\}\_\{\\\{\\exists t^\{\\prime\}\\in\[T\]\\\>\\exists y\\in\\partial N\_\{j\-1\}\(u\_\{2\}\)\_\{t^\{\\prime\}\}\\\>:\\\>I\_\{xy\}=1\\\}\}\\mid N\_\{k\_\{1\}\}\(u\_\{1\}\),N\_\{j\-1\}\(u\_\{2\}\)\\right\)=\(\|∂Nk1​\(u1\)t\|−∑i≤j−1\|∂Nk1​\(u1\)t∩∂Ni​\(u2\)t\|\)​\(1−∏t′=1T\(1−dt′​tnt\)\|∂Nj−1​\(u2\)t′\|\)\.\\displaystyle=\\left\(\|\\partial N\_\{k\_\{1\}\}\(u\_\{1\}\)\_\{t\}\|\-\\sum\_\{i\\leq j\-1\}\|\\partial N\_\{k\_\{1\}\}\(u\_\{1\}\)\_\{t\}\\cap\\partial N\_\{i\}\(u\_\{2\}\)\_\{t\}\|\\right\)\\left\(1\-\\prod\_\{t^\{\\prime\}=1\}^\{T\}\\left\(1\-\\frac\{d\_\{t^\{\\prime\}t\}\}\{n\_\{t\}\}\\right\)^\{\|\\partial N\_\{j\-1\}\(u\_\{2\}\)\_\{t^\{\\prime\}\}\|\}\\right\)\.Conditionally on∩l=0k−Lℰn,l\\cap\_\{l=0\}^\{k\-L\}\\mathcal\{E\}\_\{n,l\}withλ1\>1\\lambda\_\{1\}\>1,dt′​tnt∈\[0,1\]\\frac\{d\_\{t^\{\\prime\}t\}\}\{n\_\{t\}\}\\in\[0,1\], andDDbeing primitive, Lemma[4\.8](https://arxiv.org/html/2607.10074#S4.ThmtheoremEnv8)implies with probability at least\(1−\(2​T\+1\)​n−δ\)k≥1−k​\(2​T\+1\)​n−δ\>1−logλ1⁡n​\(2​T\+1\)​n−δ\\left\(1\-\(2T\+1\)n^\{\-\\delta\}\\right\)^\{k\}\\geq 1\-k\(2T\+1\)n^\{\-\\delta\}\>1\-\\log\_\{\\lambda\_\{1\}\}n\(2T\+1\)n^\{\-\\delta\}that

\|∂Nj−1​\(u2\)t′\|​dt′​tnt\\displaystyle\|\\partial N\_\{j\-1\}\(u\_\{2\}\)\_\{t^\{\\prime\}\}\|\\frac\{d\_\{t^\{\\prime\}t\}\}\{n\_\{t\}\}≤\(1\+ε\)​\[Dj−1\]t​\(u2\)​t′​dt′​tα​n≤\(1\+ε\)​C​λ1j−1​dt′​tα​n≤\(1\+ε\)​C​dt′​t​nκ0\+κα​n\\displaystyle\\leq\(1\+\\varepsilon\)\[D^\{j\-1\}\]\_\{t\(u\_\{2\}\)t^\{\\prime\}\}\\frac\{d\_\{t^\{\\prime\}t\}\}\{\\alpha n\}\\leq\(1\+\\varepsilon\)C\\lambda\_\{1\}^\{j\-1\}\\frac\{d\_\{t^\{\\prime\}t\}\}\{\\alpha n\}\\leq\(1\+\\varepsilon\)Cd\_\{t^\{\\prime\}t\}\\frac\{n^\{\\kappa\_\{0\}\+\\kappa\}\}\{\\alpha n\}for allt′∈\[T\]t^\{\\prime\}\\in\[T\]and some constantC\>0C\>0\. Sinceκ0\+κ<1\\kappa\_\{0\}\+\\kappa<1,\|∂Nj−1​\(u2\)t′\|​dt′​tnt≤\(1\+ε\)​C​dt′​t​nκ0\+κα​n\|\\partial N\_\{j\-1\}\(u\_\{2\}\)\_\{t^\{\\prime\}\}\|\\frac\{d\_\{t^\{\\prime\}t\}\}\{n\_\{t\}\}\\leq\(1\+\\varepsilon\)Cd\_\{t^\{\\prime\}t\}\\frac\{n^\{\\kappa\_\{0\}\+\\kappa\}\}\{\\alpha n\}vanishes, and so

\(1−dt′​tnt\)\|∂Nj−1​\(u2\)t′\|=1−\|∂Nj−1​\(u2\)t′\|​dt′​tnt​\(1\+o​\(1\)\),\\displaystyle\\left\(1\-\\frac\{d\_\{t^\{\\prime\}t\}\}\{n\_\{t\}\}\\right\)^\{\|\\partial N\_\{j\-1\}\(u\_\{2\}\)\_\{t^\{\\prime\}\}\|\}=1\-\|\\partial N\_\{j\-1\}\(u\_\{2\}\)\_\{t^\{\\prime\}\}\|\\frac\{d\_\{t^\{\\prime\}t\}\}\{n\_\{t\}\}\(1\+o\(1\)\),the identity \([3](https://arxiv.org/html/2607.10074#S7.E3)\) implies

𝔼​\(\|∂Nk1​\(u1\)t∩∂Nj​\(u2\)t\|∣Nk1​\(u1\),Nj−1​\(u2\)\)\\displaystyle\\mathbb\{E\}\(\|\\partial N\_\{k\_\{1\}\}\(u\_\{1\}\)\_\{t\}\\cap\\partial N\_\{j\}\(u\_\{2\}\)\_\{t\}\|\\mid N\_\{k\_\{1\}\}\(u\_\{1\}\),N\_\{j\-1\}\(u\_\{2\}\)\)=\(\|∂Nk1​\(u1\)t\|−∑i≤j−1\|∂Nk1​\(u1\)t∩∂Ni​\(u2\)t\|\)​\(1−∏t′=1T\(1−\|∂Nj−1​\(u2\)t′\|​dt′​tnt​\(1\+o​\(1\)\)\)\)\\displaystyle=\\left\(\|\\partial N\_\{k\_\{1\}\}\(u\_\{1\}\)\_\{t\}\|\-\\sum\_\{i\\leq j\-1\}\|\\partial N\_\{k\_\{1\}\}\(u\_\{1\}\)\_\{t\}\\cap\\partial N\_\{i\}\(u\_\{2\}\)\_\{t\}\|\\right\)\\left\(1\-\\prod\_\{t^\{\\prime\}=1\}^\{T\}\\left\(1\-\|\\partial N\_\{j\-1\}\(u\_\{2\}\)\_\{t^\{\\prime\}\}\|\\frac\{d\_\{t^\{\\prime\}t\}\}\{n\_\{t\}\}\(1\+o\(1\)\)\\right\)\\right\)=\(\|∂Nk1​\(u1\)t\|−∑i≤j−1\|∂Nk1​\(u1\)t∩∂Ni​\(u2\)t\|\)​∑t′=1T\(\|∂Nj−1​\(u2\)t′\|​dt′​tnt\)​\(1\+o​\(1\)\)\\displaystyle=\\left\(\|\\partial N\_\{k\_\{1\}\}\(u\_\{1\}\)\_\{t\}\|\-\\sum\_\{i\\leq j\-1\}\|\\partial N\_\{k\_\{1\}\}\(u\_\{1\}\)\_\{t\}\\cap\\partial N\_\{i\}\(u\_\{2\}\)\_\{t\}\|\\right\)\\sum\_\{t^\{\\prime\}=1\}^\{T\}\\left\(\|\\partial N\_\{j\-1\}\(u\_\{2\}\)\_\{t^\{\\prime\}\}\|\\frac\{d\_\{t^\{\\prime\}t\}\}\{n\_\{t\}\}\\right\)\(1\+o\(1\)\)=\(\|∂Nk1​\(u1\)t\|−∑i≤j−1\|∂Nk1​\(u1\)t∩∂Ni​\(u2\)t\|\)​\(1\+o​\(1\)\)​⟨\|∂Nj−1​\(u2\)\|→,D​et⟩nt\.\\displaystyle=\\left\(\|\\partial N\_\{k\_\{1\}\}\(u\_\{1\}\)\_\{t\}\|\-\\sum\_\{i\\leq j\-1\}\|\\partial N\_\{k\_\{1\}\}\(u\_\{1\}\)\_\{t\}\\cap\\partial N\_\{i\}\(u\_\{2\}\)\_\{t\}\|\\right\)\(1\+o\(1\)\)\\frac\{\\langle\\overrightarrow\{\|\\partial N\_\{j\-1\}\(u\_\{2\}\)\|\},De\_\{t\}\\rangle\}\{n\_\{t\}\}\.\(6\)Again by Lemma[4\.8](https://arxiv.org/html/2607.10074#S4.ThmtheoremEnv8)andDDbeing primitive, we obtain

𝔼\(\|∂Nk1\\displaystyle\\mathbb\{E\}\\big\(\|\\partial N\_\{k\_\{1\}\}\(u1\)t∩∂Nj\(u2\)t\|\|Nk1\(u1\),Nj−1\(u2\)\)≤\(1\+ε\)\[Dk1\]t​\(u1\)​tT\(1\+ε\)\[Dk\]t​\(u2\)​tα​n\\displaystyle\(u\_\{1\}\)\_\{t\}\\cap\\partial N\_\{j\}\(u\_\{2\}\)\_\{t\}\|\\big\|N\_\{k\_\{1\}\}\(u\_\{1\}\),N\_\{j\-1\}\(u\_\{2\}\)\\big\)\\leq\(1\+\\varepsilon\)\[D^\{k\_\{1\}\}\]\_\{t\(u\_\{1\}\)t\}T\(1\+\\varepsilon\)\\frac\{\[D^\{k\}\]\_\{t\(u\_\{2\}\)t\}\}\{\\alpha n\}≤T​\(1\+ε\)2​C2​λ1k1​nκ0\+κα​n≤λ1k1​n−γ7​\(⌊k⌋−⌊L⌋\)\\displaystyle\\leq T\(1\+\\varepsilon\)^\{2\}C^\{2\}\\lambda\_\{1\}^\{k\_\{1\}\}\\frac\{n^\{\\kappa\_\{0\}\+\\kappa\}\}\{\\alpha n\}\\leq\\lambda\_\{1\}^\{k\_\{1\}\}\\frac\{n^\{\-\\gamma\}\}\{7\(\\lfloor k\\rfloor\-\\lfloor L\\rfloor\)\}and so

𝔼​\(\|∂Nk1​\(u1\)t∩∂Nj​\(u2\)t\|\)\\displaystyle\\mathbb\{E\}\\big\(\|\\partial N\_\{k\_\{1\}\}\(u\_\{1\}\)\_\{t\}\\cap\\partial N\_\{j\}\(u\_\{2\}\)\_\{t\}\|\\big\)=𝔼​\(𝔼​\(\|∂Nk1​\(u1\)t∩∂Nj​\(u2\)t\|\|Nk1​\(u1\),Nj−1​\(u2\)\)\)\\displaystyle=\\mathbb\{E\}\\big\(\\mathbb\{E\}\\big\(\|\\partial N\_\{k\_\{1\}\}\(u\_\{1\}\)\_\{t\}\\cap\\partial N\_\{j\}\(u\_\{2\}\)\_\{t\}\|\\big\|N\_\{k\_\{1\}\}\(u\_\{1\}\),N\_\{j\-1\}\(u\_\{2\}\)\\big\)\\big\)≤λ1k1​n−γ7​\(⌊k⌋−⌊L⌋\)\\displaystyle\\leq\\lambda\_\{1\}^\{k\_\{1\}\}\\frac\{n^\{\-\\gamma\}\}\{7\(\\lfloor k\\rfloor\-\\lfloor L\\rfloor\)\}for allt∈\[T\]t\\in\[T\]and allL<j≤kL<j\\leq kwithC\>0C\>0and0<γ<min⁡\{1−κ0−κ,κ0\}0<\\gamma<\\min\\\{1\-\\kappa\_\{0\}\-\\kappa,\\kappa\_\{0\}\\\}for sufficiently largenn\. The factor77is chosen so thatx≥7​λx\\geq 7\\lambdawherex=λ1k1​n−γ⌊k⌋−⌊L⌋x=\\lambda\_\{1\}^\{k\_\{1\}\}\\frac\{n^\{\-\\gamma\}\}\{\\lfloor k\\rfloor\-\\lfloor L\\rfloor\}is the threshold andλ=𝔼​\(\|∂Nk1​\(u1\)t∩∂Nj​\(u2\)t\|\)\\lambda=\\mathbb\{E\}\(\|\\partial N\_\{k\_\{1\}\}\(u\_\{1\}\)\_\{t\}\\cap\\partial N\_\{j\}\(u\_\{2\}\)\_\{t\}\|\)is the mean, satisfying the condition required by Corollary 2\.4 ofJansonet al\.\([2000](https://arxiv.org/html/2607.10074#bib.bib12)\)that we use in later steps\.

Now letAAbe the event that there existt∈\[T\]t\\in\[T\]andL<j≤kL<j\\leq ksuch that

\|∂Nk1​\(u1\)t∩∂Nj​\(u2\)t\|≥λ1k1​n−γ⌊k⌋−⌊L⌋\.\\left\|\\partial N\_\{k\_\{1\}\}\(u\_\{1\}\)\_\{t\}\\cap\\partial N\_\{j\}\(u\_\{2\}\)\_\{t\}\\right\|\\geq\\lambda\_\{1\}^\{k\_\{1\}\}\\frac\{n^\{\-\\gamma\}\}\{\\lfloor k\\rfloor\-\\lfloor L\\rfloor\}\.LetBBbe the event that

𝔼​\(\|∂Nk1​\(u1\)t∩∂Nj​\(u2\)t\|\)≤λ1k1​n−γ7​\(⌊k⌋−⌊L⌋\)\\mathbb\{E\}\(\|\\partial N\_\{k\_\{1\}\}\(u\_\{1\}\)\_\{t\}\\cap\\partial N\_\{j\}\(u\_\{2\}\)\_\{t\}\|\)\\leq\\lambda\_\{1\}^\{k\_\{1\}\}\\frac\{n^\{\-\\gamma\}\}\{7\(\\lfloor k\\rfloor\-\\lfloor L\\rfloor\)\}for allt∈\[T\]t\\in\[T\]and allL<j≤kL<j\\leq k\. Hence,

ℙ​\(A\)≤ℙ​\(A∣B\)\+ℙ​\(Bc\)<ℙ​\(A∣B\)\+logλ1⁡n​\(2​T\+1\)​n−δ\.\\mathbb\{P\}\(A\)\\leq\\mathbb\{P\}\(A\\mid B\)\+\\mathbb\{P\}\(B^\{c\}\)<\\mathbb\{P\}\(A\\mid B\)\+\\log\_\{\\lambda\_\{1\}\}n\(2T\+1\)n^\{\-\\delta\}\.By Theorem 2\.8 and Corollary 2\.4 fromJansonet al\.\([2000](https://arxiv.org/html/2607.10074#bib.bib12)\)with union bound,

ℙ​\(A∣B\)\\displaystyle\\mathbb\{P\}\(A\\mid B\)≤T​\(⌊k⌋−⌊L⌋\)​exp⁡\(−λ1k1​n−γ⌊k⌋−⌊L⌋\)\\displaystyle\\leq T\(\\lfloor k\\rfloor\-\\lfloor L\\rfloor\)\\exp\\left\(\-\\lambda\_\{1\}^\{k\_\{1\}\}\\frac\{n^\{\-\\gamma\}\}\{\\lfloor k\\rfloor\-\\lfloor L\\rfloor\}\\right\)<T​\(\(κ0\+κ\)​logλ1⁡n−κ0​logλ1⁡n\+1\)​exp⁡\(−nκ0−γ⌊k⌋−⌊L⌋\)\\displaystyle<T\(\(\\kappa\_\{0\}\+\\kappa\)\\log\_\{\\lambda\_\{1\}\}n\-\\kappa\_\{0\}\\log\_\{\\lambda\_\{1\}\}n\+1\)\\exp\\left\(\-\\frac\{n^\{\\kappa\_\{0\}\-\\gamma\}\}\{\\lfloor k\\rfloor\-\\lfloor L\\rfloor\}\\right\)=T​\(κ​logλ1⁡n\+1\)​exp⁡\(−nγ′\)\\displaystyle=T\(\\kappa\\log\_\{\\lambda\_\{1\}\}n\+1\)\\exp\\left\(\-n^\{\\gamma^\{\\prime\}\}\\right\)forγ′=κ0−γ\>0\\gamma^\{\\prime\}=\\kappa\_\{0\}\-\\gamma\>0\. It follows that

ℙ​\(Ac∣Nk1​\(u1\),Nk2−1​\(u2\)\)\\displaystyle\\mathbb\{P\}\\Bigg\(A^\{c\}\\mid N\_\{k\_\{1\}\}\(u\_\{1\}\),N\_\{k\_\{2\}\-1\}\(u\_\{2\}\)\\Bigg\)\>1−T​\(κ​logλ1⁡n\+1\)​exp⁡\(−nγ′\)−logλ1⁡n​\(2​T\+1\)​n−δ\\displaystyle\>1\-T\(\\kappa\\log\_\{\\lambda\_\{1\}\}n\+1\)\\exp\\left\(\-n^\{\\gamma^\{\\prime\}\}\\right\)\-\\log\_\{\\lambda\_\{1\}\}n\(2T\+1\)n^\{\-\\delta\}≥1−logλ1⁡n​\(2​T\+2\)​n−δ\\displaystyle\\geq 1\-\\log\_\{\\lambda\_\{1\}\}n\(2T\+2\)n^\{\-\\delta\}\(7\)for sufficiently largennasT​\(κ​logλ1⁡n\+1\)​exp⁡\(−nγ′\)T\(\\kappa\\log\_\{\\lambda\_\{1\}\}n\+1\)\\exp\\left\(\-n^\{\\gamma^\{\\prime\}\}\\right\)vanishes faster thanlogλ1⁡n​\(2​T\+1\)​n−δ\\log\_\{\\lambda\_\{1\}\}n\(2T\+1\)n^\{\-\\delta\}\.

Sincek1\>κ0​logλ1⁡nk\_\{1\}\>\\kappa\_\{0\}\\log\_\{\\lambda\_\{1\}\}n, Markov’s inequality and Lemma[4\.6](https://arxiv.org/html/2607.10074#S4.ThmtheoremEnv6)imply that there existγ′′∈\(κ0,k1logλ1⁡n\)\\gamma^\{\\prime\\prime\}\\in\\left\(\\kappa\_\{0\},\\frac\{k\_\{1\}\}\{\\log\_\{\\lambda\_\{1\}\}n\}\\right\)andδ′\>0\\delta^\{\\prime\}\>0such that

ℙ​\(\|NL​\(u2\)t\|≥nγ′′\)≤O​\(λ1L\)nγ′′=O​\(nκ0\)nγ′′≤n−δ′\\mathbb\{P\}\(\|N\_\{L\}\(u\_\{2\}\)\_\{t\}\|\\geq n^\{\\gamma^\{\\prime\\prime\}\}\)\\leq\\frac\{O\(\\lambda\_\{1\}^\{L\}\)\}\{n^\{\\gamma^\{\\prime\\prime\}\}\}=\\frac\{O\\left\(n^\{\\kappa\_\{0\}\}\\right\)\}\{n^\{\\gamma^\{\\prime\\prime\}\}\}\\leq n^\{\-\\delta^\{\\prime\}\}fort∈\[T\]t\\in\[T\]with sufficiently largenn\. Therefore,

ℙ\(\|NL\\displaystyle\\mathbb\{P\}\(\|N\_\{L\}\(u2\)t\|≤nγ′′fort∈\[T\]\)=1−ℙ\(∃t∈\[T\]s\.t\.\|NL\(u2\)t\|≥nγ′′\)\\displaystyle\(u\_\{2\}\)\_\{t\}\|\\leq n^\{\\gamma^\{\\prime\\prime\}\}\\text\{ for \}t\\in\[T\]\)=1\-\\mathbb\{P\}\(\\exists t\\in\[T\]s\.t\.\\left\|N\_\{L\}\(u\_\{2\}\)\_\{t\}\\right\|\\geq n^\{\\gamma^\{\\prime\\prime\}\}\)≥1−∑t=1Tℙ​\(\|NL​\(u2\)t\|≥nγ′′\)≥1−T​n−δ′\.\\displaystyle\\geq 1\-\\sum\_\{t=1\}^\{T\}\\mathbb\{P\}\(\|N\_\{L\}\(u\_\{2\}\)\_\{t\}\|\\geq n^\{\\gamma^\{\\prime\\prime\}\}\)\\geq 1\-Tn^\{\-\\delta^\{\\prime\}\}\.\(8\)Combining \([6](https://arxiv.org/html/2607.10074#S7.E6)\), \([7](https://arxiv.org/html/2607.10074#S7.E7)\), \([8](https://arxiv.org/html/2607.10074#S7.E8)\) with Proposition[4\.3](https://arxiv.org/html/2607.10074#S4.ThmtheoremEnv3)andDDbeing primitive, we have w\.h\.p\. that

𝔼​\(\|∂Nk1​\(u1\)t∩∂Nk2​\(u2\)t\|\|Nk1​\(u1\),Nk2−1​\(u2\)\)\\displaystyle\\mathbb\{E\}\\big\(\|\\partial N\_\{k\_\{1\}\}\(u\_\{1\}\)\_\{t\}\\cap\\partial N\_\{k\_\{2\}\}\(u\_\{2\}\)\_\{t\}\|\\big\|N\_\{k\_\{1\}\}\(u\_\{1\}\),N\_\{k\_\{2\}\-1\}\(u\_\{2\}\)\\big\)≥\(\|∂Nk1​\(u1\)t\|−∑L<j≤k2−1\|∂Nk1​\(u1\)t∩∂Nj​\(u2\)t\|−\|NL​\(u2\)t\|\)​\(1\+o​\(1\)\)​⟨\|∂Nk2−1​\(u2\)\|→,D​et⟩nt\\displaystyle\{\\geq\\left\(\|\\partial N\_\{k\_\{1\}\}\(u\_\{1\}\)\_\{t\}\|\-\\sum\_\{L<j\\leq k\_\{2\}\-1\}\|\\partial N\_\{k\_\{1\}\}\(u\_\{1\}\)\_\{t\}\\cap\\partial N\_\{j\}\(u\_\{2\}\)\_\{t\}\|\-\|N\_\{L\}\(u\_\{2\}\)\_\{t\}\|\\right\)\(1\+o\(1\)\)\\frac\{\\langle\\overrightarrow\{\|\\partial N\_\{k\_\{2\}\-1\}\(u\_\{2\}\)\|\},De\_\{t\}\\rangle\}\{n\_\{t\}\}\}≥\(\(1−ε\)​\[Dk1\]t​\(u1\)​t−\(⌊k2⌋−1−⌊L⌋\)​λ1k1​n−γ⌊k⌋−⌊L⌋−nγ′′\)​⟨\(1−ε\)​et​\(u2\)⊤​Dk2−1,D​et⟩nt\\displaystyle\\geq\\left\(\(1\-\\varepsilon\)\[D^\{k\_\{1\}\}\]\_\{t\(u\_\{1\}\)t\}\-\(\\lfloor k\_\{2\}\\rfloor\-1\-\\lfloor L\\rfloor\)\\lambda\_\{1\}^\{k\_\{1\}\}\\frac\{n^\{\-\\gamma\}\}\{\\lfloor k\\rfloor\-\\lfloor L\\rfloor\}\-n^\{\\gamma^\{\\prime\\prime\}\}\\right\)\\frac\{\\langle\(1\-\\varepsilon\)e\_\{t\(u\_\{2\}\)\}^\{\\top\}D^\{k\_\{2\}\-1\},De\_\{t\}\\rangle\}\{n\_\{t\}\}≥\(\(1−ε\)​\[Dk1\]t​\(u1\)​t−λ1k1​n−γ−nγ′′\)​\(1−ε\)​\[Dk2\]t​\(u2\)​tnt\\displaystyle\\geq\\left\(\(1\-\\varepsilon\)\[D^\{k\_\{1\}\}\]\_\{t\(u\_\{1\}\)t\}\-\\lambda\_\{1\}^\{k\_\{1\}\}n^\{\-\\gamma\}\-n^\{\\gamma^\{\\prime\\prime\}\}\\right\)\\frac\{\(1\-\\varepsilon\)\[D^\{k\_\{2\}\}\]\_\{t\(u\_\{2\}\)t\}\}\{n\_\{t\}\}≥\(\(1−ε\)​c​λ1k1−λ1k1​n−γ−nγ′′\)​\(1−ε\)​c​λ1k2nt≥\(1−ε\)2​c2​λ1k1\+k22​nt\\displaystyle\\geq\\left\(\(1\-\\varepsilon\)c\\lambda\_\{1\}^\{k\_\{1\}\}\-\\lambda\_\{1\}^\{k\_\{1\}\}n^\{\-\\gamma\}\-n^\{\\gamma^\{\\prime\\prime\}\}\\right\)\\frac\{\(1\-\\varepsilon\)c\\lambda\_\{1\}^\{k\_\{2\}\}\}\{n\_\{t\}\}\\geq\\frac\{\(1\-\\varepsilon\)^\{2\}c^\{2\}\\lambda\_\{1\}^\{k\_\{1\}\+k\_\{2\}\}\}\{2n\_\{t\}\}\(9\)and

𝔼​\(\|∂Nk1​\(u1\)t∩∂Nk2​\(u2\)t\|\|Nk1​\(u1\),Nk2−1​\(u2\)\)\\displaystyle\\mathbb\{E\}\\big\(\|\\partial N\_\{k\_\{1\}\}\(u\_\{1\}\)\_\{t\}\\cap\\partial N\_\{k\_\{2\}\}\(u\_\{2\}\)\_\{t\}\|\\big\|N\_\{k\_\{1\}\}\(u\_\{1\}\),N\_\{k\_\{2\}\-1\}\(u\_\{2\}\)\\big\)≤\|∂Nk1​\(u1\)t\|​\(1\+o​\(1\)\)​⟨\|∂Nk2−1​\(u2\)\|→,D​et⟩nt≤\(1\+ε\)​\[Dk1\]t​\(u1\)​t​⟨\(1\+ε\)​et​\(u2\)⊤​Dk2−1,D​et⟩nt\\displaystyle\\leq\|\\partial N\_\{k\_\{1\}\}\(u\_\{1\}\)\_\{t\}\|\(1\+o\(1\)\)\\frac\{\\langle\\overrightarrow\{\|\\partial N\_\{k\_\{2\}\-1\}\(u\_\{2\}\)\|\},De\_\{t\}\\rangle\}\{n\_\{t\}\}\\leq\(1\+\\varepsilon\)\[D^\{k\_\{1\}\}\]\_\{t\(u\_\{1\}\)t\}\\frac\{\\langle\(1\+\\varepsilon\)e\_\{t\(u\_\{2\}\)\}^\{\\top\}D^\{k\_\{2\}\-1\},De\_\{t\}\\rangle\}\{n\_\{t\}\}=\(1\+ε\)2​\[Dk1\]t​\(u1\)​t​\[Dk2\]t​\(u2\)​tnt≤\(1\+ε\)2​C2​λ1k1\+k2nt\\displaystyle=\(1\+\\varepsilon\)^\{2\}\[D^\{k\_\{1\}\}\]\_\{t\(u\_\{1\}\)t\}\\frac\{\[D^\{k\_\{2\}\}\]\_\{t\(u\_\{2\}\)t\}\}\{n\_\{t\}\}\\leq\\frac\{\(1\+\\varepsilon\)^\{2\}C^\{2\}\\lambda\_\{1\}^\{k\_\{1\}\+k\_\{2\}\}\}\{n\_\{t\}\}\(10\)for somec,C\>0c,C\>0, where the last ”≥\\geq” holds sinceλ1k1​n−γ\\lambda\_\{1\}^\{k\_\{1\}\}n^\{\-\\gamma\}andnγ′′n^\{\\gamma^\{\\prime\\prime\}\}grow strictly slower than\(1−ε\)​c​λ1k1\(1\-\\varepsilon\)c\\lambda\_\{1\}^\{k\_\{1\}\}asγ\>0\\gamma\>0andγ′′<k1logλ1⁡n\\gamma^\{\\prime\\prime\}<\\frac\{k\_\{1\}\}\{\\log\_\{\\lambda\_\{1\}\}n\}\. Here, the two regimesj≤Lj\\leq Landj\>Lj\>Lare controlled by different tools: forj\>Lj\>L, the eventℰn,l−L\\mathcal\{E\}\_\{n,l\-L\}from Lemma[4\.8](https://arxiv.org/html/2607.10074#S4.ThmtheoremEnv8)is in force and the intersection layers are bounded using the eventAcA^\{c\}; forj≤Lj\\leq L, the branching process approximation is not yet running, so the cruder Markov bound \([8](https://arxiv.org/html/2607.10074#S7.E8)\) is used instead\. The bounds in \([9](https://arxiv.org/html/2607.10074#S7.E9)\) and \([10](https://arxiv.org/html/2607.10074#S7.E10)\) hold for allt∈\[T\]t\\in\[T\]w\.h\.p\., and we denote this eventSS\.

Letρ\>0\\rho\>0andRRdenote the event that

\|∂Nk1​\(u1\)t∩∂Nk2​\(u2\)t\|∉\(\(1−ε\)3​c2​λ1k1\+k22​nt,\(1\+ε\)3​C2​λ1k1\+k2nt\)\|\\partial N\_\{k\_\{1\}\}\(u\_\{1\}\)\_\{t\}\\cap\\partial N\_\{k\_\{2\}\}\(u\_\{2\}\)\_\{t\}\|\\notin\\left\(\\frac\{\(1\-\\varepsilon\)^\{3\}c^\{2\}\\lambda\_\{1\}^\{k\_\{1\}\+k\_\{2\}\}\}\{2n\_\{t\}\},\\frac\{\(1\+\\varepsilon\)^\{3\}C^\{2\}\\lambda\_\{1\}^\{k\_\{1\}\+k\_\{2\}\}\}\{n\_\{t\}\}\\right\)for allk1,k2,tk\_\{1\},k\_\{2\},tsuch thatk1\+k2≥\(1\+ρ\)​logλ1⁡ntk\_\{1\}\+k\_\{2\}\\geq\(1\+\\rho\)\\log\_\{\\lambda\_\{1\}\}n\_\{t\}\. It follows that

ℙ​\(R∣Nk1​\(u1\),Nk2−1​\(u2\)\)≤ℙ​\(R∣S,Nk1​\(u1\),Nk2−1​\(u2\)\)\+ℙ​\(Sc∣Nk1​\(u1\),Nk2−1​\(u2\)\)\\displaystyle\\mathbb\{P\}\(R\\mid N\_\{k\_\{1\}\}\(u\_\{1\}\),N\_\{k\_\{2\}\-1\}\(u\_\{2\}\)\)\\leq\\mathbb\{P\}\(R\\mid S,N\_\{k\_\{1\}\}\(u\_\{1\}\),N\_\{k\_\{2\}\-1\}\(u\_\{2\}\)\)\+\\mathbb\{P\}\(S^\{c\}\\mid N\_\{k\_\{1\}\}\(u\_\{1\}\),N\_\{k\_\{2\}\-1\}\(u\_\{2\}\)\)Using Chernoff\-Hoeffding bound \([Dubhashi and Panconesi](https://arxiv.org/html/2607.10074#bib.bib265),[2009](https://arxiv.org/html/2607.10074#bib.bib265), Theorem 1\.1\) and union bound,

ℙ\\displaystyle\\mathbb\{P\}\(R∣S,Nk1​\(u1\),Nk2−1​\(u2\)\)≤T​\(⌊k⌋−⌈L⌉\+1\)2​2​exp⁡\(−ε23​\(1−ε\)2​c2​λ1k1\+k22​nt\)\\displaystyle\(R\\mid S,N\_\{k\_\{1\}\}\(u\_\{1\}\),N\_\{k\_\{2\}\-1\}\(u\_\{2\}\)\)\\leq T\(\\lfloor k\\rfloor\-\\left\\lceil L\\right\\rceil\+1\)^\{2\}2\\exp\\left\(\-\\frac\{\\varepsilon^\{2\}\}\{3\}\\frac\{\(1\-\\varepsilon\)^\{2\}c^\{2\}\\lambda\_\{1\}^\{k\_\{1\}\+k\_\{2\}\}\}\{2n\_\{t\}\}\\right\)≤T​\(\(κ0\+κ\)​logλ1⁡n−κ0​logλ1⁡n\+1\)2​2​exp⁡\(−ε23​\(1−ε\)2​c2​n1\+ρ2​n\)\\displaystyle\\leq T\(\(\\kappa\_\{0\}\+\\kappa\)\\log\_\{\\lambda\_\{1\}\}n\-\\kappa\_\{0\}\\log\_\{\\lambda\_\{1\}\}n\+1\)^\{2\}2\\exp\\left\(\-\\frac\{\\varepsilon^\{2\}\}\{3\}\\frac\{\(1\-\\varepsilon\)^\{2\}c^\{2\}n^\{1\+\\rho\}\}\{2n\}\\right\)=T​\(κ​logλ1⁡n\+1\)2​2​exp⁡\(−ε23​\(1−ε\)2​c2​nρ2\)\\displaystyle=T\(\\kappa\\log\_\{\\lambda\_\{1\}\}n\+1\)^\{2\}2\\exp\\left\(\-\\frac\{\\varepsilon^\{2\}\}\{3\}\\frac\{\(1\-\\varepsilon\)^\{2\}c^\{2\}n^\{\\rho\}\}\{2\}\\right\)Therefore,

𝔼​\(𝟏\{Rc\}\|Nk1​\(u1\),Nk2−1​\(u2\)\)\\displaystyle\\mathbb\{E\}\(\\mathbf\{1\}\_\{\\\{R^\{c\}\\\}\}\|N\_\{k\_\{1\}\}\(u\_\{1\}\),N\_\{k\_\{2\}\-1\}\(u\_\{2\}\)\)=ℙ​\(Rc∣Nk1​\(u1\),Nk2−1​\(u2\)\)\\displaystyle=\\mathbb\{P\}\\left\(R^\{c\}\\mid N\_\{k\_\{1\}\}\(u\_\{1\}\),N\_\{k\_\{2\}\-1\}\(u\_\{2\}\)\\right\)≥ℙ​\(S∣Nk1​\(u1\),Nk2−1​\(u2\)\)−ℙ​\(R∣S,Nk1​\(u1\),Nk2−1​\(u2\)\)\\displaystyle\\geq\\mathbb\{P\}\(S\\mid N\_\{k\_\{1\}\}\(u\_\{1\}\),N\_\{k\_\{2\}\-1\}\(u\_\{2\}\)\)\-\\mathbb\{P\}\(R\\mid S,N\_\{k\_\{1\}\}\(u\_\{1\}\),N\_\{k\_\{2\}\-1\}\(u\_\{2\}\)\)where the right\-hand side converges to 1\. Taking expectation on both sides, we get

\|∂Nk1​\(u1\)t∩∂Nk2​\(u2\)t\|∈\(\(1−ε\)3​c2​λ1k1\+k22​nt,\(1\+ε\)3​C2​λ1k1\+k2nt\)\|\\partial N\_\{k\_\{1\}\}\(u\_\{1\}\)\_\{t\}\\cap\\partial N\_\{k\_\{2\}\}\(u\_\{2\}\)\_\{t\}\|\\in\\left\(\\frac\{\(1\-\\varepsilon\)^\{3\}c^\{2\}\\lambda\_\{1\}^\{k\_\{1\}\+k\_\{2\}\}\}\{2n\_\{t\}\},\\frac\{\(1\+\\varepsilon\)^\{3\}C^\{2\}\\lambda\_\{1\}^\{k\_\{1\}\+k\_\{2\}\}\}\{n\_\{t\}\}\\right\)for allt∈\[T\]t\\in\[T\]andL≤k1,k2≤kL\\leq k\_\{1\},k\_\{2\}\\leq ksatisfyingk1\+k2≥\(1\+ρ\)​logλ1⁡ntk\_\{1\}\+k\_\{2\}\\geq\(1\+\\rho\)\\log\_\{\\lambda\_\{1\}\}n\_\{t\}w\.h\.p\.

### 7\.6Proof of Theorem[4\.1](https://arxiv.org/html/2607.10074#S4.ThmtheoremEnv1)

Ifu1u\_\{1\}andu2u\_\{2\}are not in the same connected component,d​\(u1,u2\)=∞d\(u\_\{1\},u\_\{2\}\)=\\inftyand eitherd​\(u1,s\)=∞d\(u\_\{1\},s\)=\\inftyord​\(u2,s\)=∞d\(u\_\{2\},s\)=\\inftyfor any landmarksssampled in the local step \(see Section[2\.2](https://arxiv.org/html/2607.10074#S2.SS2)\)\. Hence,d¯​\(u1,u2\)=∞\\underline\{d\}\(u\_\{1\},u\_\{2\}\)=\\infty\. Now we focus on the case thatu1u\_\{1\}andu2u\_\{2\}are in the same connected component \(i\.e\.,u1↔u2u\_\{1\}\\leftrightarrow u\_\{2\}\)\.

Letk1=ε′​d​\(u1,u2\)k\_\{1\}=\\varepsilon^\{\\prime\}d\(u\_\{1\},u\_\{2\}\)andk2=\(1−ε\+ε′\)​d​\(u1,u2\)k\_\{2\}=\(1\-\\varepsilon\+\\varepsilon^\{\\prime\}\)d\(u\_\{1\},u\_\{2\}\)with0<ε′=min⁡\{ε2,ε−θ\}−ε′′<min⁡\{ε2,ε−θ\}0<\\varepsilon^\{\\prime\}=\\min\\left\\\{\\frac\{\\varepsilon\}\{2\},\\varepsilon\-\\theta\\right\\\}\-\\varepsilon^\{\\prime\\prime\}<\\min\\left\\\{\\frac\{\\varepsilon\}\{2\},\\varepsilon\-\\theta\\right\\\}for someε′′∈\(0,min⁡\{ε2,ε−θ\}\)\\varepsilon^\{\\prime\\prime\}\\in\\left\(0,\\min\\left\\\{\\frac\{\\varepsilon\}\{2\},\\varepsilon\-\\theta\\right\\\}\\right\)\(to be chosen later\)\. Sinceε′<ε2\\varepsilon^\{\\prime\}<\\frac\{\\varepsilon\}\{2\},k1\+k2<d​\(u1,u2\)k\_\{1\}\+k\_\{2\}<d\(u\_\{1\},u\_\{2\}\), and soNk1​\(u1\)∩Nk2​\(u2\)=∅N\_\{k\_\{1\}\}\(u\_\{1\}\)\\cap N\_\{k\_\{2\}\}\(u\_\{2\}\)=\\varnothing\.

LetSi​jS\_\{ij\}be the landmark set of sizeMiM^\{i\}sampled in thejj\-th round andZi​jZ\_\{ij\}denote the event thatSi​j∩Nk1​\(u1\)≠∅S\_\{ij\}\\cap N\_\{k\_\{1\}\}\(u\_\{1\}\)\\neq\\varnothingbutSi​j∩Nk2​\(u2\)=∅S\_\{ij\}\\cap N\_\{k\_\{2\}\}\(u\_\{2\}\)=\\varnothing\. IfZi​jZ\_\{ij\}happens for somei≤ri\\leq randj≤Rj\\leq R, thend​\(u1,Si​j\)≤k1d\(u\_\{1\},S\_\{ij\}\)\\leq k\_\{1\}andd​\(u2,Si​j\)≥k2d\(u\_\{2\},S\_\{ij\}\)\\geq k\_\{2\}, and consequently,d¯​\(u1,u2\)≥k2−k1=\(1−ε\)​d​\(u1,u2\)\\underline\{d\}\(u\_\{1\},u\_\{2\}\)\\geq k\_\{2\}\-k\_\{1\}=\(1\-\\varepsilon\)d\(u\_\{1\},u\_\{2\}\)\. Thus, denotingZ=∪i≤r,j≤RZi​jZ=\\cup\_\{i\\leq r,j\\leq R\}Z\_\{ij\}, it suffices to prove thatℙ\(Z∣u1↔u2\)→ℙ1\\mathbb\{P\}\(Z\\mid u\_\{1\}\\leftrightarrow u\_\{2\}\)\\xrightarrow\{\\scriptscriptstyle\\mathbb\{P\}\}1\. Since

ℙ\(u1↔u2butu1,u2∉𝒞\(1\)∣G\)=1n2∑i≥2\|𝒞\(i\)\|2≤\|𝒞\(2\)\|n→ℙ0\\mathbb\{P\}\(u\_\{1\}\\leftrightarrow u\_\{2\}\\text\{ but \}u\_\{1\},u\_\{2\}\\notin\\mathcal\{C\}\_\{\\scriptscriptstyle\(1\)\}\\mid G\)=\\frac\{1\}\{n^\{2\}\}\\sum\_\{i\\geq 2\}\|\\mathcal\{C\}\_\{\\scriptscriptstyle\(i\)\}\|^\{2\}\\leq\\frac\{\|\\mathcal\{C\}\_\{\\scriptscriptstyle\(2\)\}\|\}\{n\}\\xrightarrow\{\\scriptscriptstyle\\mathbb\{P\}\}0by Theorems 3\.1 and 3\.12 fromBollobáset al\.\([2007](https://arxiv.org/html/2607.10074#bib.bib264)\), it suffices to show thatℙ​\(Z∣u1,u2∈𝒞\(1\)\)→ℙ1\\mathbb\{P\}\(Z\\mid u\_\{1\},u\_\{2\}\\in\\mathcal\{C\}\_\{\\scriptscriptstyle\(1\)\}\)\\xrightarrow\{\\scriptscriptstyle\\mathbb\{P\}\}1\(or equivalentlyℙ​\(Zc∣u1,u2∈𝒞\(1\)\)→ℙ0\\mathbb\{P\}\(Z^\{c\}\\mid u\_\{1\},u\_\{2\}\\in\\mathcal\{C\}\_\{\\scriptscriptstyle\(1\)\}\)\\xrightarrow\{\\scriptscriptstyle\\mathbb\{P\}\}0\)\.

Conditioning throughout on the eventX=\{u1,u2∈𝒞\(1\)\}X=\\\{u\_\{1\},u\_\{2\}\\in\\mathcal\{C\}\_\{\\scriptscriptstyle\(1\)\}\\\}, we have for each\(i,j\)\(i,j\)that

ℙ​\(Zi​j∣X\)​𝟏X\\displaystyle\\mathbb\{P\}\(Z\_\{ij\}\\mid X\)\\mathbf\{1\}\_\{X\}=\[\(1−\|Nk2​\(u2\)\|n\)Mi−\(1−\|Nk1​\(u1\)\|\+\|Nk2​\(u2\)\|n\)Mi\]​𝟏X,\\displaystyle=\\left\[\\bigg\(1\-\\frac\{\|N\_\{k\_\{2\}\}\(u\_\{2\}\)\|\}\{n\}\\bigg\)^\{M^\{i\}\}\-\\bigg\(1\-\\frac\{\|N\_\{k\_\{1\}\}\(u\_\{1\}\)\|\+\|N\_\{k\_\{2\}\}\(u\_\{2\}\)\|\}\{n\}\\bigg\)^\{M^\{i\}\}\\right\]\\mathbf\{1\}\_\{X\},sinceℙ​\(Ac∩B\)=ℙ​\(B\)−ℙ​\(A∩B\)\\mathbb\{P\}\(A^\{c\}\\cap B\)=\\mathbb\{P\}\(B\)\-\\mathbb\{P\}\(A\\cap B\)\. By independence ofZi​jZ\_\{ij\}’s,

ℙ​\(Zc∣X\)​𝟏X\\displaystyle\\mathbb\{P\}\(Z^\{c\}\\mid X\)\\mathbf\{1\}\_\{X\}\(11\)=\(∏i=0r\(1−\(1−\|Nk2​\(u2\)\|n\)Mi\+\(1−\|Nk1​\(u1\)\|\+\|Nk2​\(u2\)\|n\)Mi\)\)R​𝟏X\\displaystyle=\\bigg\(\\prod\_\{i=0\}^\{r\}\\bigg\(1\-\\bigg\(1\-\\frac\{\|N\_\{k\_\{2\}\}\(u\_\{2\}\)\|\}\{n\}\\bigg\)^\{M^\{i\}\}\+\\bigg\(1\-\\frac\{\|N\_\{k\_\{1\}\}\(u\_\{1\}\)\|\+\|N\_\{k\_\{2\}\}\(u\_\{2\}\)\|\}\{n\}\\bigg\)^\{M^\{i\}\}\\bigg\)\\bigg\)^\{R\}\\mathbf\{1\}\_\{X\}≤exp⁡\(−R​∑i=0r\(\(1−\|Nk2​\(u2\)\|n\)Mi−\(1−\|Nk1​\(u1\)\|\+\|Nk2​\(u2\)\|n\)Mi\)\)​𝟏X\\displaystyle\\leq\\exp\\bigg\(\-R\\sum\_\{i=0\}^\{r\}\\bigg\(\\bigg\(1\-\\frac\{\|N\_\{k\_\{2\}\}\(u\_\{2\}\)\|\}\{n\}\\bigg\)^\{M^\{i\}\}\-\\bigg\(1\-\\frac\{\|N\_\{k\_\{1\}\}\(u\_\{1\}\)\|\+\|N\_\{k\_\{2\}\}\(u\_\{2\}\)\|\}\{n\}\\bigg\)^\{M^\{i\}\}\\bigg\)\\bigg\)\\mathbf\{1\}\_\{X\}=exp⁡\(−R​∑i=0r\|Nk1​\(u1\)\|n​∑j=0Mi−1\(1−\|Nk2​\(u2\)\|n\)Mi−1−j​\(1−\|Nk1​\(u1\)\|\+\|Nk2​\(u2\)\|n\)j\)​𝟏X\\displaystyle=\\exp\\bigg\(\-R\\sum\_\{i=0\}^\{r\}\\frac\{\|N\_\{k\_\{1\}\}\(u\_\{1\}\)\|\}\{n\}\\sum\_\{j=0\}^\{M^\{i\}\-1\}\\bigg\(1\-\\frac\{\|N\_\{k\_\{2\}\}\(u\_\{2\}\)\|\}\{n\}\\bigg\)^\{M^\{i\}\-1\-j\}\\bigg\(1\-\\frac\{\|N\_\{k\_\{1\}\}\(u\_\{1\}\)\|\+\|N\_\{k\_\{2\}\}\(u\_\{2\}\)\|\}\{n\}\\bigg\)^\{j\}\\bigg\)\\mathbf\{1\}\_\{X\}≤exp⁡\(−R​\|Nk1​\(u1\)\|n​∑i=0rMi​\(1−\|Nk1​\(u1\)\|\+\|Nk2​\(u2\)\|n\)Mi−1\)​𝟏X\\displaystyle\\leq\\exp\\bigg\(\-R\\frac\{\|N\_\{k\_\{1\}\}\(u\_\{1\}\)\|\}\{n\}\\sum\_\{i=0\}^\{r\}M^\{i\}\\bigg\(1\-\\frac\{\|N\_\{k\_\{1\}\}\(u\_\{1\}\)\|\+\|N\_\{k\_\{2\}\}\(u\_\{2\}\)\|\}\{n\}\\bigg\)^\{M^\{i\}\-1\}\\bigg\)\\mathbf\{1\}\_\{X\}≤exp⁡\(−R​\|∂Nk1​\(u1\)\|n​Mr​\(1−\|Nk1​\(u1\)\|\+\|Nk2​\(u2\)\|n\)Mr−1\)​𝟏X\\displaystyle\\leq\\exp\\bigg\(\-R\\frac\{\|\\partial N\_\{k\_\{1\}\}\(u\_\{1\}\)\|\}\{n\}M^\{r\}\\bigg\(1\-\\frac\{\|N\_\{k\_\{1\}\}\(u\_\{1\}\)\|\+\|N\_\{k\_\{2\}\}\(u\_\{2\}\)\|\}\{n\}\\bigg\)^\{M^\{r\}\-1\}\\bigg\)\\mathbf\{1\}\_\{X\}\(12\)where the first inequality uses1−x≤exp⁡\(−x\)1\-x\\leq\\exp\(\-x\)and the last inequality follows from∑i=0rMi=Mr\+1−1M−1\>Mr\+1−MrM−1=Mr\\sum\_\{i=0\}^\{r\}M^\{i\}=\\frac\{M^\{r\+1\}\-1\}\{M\-1\}\>\\frac\{M^\{r\+1\}\-M^\{r\}\}\{M\-1\}=M^\{r\}\.

Conditionally onu1,u2u\_\{1\},u\_\{2\}being in the same connected component, Theorem 6\.2 from\\swapHofstadvan der \([2024b](https://arxiv.org/html/2607.10074#bib.bib10)\)implies thatd​\(u1,u2\)/logλ1⁡n→ℙ1d\(u\_\{1\},u\_\{2\}\)/\\log\_\{\\lambda\_\{1\}\}n\\xrightarrow\{\\scriptscriptstyle\\mathbb\{P\}\}1\. In other words,

\(1−ϵ\)​logλ1⁡n≤d​\(u1,u2\)≤\(1\+ϵ\)​logλ1⁡nw\.h\.p\.\(1\-\\epsilon\)\\log\_\{\\lambda\_\{1\}\}n\\leq d\(u\_\{1\},u\_\{2\}\)\\leq\(1\+\\epsilon\)\\log\_\{\\lambda\_\{1\}\}n\\quad\\text\{w\.h\.p\.\}for any fixedϵ\>0\\epsilon\>0\. Choosingϵ\\epsilonsmall enough so thatε′​ϵ<1−ε′\\varepsilon^\{\\prime\}\\epsilon<1\-\\varepsilon^\{\\prime\}, we then obtain

k1≤ε′​\(1\+ϵ\)​logλ1⁡n<logλ1⁡nw\.h\.p\.,k\_\{1\}\\leq\\varepsilon^\{\\prime\}\(1\+\\epsilon\)\\log\_\{\\lambda\_\{1\}\}n<\\log\_\{\\lambda\_\{1\}\}n\\quad\\text\{w\.h\.p\.\},which allows us to apply Proposition[4\.3](https://arxiv.org/html/2607.10074#S4.ThmtheoremEnv3)on\|∂Nk1​\(u1\)\|\|\\partial N\_\{k\_\{1\}\}\(u\_\{1\}\)\|\. By Proposition[4\.3](https://arxiv.org/html/2607.10074#S4.ThmtheoremEnv3)andDDbeing primitive, there existsc\>0c\>0such that

\|∂Nk1​\(u1\)\|≥∑t=1T\(1−ε\)​\[Dk1\]t​\(u1\)​t≥T​\(1−ε\)​c​λ1k1≥T​\(1−ε\)​c⋅nε′​\(1−ϵ\)w\.h\.p\.\\displaystyle\|\\partial N\_\{k\_\{1\}\}\(u\_\{1\}\)\|\\geq\\sum\_\{t=1\}^\{T\}\(1\-\\varepsilon\)\[D^\{k\_\{1\}\}\]\_\{t\(u\_\{1\}\)t\}\\geq T\(1\-\\varepsilon\)c\\lambda\_\{1\}^\{k\_\{1\}\}\\geq T\(1\-\\varepsilon\)c\\cdot n^\{\\varepsilon^\{\\prime\}\(1\-\\epsilon\)\}\\quad\\text\{w\.h\.p\.\}\(13\)
Next, choosingϵ\\epsilonsmall enough so thatϵ​\(1−ε\+ε′\)<\(1−θ\)−\(1−ε\+ε′\)\\epsilon\(1\-\\varepsilon\+\\varepsilon^\{\\prime\}\)<\(1\-\\theta\)\-\(1\-\\varepsilon\+\\varepsilon^\{\\prime\}\), we have that

k2≤\(1−ε\+ε′\)​\(1\+ϵ\)​logλ1⁡n<\(1−θ\)​logλ1⁡nw\.h\.p\.,k\_\{2\}\\leq\(1\-\\varepsilon\+\\varepsilon^\{\\prime\}\)\(1\+\\epsilon\)\\log\_\{\\lambda\_\{1\}\}n<\\left\(1\-\\theta\\right\)\\log\_\{\\lambda\_\{1\}\}n\\quad\\text\{w\.h\.p\.\},and so there existsγ∈\(0,1−θ\)\\gamma\\in\\left\(0,1\-\\theta\\right\)such thatk1<k2<γ​logλ1⁡nk\_\{1\}<k\_\{2\}<\\gamma\\log\_\{\\lambda\_\{1\}\}n\. By Markov’s inequality and Lemma[4\.6](https://arxiv.org/html/2607.10074#S4.ThmtheoremEnv6), there existsδ\>0\\delta\>0such that

ℙ​\(\|Nki​\(ui\)t\|≥nγ\)≤O​\(λ1ki\)nγ≤n−δ\\mathbb\{P\}\(\|N\_\{k\_\{i\}\}\(u\_\{i\}\)\_\{t\}\|\\geq n^\{\\gamma\}\)\\leq\\frac\{O\(\\lambda\_\{1\}^\{k\_\{i\}\}\)\}\{n^\{\\gamma\}\}\\leq n^\{\-\\delta\}fori=1,2i=1,2andt∈\[T\]t\\in\[T\]with sufficiently largenn\. Therefore,

ℙ\(\|Nki\(ui\)t\|\\displaystyle\\mathbb\{P\}\(\|N\_\{k\_\{i\}\}\(u\_\{i\}\)\_\{t\}\|≤nγfori=1,2andt∈\[T\]\)\\displaystyle\\leq n^\{\\gamma\}\\text\{ for \}i=1,2\\text\{ and \}t\\in\[T\]\)=1−ℙ\(∃i∈\{1,2\},t∈\[T\]s\.t\.\|Nki\(ui\)t\|≥nγ\)\\displaystyle=1\-\\mathbb\{P\}\(\\exists i\\in\\\{1,2\\\},t\\in\[T\]s\.t\.\\left\|N\_\{k\_\{i\}\}\(u\_\{i\}\)\_\{t\}\\right\|\\geq n^\{\\gamma\}\)≥1−∑i=1,2∑t=1Tℙ​\(\|Nki​\(ui\)t\|≥nγ\)≥1−2​T​n−δ,\\displaystyle\\geq 1\-\\sum\_\{i=1,2\}\\sum\_\{t=1\}^\{T\}\\mathbb\{P\}\(\|N\_\{k\_\{i\}\}\(u\_\{i\}\)\_\{t\}\|\\geq n^\{\\gamma\}\)\\geq 1\-2Tn^\{\-\\delta\},and so

\|Nki​\(ui\)\|≤T​nγfor​i=1,2​w\.h\.p\.\\displaystyle\|N\_\{k\_\{i\}\}\(u\_\{i\}\)\|\\leq Tn^\{\\gamma\}\\quad\\text\{for \}i=1,2\\text\{ w\.h\.p\.\}\(14\)
From \([13](https://arxiv.org/html/2607.10074#S7.E13)\) and \([14](https://arxiv.org/html/2607.10074#S7.E14)\),

ℙ​\(Zc∣X\)​𝟏X\\displaystyle\\mathbb\{P\}\(Z^\{c\}\\mid X\)\\mathbf\{1\}\_\{X\}≤exp⁡\(−R​T​\(1−ε\)​c⋅nε′​\(1−ϵ\)n​Mθlog⁡M​log⁡n−1​\(1−2​T​nγn\)Mθlog⁡M​log⁡n\)\\displaystyle\\leq\\exp\\bigg\(\-R\\frac\{T\(1\-\\varepsilon\)c\\cdot n^\{\\varepsilon^\{\\prime\}\(1\-\\epsilon\)\}\}\{n\}M^\{\\frac\{\\theta\}\{\\log M\}\\log n\-1\}\\bigg\(1\-\\frac\{2Tn^\{\\gamma\}\}\{n\}\\bigg\)^\{M^\{\\frac\{\\theta\}\{\\log M\}\\log n\}\}\\bigg\)=exp⁡\(−R​T​\(1−ε\)​c⋅n\(min⁡\{ε2,ε−θ\}−ε′′\)​\(1−ϵ\)n​M​nθ​\(1−2​T​nγn\)nθ\)\.\\displaystyle=\\exp\\bigg\(\-R\\frac\{T\(1\-\\varepsilon\)c\\cdot n^\{\\left\(\\min\\left\\\{\\frac\{\\varepsilon\}\{2\},\\varepsilon\-\\theta\\right\\\}\-\\varepsilon^\{\\prime\\prime\}\\right\)\(1\-\\epsilon\)\}\}\{nM\}n^\{\\theta\}\\bigg\(1\-\\frac\{2Tn^\{\\gamma\}\}\{n\}\\bigg\)^\{n^\{\\theta\}\}\\bigg\)\.Sinceγ\+θ<1\\gamma\+\\theta<1,\(1−2​T​nγn\)nθ≥1−2​T​nγ\+θn→1\\bigg\(1\-\\frac\{2Tn^\{\\gamma\}\}\{n\}\\bigg\)^\{n^\{\\theta\}\}\\geq 1\-\\frac\{2Tn^\{\\gamma\+\\theta\}\}\{n\}\\to 1asn→∞n\\to\\infty\. Sinceϵ\\epsiloncan be chosen small enough so that−ε′′​ϵ\+ε′′\+ϵ​min⁡\{ε2,ε−θ\}<ς\-\\varepsilon^\{\\prime\\prime\}\\epsilon\+\\varepsilon^\{\\prime\\prime\}\+\\epsilon\\min\\left\\\{\\frac\{\\varepsilon\}\{2\},\\varepsilon\-\\theta\\right\\\}<\\varsigmafor anyς\>0\\varsigma\>0,

R=Ω​\(n1−θ−min⁡\{ε2,ε−θ\}\+ς\)R=\\Omega\\left\(n^\{1\-\\theta\-\\min\\left\\\{\\frac\{\\varepsilon\}\{2\},\\varepsilon\-\\theta\\right\\\}\+\\varsigma\}\\right\)is sufficient for the final bound to tend to 0\. By Assumption[3\.2](https://arxiv.org/html/2607.10074#S3.ThmtheoremEnv2),ℙ​\(x∈𝒞\(1\)\)\\mathbb\{P\}\(x\\in\\mathcal\{C\}\_\{\(1\)\}\)is strictly bounded away from zero\(Bollobáset al\.[2007](https://arxiv.org/html/2607.10074#bib.bib264), Theorem 3\.1\), soℙ​\(X\)\\mathbb\{P\}\(X\)does not vanish\. Then sinceℙ​\(Zc∣X\)​𝟏X\\mathbb\{P\}\(Z^\{c\}\\mid X\)\\mathbf\{1\}\_\{X\}vanishes,ℙ​\(Zc∣X\)\\mathbb\{P\}\(Z^\{c\}\\mid X\)thus vanishes\.

### 7\.7Proof of Theorem[4\.2](https://arxiv.org/html/2607.10074#S4.ThmtheoremEnv2)

Ifu1u\_\{1\}andu2u\_\{2\}are not in the same connected component,d​\(u1,u2\)=∞d\(u\_\{1\},u\_\{2\}\)=\\inftyand eitherd​\(u1,s\)=∞d\(u\_\{1\},s\)=\\inftyord​\(u2,s\)=∞d\(u\_\{2\},s\)=\\inftyfor any landmarksssampled in the local step \(see Section[2\.2](https://arxiv.org/html/2607.10074#S2.SS2)\)\. Hence,d¯​\(u1,u2\)=∞\\bar\{d\}\(u\_\{1\},u\_\{2\}\)=\\infty\. Now we focus on the case thatu1u\_\{1\}andu2u\_\{2\}are in the same connected component \(i\.e\.,u1↔u2u\_\{1\}\\leftrightarrow u\_\{2\}\)\.

Letk=ε′​d​\(u1,u2\)k=\\varepsilon^\{\\prime\}d\(u\_\{1\},u\_\{2\}\)with0<ε′=1\+ε2−ε′′<1\+ε20<\\varepsilon^\{\\prime\}=\\frac\{1\+\\varepsilon\}\{2\}\-\\varepsilon^\{\\prime\\prime\}<\\frac\{1\+\\varepsilon\}\{2\}for someε′′∈\(0,1\+ε2\)\\varepsilon^\{\\prime\\prime\}\\in\\left\(0,\\frac\{1\+\\varepsilon\}\{2\}\\right\)\(to be chosen later\)\.

LetSi​jS\_\{ij\}be the landmark set of sizeMiM^\{i\}sampled in thejj\-th round andZi​jZ\_\{ij\}be the event thatSi​jS\_\{ij\}contains at least one landmark node inNk​\(u1\)∩Nk​\(u2\)N\_\{k\}\(u\_\{1\}\)\\cap N\_\{k\}\(u\_\{2\}\)and none in\(Nk​\(u1\)∪Nk​\(u2\)\)∖\(Nk​\(u1\)∩Nk​\(u2\)\)\(N\_\{k\}\(u\_\{1\}\)\\cup N\_\{k\}\(u\_\{2\}\)\)\\setminus\(N\_\{k\}\(u\_\{1\}\)\\cap N\_\{k\}\(u\_\{2\}\)\)\. IfZi​jZ\_\{ij\}happens for somei≤ri\\leq randj≤Rj\\leq R, the landmarks in the intersection will be the common landmarks for calculatingd¯​\(u1,u2\)\\bar\{d\}\(u\_\{1\},u\_\{2\}\), and sod¯​\(u1,u2\)≤2​k≤\(1\+ε\)​d​\(u1,u2\)\\bar\{d\}\(u\_\{1\},u\_\{2\}\)\\leq 2k\\leq\(1\+\\varepsilon\)d\(u\_\{1\},u\_\{2\}\)\. Thus, denotingZ=∪i≤r,j≤RZi​jZ=\\cup\_\{i\\leq r,j\\leq R\}Z\_\{ij\}, it suffices to prove thatℙ\(Z∣u1↔u2\)→ℙ1\\mathbb\{P\}\(Z\\mid u\_\{1\}\\leftrightarrow u\_\{2\}\)\\xrightarrow\{\\scriptscriptstyle\\mathbb\{P\}\}1\. Since

ℙ\(u1↔u2butu1,u2∉𝒞\(1\)∣G\)=1n2∑i≥2\|𝒞\(i\)\|2≤\|𝒞\(2\)\|n→ℙ0\\mathbb\{P\}\(u\_\{1\}\\leftrightarrow u\_\{2\}\\text\{ but \}u\_\{1\},u\_\{2\}\\notin\\mathcal\{C\}\_\{\\scriptscriptstyle\(1\)\}\\mid G\)=\\frac\{1\}\{n^\{2\}\}\\sum\_\{i\\geq 2\}\|\\mathcal\{C\}\_\{\\scriptscriptstyle\(i\)\}\|^\{2\}\\leq\\frac\{\|\\mathcal\{C\}\_\{\\scriptscriptstyle\(2\)\}\|\}\{n\}\\xrightarrow\{\\scriptscriptstyle\\mathbb\{P\}\}0by Theorems 3\.1 and 3\.12 fromBollobáset al\.\([2007](https://arxiv.org/html/2607.10074#bib.bib264)\), it suffices to show thatℙ​\(Z∣u1,u2∈𝒞\(1\)\)→ℙ1\\mathbb\{P\}\(Z\\mid u\_\{1\},u\_\{2\}\\in\\mathcal\{C\}\_\{\\scriptscriptstyle\(1\)\}\)\\xrightarrow\{\\scriptscriptstyle\\mathbb\{P\}\}1\(or equivalentlyℙ​\(Zc∣u1,u2∈𝒞\(1\)\)→ℙ0\\mathbb\{P\}\(Z^\{c\}\\mid u\_\{1\},u\_\{2\}\\in\\mathcal\{C\}\_\{\\scriptscriptstyle\(1\)\}\)\\xrightarrow\{\\scriptscriptstyle\\mathbb\{P\}\}0\)\.

Conditioning throughout on the eventX=\{u1,u2∈𝒞\(1\)\}X=\\\{u\_\{1\},u\_\{2\}\\in\\mathcal\{C\}\_\{\\scriptscriptstyle\(1\)\}\\\}, we have for each \(i,j\) that

ℙ\(Zi​j\\displaystyle\\mathbb\{P\}\(Z\_\{ij\}∣X\)𝟏X\\displaystyle\\mid X\)\\mathbf\{1\}\_\{X\}=\|Nk​\(u1\)∩Nk​\(u2\)\|n​\(\|Nk​\(u1\)∩Nk​\(u2\)\|n\+1−\|Nk​\(u1\)∪Nk​\(u2\)\|n\)\|Si​j\|−1​𝟏X\.\\displaystyle=\\frac\{\|N\_\{k\}\(u\_\{1\}\)\\cap N\_\{k\}\(u\_\{2\}\)\|\}\{n\}\\left\(\\frac\{\|N\_\{k\}\(u\_\{1\}\)\\cap N\_\{k\}\(u\_\{2\}\)\|\}\{n\}\+1\-\\frac\{\|N\_\{k\}\(u\_\{1\}\)\\cup N\_\{k\}\(u\_\{2\}\)\|\}\{n\}\\right\)^\{\|S\_\{ij\}\|\-1\}\\mathbf\{1\}\_\{X\}\.By independence ofZi​jZ\_\{ij\}’s,

ℙ​\(Zc∣X\)​𝟏X\\displaystyle\\mathbb\{P\}\(Z^\{c\}\\mid X\)\\mathbf\{1\}\_\{X\}=\(∏i=0r\(1−\|Nk​\(u1\)∩Nk​\(u2\)\|n​\(\|Nk​\(u1\)∩Nk​\(u2\)\|n\+1−\|Nk​\(u1\)∪Nk​\(u2\)\|n\)\|Si​j\|−1\)\)R​𝟏X\\displaystyle=\\left\(\\prod\_\{i=0\}^\{r\}\\left\(1\-\\frac\{\|N\_\{k\}\(u\_\{1\}\)\\cap N\_\{k\}\(u\_\{2\}\)\|\}\{n\}\\left\(\\frac\{\|N\_\{k\}\(u\_\{1\}\)\\cap N\_\{k\}\(u\_\{2\}\)\|\}\{n\}\+1\-\\frac\{\|N\_\{k\}\(u\_\{1\}\)\\cup N\_\{k\}\(u\_\{2\}\)\|\}\{n\}\\right\)^\{\|S\_\{ij\}\|\-1\}\\right\)\\right\)^\{R\}\\mathbf\{1\}\_\{X\}≤exp⁡\(−R​∑i=0r\|∂Nk​\(u1\)t∩∂Nk​\(u2\)t\|n​\(\|∂Nk​\(u1\)t∩∂Nk​\(u2\)t\|n\+1−\|Nk​\(u1\)\|\+\|Nk​\(u2\)\|n\)Mi−1\)​𝟏X\\displaystyle\\leq\\exp\\left\(\-R\\sum\_\{i=0\}^\{r\}\\frac\{\|\\partial N\_\{k\}\(u\_\{1\}\)\_\{t\}\\cap\\partial N\_\{k\}\(u\_\{2\}\)\_\{t\}\|\}\{n\}\\left\(\\frac\{\|\\partial N\_\{k\}\(u\_\{1\}\)\_\{t\}\\cap\\partial N\_\{k\}\(u\_\{2\}\)\_\{t\}\|\}\{n\}\+1\-\\frac\{\|N\_\{k\}\(u\_\{1\}\)\|\+\|N\_\{k\}\(u\_\{2\}\)\|\}\{n\}\\right\)^\{M^\{i\}\-1\}\\right\)\\mathbf\{1\}\_\{X\}for any node typet∈\[T\]t\\in\[T\], since1−x≤exp⁡\(−x\)1\-x\\leq\\exp\(\-x\)forx≥0x\\geq 0\.

Conditionally onu1,u2u\_\{1\},u\_\{2\}being in the same connected component, Theorem 6\.2 from\\swapHofstadvan der \([2024b](https://arxiv.org/html/2607.10074#bib.bib10)\)implies thatd​\(u1,u2\)/logλ1⁡n→ℙ1d\(u\_\{1\},u\_\{2\}\)/\\log\_\{\\lambda\_\{1\}\}n\\xrightarrow\{\\scriptscriptstyle\\mathbb\{P\}\}1\. In other words,

\(1−ϵ\)​logλ1⁡n≤d​\(u1,u2\)≤\(1\+ϵ\)​logλ1⁡nw\.h\.p\.\(1\-\\epsilon\)\\log\_\{\\lambda\_\{1\}\}n\\leq d\(u\_\{1\},u\_\{2\}\)\\leq\(1\+\\epsilon\)\\log\_\{\\lambda\_\{1\}\}n\\quad\\text\{w\.h\.p\.\}for any fixedϵ\>0\\epsilon\>0\. Choosingϵ\\epsilonsmall enough so thatε′​ϵ<1−ε′\\varepsilon^\{\\prime\}\\epsilon<1\-\\varepsilon^\{\\prime\}, we then obtain

k≤ε′​\(1\+ϵ\)​logλ1⁡n<logλ1⁡nw\.h\.p\.k\\leq\\varepsilon^\{\\prime\}\(1\+\\epsilon\)\\log\_\{\\lambda\_\{1\}\}n<\\log\_\{\\lambda\_\{1\}\}n\\quad\\text\{w\.h\.p\.\}This allows us to use Proposition[4\.3](https://arxiv.org/html/2607.10074#S4.ThmtheoremEnv3)\.

Since\(1\+ε\)​logλ1⁡n\>logλ1⁡nt\(1\+\\varepsilon\)\\log\_\{\\lambda\_\{1\}\}n\>\\log\_\{\\lambda\_\{1\}\}n\_\{t\}, we can chooseε′′,ϵ\\varepsilon^\{\\prime\\prime\},\\epsilonsmall enough so that

\(−2​ε′′​ϵ\+2​ε′′\+ϵ​\(1\+ε\)\)​logλ1⁡n<\(1\+ε\)​logλ1⁡n−logλ1⁡nt\\left\(\-2\\varepsilon^\{\\prime\\prime\}\\epsilon\+2\\varepsilon^\{\\prime\\prime\}\+\\epsilon\(1\+\\varepsilon\)\\right\)\\log\_\{\\lambda\_\{1\}\}n<\(1\+\\varepsilon\)\\log\_\{\\lambda\_\{1\}\}n\-\\log\_\{\\lambda\_\{1\}\}n\_\{t\}and obtain

2​k≥2​\(1\+ε2−ε′′\)​\(1−ϵ\)​logλ1⁡n\>logλ1⁡nt,2k\\geq 2\\left\(\\frac\{1\+\\varepsilon\}\{2\}\-\\varepsilon^\{\\prime\\prime\}\\right\)\(1\-\\epsilon\)\\log\_\{\\lambda\_\{1\}\}n\>\\log\_\{\\lambda\_\{1\}\}n\_\{t\},which allows us to also use Proposition[4\.4](https://arxiv.org/html/2607.10074#S4.ThmtheoremEnv4)\.

ChoosingL=κ0​logλ1⁡n<min⁡\{k,γ​logλ1⁡n\}L=\\kappa\_\{0\}\\log\_\{\\lambda\_\{1\}\}n<\\min\\\{k,\\gamma\\log\_\{\\lambda\_\{1\}\}n\\\}for someκ0∈\(0,γ\)\\kappa\_\{0\}\\in\(0,\\gamma\)andγ∈\(0,1−θ\)\\gamma\\in\\left\(0,1\-\\theta\\right\), we obtain from Markov’s inequality and Lemma[4\.6](https://arxiv.org/html/2607.10074#S4.ThmtheoremEnv6)that

ℙ​\(\|NL​\(ui\)t\|≥nγ\)≤O​\(λ1L\)nγ≤n−δ\\mathbb\{P\}\(\|N\_\{L\}\(u\_\{i\}\)\_\{t\}\|\\geq n^\{\\gamma\}\)\\leq\\frac\{O\(\\lambda\_\{1\}^\{L\}\)\}\{n^\{\\gamma\}\}\\leq n^\{\-\\delta\}for alli=1,2i=1,2andt∈\[T\]t\\in\[T\]with someδ\>0\\delta\>0and sufficiently largenn\. Therefore,

ℙ\(\|NL\(ui\)t\|≤\\displaystyle\\mathbb\{P\}\(\|N\_\{L\}\(u\_\{i\}\)\_\{t\}\|\\leqnγfort∈\[T\]andi=1,2\)\\displaystyle n^\{\\gamma\}\\text\{ for \}t\\in\[T\]\\text\{ and \}i=1,2\)=1−ℙ\(∃t∈\[T\],i∈\{1,2\}s\.t\.\|NL\(ui\)t\|≥nγ\)\\displaystyle=1\-\\mathbb\{P\}\(\\exists t\\in\[T\],i\\in\\\{1,2\\\}s\.t\.\\left\|N\_\{L\}\(u\_\{i\}\)\_\{t\}\\right\|\\geq n^\{\\gamma\}\)≥1−∑i=1,2∑t=1Tℙ​\(\|NL​\(ui\)t\|≥nγ\)≥1−2​T​n−δ,\\displaystyle\\geq 1\-\\sum\_\{i=1,2\}\\sum\_\{t=1\}^\{T\}\\mathbb\{P\}\(\|N\_\{L\}\(u\_\{i\}\)\_\{t\}\|\\geq n^\{\\gamma\}\)\\geq 1\-2Tn^\{\-\\delta\},and so\|NL​\(ui\)\|≤T​nγ\|N\_\{L\}\(u\_\{i\}\)\|\\leq Tn^\{\\gamma\}fori=1,2i=1,2w\.h\.p\. Then by Proposition[4\.3](https://arxiv.org/html/2607.10074#S4.ThmtheoremEnv3)andDDbeing primitive, there existsC\>0C\>0such that

\|Nk​\(ui\)\|\\displaystyle\|N\_\{k\}\(u\_\{i\}\)\|=\|NL​\(ui\)\|\+∑l=L\+1k\|∂Nl​\(ui\)\|≤T​nγ\+∑l=L\+1k∑t=1T\(1\+ε\)​\[Dl\]t​\(ui\)​t\\displaystyle=\|N\_\{L\}\(u\_\{i\}\)\|\+\\sum\_\{l=L\+1\}^\{k\}\|\\partial N\_\{l\}\(u\_\{i\}\)\|\\leq Tn^\{\\gamma\}\+\\sum\_\{l=L\+1\}^\{k\}\\sum\_\{t=1\}^\{T\}\(1\+\\varepsilon\)\[D^\{l\}\]\_\{t\(u\_\{i\}\)t\}≤T​nγ\+∑l=L\+1k∑t=1T\(1\+ε\)​C​λ1l<T​nγ\+logλ1⁡n​T​\(1\+ε\)​C​λ1k\\displaystyle\\leq Tn^\{\\gamma\}\+\\sum\_\{l=L\+1\}^\{k\}\\sum\_\{t=1\}^\{T\}\(1\+\\varepsilon\)C\\lambda\_\{1\}^\{l\}<Tn^\{\\gamma\}\+\\log\_\{\\lambda\_\{1\}\}nT\(1\+\\varepsilon\)C\\lambda\_\{1\}^\{k\}fori=1,2i=1,2w\.h\.p\. Combining these with Proposition[4\.4](https://arxiv.org/html/2607.10074#S4.ThmtheoremEnv4)andnt≤nn\_\{t\}\\leq n, we have w\.h\.p\. that

ℙ​\(Zc∣X\)​𝟏X\\displaystyle\\mathbb\{P\}\(Z^\{c\}\\mid X\)\\mathbf\{1\}\_\{X\}≤exp⁡\(−R​\(1−ε\)3​c2​λ12​k2​n2​∑i=0r\(\(1−ε\)3​c2​λ12​k2​n2\+1−2​T​nγ\+2​logλ1⁡n​T​\(1\+ε\)​C​λ1kn\)Mi−1\)\\displaystyle\\leq\\exp\\left\(\-R\\frac\{\(1\-\\varepsilon\)^\{3\}c^\{2\}\\lambda\_\{1\}^\{2k\}\}\{2n^\{2\}\}\\sum\_\{i=0\}^\{r\}\\left\(\\frac\{\(1\-\\varepsilon\)^\{3\}c^\{2\}\\lambda\_\{1\}^\{2k\}\}\{2n^\{2\}\}\+1\-\\frac\{2Tn^\{\\gamma\}\+2\\log\_\{\\lambda\_\{1\}\}nT\(1\+\\varepsilon\)C\\lambda\_\{1\}^\{k\}\}\{n\}\\right\)^\{M^\{i\}\-1\}\\right\)
Sinceγ<1\\gamma<1and0<λ1k<n0<\\lambda\_\{1\}^\{k\}<n,

0<2​T​nγ\+2​logλ1⁡n​T​\(1\+ε\)​C​λ1kn−\(1−ε\)3​c2​λ12​k2​n2<10<\\frac\{2Tn^\{\\gamma\}\+2\\log\_\{\\lambda\_\{1\}\}nT\(1\+\\varepsilon\)C\\lambda\_\{1\}^\{k\}\}\{n\}\-\\frac\{\(1\-\\varepsilon\)^\{3\}c^\{2\}\\lambda\_\{1\}^\{2k\}\}\{2n^\{2\}\}<1for sufficiently largenn\. Also since∑i=0r\(Mi−1\)<∑i=0rMi=Mr\+1−1M−1≤nθ​MM−1\\sum\_\{i=0\}^\{r\}\(M^\{i\}\-1\)<\\sum\_\{i=0\}^\{r\}M^\{i\}=\\frac\{M^\{r\+1\}\-1\}\{M\-1\}\\leq\\frac\{n^\{\\theta\}M\}\{M\-1\},

ℙ​\(Zc∣X\)​𝟏X\\displaystyle\\mathbb\{P\}\(Z^\{c\}\\mid X\)\\mathbf\{1\}\_\{X\}≤exp⁡\(−R​\(1−ε\)3​c2​λ12​k2​n2​∑i=0r\(1−\(2​T​nγn\+2​logλ1⁡n​T​\(1\+ε\)​C​λ1kn−\(1−ε\)3​c2​λ12​k2​n2\)​\(Mi−1\)\)\)\\displaystyle\\leq\\exp\\left\(\-R\\frac\{\(1\-\\varepsilon\)^\{3\}c^\{2\}\\lambda\_\{1\}^\{2k\}\}\{2n^\{2\}\}\\sum\_\{i=0\}^\{r\}\\left\(1\-\\left\(\\frac\{2Tn^\{\\gamma\}\}\{n\}\+\\frac\{2\\log\_\{\\lambda\_\{1\}\}nT\(1\+\\varepsilon\)C\\lambda\_\{1\}^\{k\}\}\{n\}\-\\frac\{\(1\-\\varepsilon\)^\{3\}c^\{2\}\\lambda\_\{1\}^\{2k\}\}\{2n^\{2\}\}\\right\)\(M^\{i\}\-1\)\\right\)\\right\)<exp⁡\(−R​\(1−ε\)3​c2​λ12​k2​n2​\(r−\(2​T​nγn\+2​logλ1⁡n​T​\(1\+ε\)​C​λ1kn−\(1−ε\)3​c2​λ12​k2​n2\)​nθ​MM−1\)\)\.\\displaystyle<\\exp\\left\(\-R\\frac\{\(1\-\\varepsilon\)^\{3\}c^\{2\}\\lambda\_\{1\}^\{2k\}\}\{2n^\{2\}\}\\left\(r\-\\left\(\\frac\{2Tn^\{\\gamma\}\}\{n\}\+\\frac\{2\\log\_\{\\lambda\_\{1\}\}nT\(1\+\\varepsilon\)C\\lambda\_\{1\}^\{k\}\}\{n\}\-\\frac\{\(1\-\\varepsilon\)^\{3\}c^\{2\}\\lambda\_\{1\}^\{2k\}\}\{2n^\{2\}\}\\right\)\\frac\{n^\{\\theta\}M\}\{M\-1\}\\right\)\\right\)\.Sinceγ<1−θ\\gamma<1\-\\theta,0<2​T​nγ\+θn→00<\\frac\{2Tn^\{\\gamma\+\\theta\}\}\{n\}\\to 0asn→∞n\\to\\infty\. Since we can chooseε′′,ϵ\\varepsilon^\{\\prime\\prime\},\\epsilonsmall enough so that

−ε′′​ϵ−ε′′\+ϵ​1\+ε2<1−ε2−θ,\-\\varepsilon^\{\\prime\\prime\}\\epsilon\-\\varepsilon^\{\\prime\\prime\}\+\\epsilon\\frac\{1\+\\varepsilon\}\{2\}<\\frac\{1\-\\varepsilon\}\{2\}\-\\theta,we then have

0<λ12​kn2​nθ<λ1kn​nθ≤n\(1\+ε2−ε′′\)​\(1\+ϵ\)\+θn<10<\\frac\{\\lambda\_\{1\}^\{2k\}\}\{n^\{2\}\}n^\{\\theta\}<\\frac\{\\lambda\_\{1\}^\{k\}\}\{n\}n^\{\\theta\}\\leq\\frac\{n^\{\\left\(\\frac\{1\+\\varepsilon\}\{2\}\-\\varepsilon^\{\\prime\\prime\}\\right\)\(1\+\\epsilon\)\+\\theta\}\}\{n\}<1for sufficiently largenn\. Then w\.h\.p\.,

ℙ​\(Zc∣X\)​𝟏X\\displaystyle\\mathbb\{P\}\(Z^\{c\}\\mid X\)\\mathbf\{1\}\_\{X\}<exp⁡\(−R​\(1−ε\)3​c2​n2​ε′​\(1−ϵ\)2​n2​\(θlog⁡M​log⁡n−1−\(1\+1−0\)\)\)\\displaystyle<\\exp\\left\(\-R\\frac\{\(1\-\\varepsilon\)^\{3\}c^\{2\}n^\{2\\varepsilon^\{\\prime\}\(1\-\\epsilon\)\}\}\{2n^\{2\}\}\\left\(\\frac\{\\theta\}\{\\log M\}\\log n\-1\-\\left\(1\+1\-0\\right\)\\right\)\\right\)<exp⁡\(−R​\(1−ε\)3​c2​n2​\(1\+ε2−ε′′\)​\(1−ϵ\)2​n2​θ2​log⁡M​log⁡n\)\\displaystyle<\\exp\\left\(\-R\\frac\{\(1\-\\varepsilon\)^\{3\}c^\{2\}n^\{2\\left\(\\frac\{1\+\\varepsilon\}\{2\}\-\\varepsilon^\{\\prime\\prime\}\\right\)\(1\-\\epsilon\)\}\}\{2n^\{2\}\}\\frac\{\\theta\}\{2\\log M\}\\log n\\right\)for somec\>0c\>0\. Sinceε′′\\varepsilon^\{\\prime\\prime\}andϵ\\epsiloncan be chosen to be small enough so that2​\(−ε′′​ϵ\+ε′′\+ϵ​1\+ε2\)<ς2\\left\(\-\\varepsilon^\{\\prime\\prime\}\\epsilon\+\\varepsilon^\{\\prime\\prime\}\+\\epsilon\\frac\{1\+\\varepsilon\}\{2\}\\right\)<\\varsigmafor anyς\>0\\varsigma\>0,

R=Ω​\(n2−2​1\+ε2\+ς\)=Ω​\(n1−ε\+ς\)R=\\Omega\\left\(n^\{2\-2\\frac\{1\+\\varepsilon\}\{2\}\+\\varsigma\}\\right\)=\\Omega\\left\(n^\{1\-\\varepsilon\+\\varsigma\}\\right\)is sufficient for the final bound to tend to 0\. By Assumption[3\.2](https://arxiv.org/html/2607.10074#S3.ThmtheoremEnv2),ℙ​\(x∈𝒞\(1\)\)\\mathbb\{P\}\(x\\in\\mathcal\{C\}\_\{\(1\)\}\)is strictly bounded away from zero\(Bollobáset al\.[2007](https://arxiv.org/html/2607.10074#bib.bib264), Theorem 3\.1\), soℙ​\(X\)\\mathbb\{P\}\(X\)does not vanish\. Then sinceℙ​\(Zc∣X\)​𝟏X\\mathbb\{P\}\(Z^\{c\}\\mid X\)\\mathbf\{1\}\_\{X\}vanishes,ℙ​\(Zc∣X\)\\mathbb\{P\}\(Z^\{c\}\\mid X\)thus vanishes\.

### 7\.8Proof of Theorem[4\.5](https://arxiv.org/html/2607.10074#S4.ThmtheoremEnv5)

#### Lower\-Bound Distortion\.

Suppose Proposition[4\.3](https://arxiv.org/html/2607.10074#S4.ThmtheoremEnv3)holds with probability at least1−n−4​δ1\-n^\{\-4\\delta\}instead of w\.h\.p\. In that case,

ℙ\(d¯​\(u,v\)d​\(u,v\)<1−ε\|u↔v\)≤n−4​δ\.\\mathbb\{P\}\\left\(\\frac\{\\underline\{d\}\(u,v\)\}\{d\(u,v\)\}<1\-\\varepsilon\\;\\middle\|\\;u\\leftrightarrow v\\right\)\\leq n^\{\-4\\delta\}\.Therefore, we have:

ℙ\(d¯\(u,v\)<\(1−ε\)d\(u,v\),u↔v\)\\displaystyle\\mathbb\{P\}\\left\(\\underline\{d\}\(u,v\)<\(1\-\\varepsilon\)d\(u,v\),\\;u\\leftrightarrow v\\right\)≤ℙ\(u↔v\)n−4​δ=𝔼\[∑i\|𝒞\(i\)\|2n2\]n−4​δ≤n−4​δ,\\displaystyle\\leq\\mathbb\{P\}\(u\\leftrightarrow v\)n^\{\-4\\delta\}=\\mathbb\{E\}\\left\[\\frac\{\\sum\_\{i\}\|\\mathcal\{C\}\_\{\(i\)\}\|^\{2\}\}\{n^\{2\}\}\\right\]n^\{\-4\\delta\}\\leq n^\{\-4\\delta\},\(15\)where in the final step we use𝔼​\[∑i\|𝒞\(i\)\|2n2\]≤𝔼​\[\|𝒞\(1\)\|n\]→ζ<1\\mathbb\{E\}\\left\[\\frac\{\\sum\_\{i\}\|\\mathcal\{C\}\_\{\(i\)\}\|^\{2\}\}\{n^\{2\}\}\\right\]\\leq\\mathbb\{E\}\\left\[\\frac\{\|\\mathcal\{C\}\_\{\(1\)\}\|\}\{n\}\\right\]\\to\\zeta<1because\|𝒞\(1\)\|=Θ​\(n\)\|\\mathcal\{C\}\_\{\(1\)\}\|=\\Theta\(n\)and\|𝒞\(i\)\|=O​\(log⁡n\)\|\\mathcal\{C\}\_\{\(i\)\}\|=O\(\\log n\)fori\>1i\>1w\.h\.p\.\(Bollobáset al\.[2007](https://arxiv.org/html/2607.10074#bib.bib264), Theorems 3\.1 and 3\.12\)\.

LetY=\{\(i,j\):d¯​\(i,j\)≥\(1−ε\)​d​\(i,j\),i↔j\}Y=\\left\\\{\(i,j\):\\underline\{d\}\(i,j\)\\geq\(1\-\\varepsilon\)d\(i,j\),\\;i\\leftrightarrow j\\right\\\}denote the set of node pairs where the lower\-bound distance approximation succeeds\. To evaluate the global average distortion over all connected pairs, we decompose the sum overYYand its complementYcY^\{c\}:

1\|U\|​∑i↔j\\displaystyle\\frac\{1\}\{\|U\|\}\\sum\_\{i\\leftrightarrow j\}d¯​\(i,j\)d​\(i,j\)=1\|U\|​∑\(i,j\)∈Yd¯​\(i,j\)d​\(i,j\)\+1\|U\|​∑\(i,j\)∈Yc,i↔jd¯​\(i,j\)d​\(i,j\)\\displaystyle\\frac\{\\underline\{d\}\(i,j\)\}\{d\(i,j\)\}=\\frac\{1\}\{\|U\|\}\\sum\_\{\(i,j\)\\in Y\}\\frac\{\\underline\{d\}\(i,j\)\}\{d\(i,j\)\}\+\\frac\{1\}\{\|U\|\}\\sum\_\{\(i,j\)\\in Y^\{c\},\\,i\\leftrightarrow j\}\\frac\{\\underline\{d\}\(i,j\)\}\{d\(i,j\)\}≥1\|U\|​∑\(i,j\)∈Y\(1−ε\)\+0=\(1−ε\)​\(1−\|Yc\|\|U\|\)\.\\displaystyle\\geq\\frac\{1\}\{\|U\|\}\\sum\_\{\(i,j\)\\in Y\}\(1\-\\varepsilon\)\+0=\(1\-\\varepsilon\)\\left\(1\-\\frac\{\|Y^\{c\}\|\}\{\|U\|\}\\right\)\.
To show1\|U\|​∑u1↔u2d¯​\(u1,u2\)d​\(u1,u2\)≥1−ε\\frac\{1\}\{\|U\|\}\\sum\_\{u\_\{1\}\\leftrightarrow u\_\{2\}\}\\frac\{\\underline\{d\}\(u\_\{1\},u\_\{2\}\)\}\{d\(u\_\{1\},u\_\{2\}\)\}\\geq 1\-\\varepsilonw\.h\.p\., it suffices to show\|Yc\|\|U\|→ℙ0\\frac\{\|Y^\{c\}\|\}\{\|U\|\}\\xrightarrow\{\\mathbb\{P\}\}0, and we proceed by contradiction\. Supposelimn→∞ℙ​\(\|Yc\|\|U\|\>δ\)\>0\\lim\_\{n\\to\\infty\}\\mathbb\{P\}\\left\(\\frac\{\|Y^\{c\}\|\}\{\|U\|\}\>\\delta\\right\)\>0\. Then there exists a subsequence\(nl\)l≥1\(n\_\{l\}\)\_\{l\\geq 1\}such thatℙ​\(\|Yc\|\|U\|\>δ\)≥ε′\\mathbb\{P\}\\left\(\\frac\{\|Y^\{c\}\|\}\{\|U\|\}\>\\delta\\right\)\\geq\\varepsilon^\{\\prime\}for some constantε′\>0\\varepsilon^\{\\prime\}\>0\. By selecting a pair of vertices\(u,v\)\(u,v\)uniformly at random from the graph, the joint event\{d¯​\(u,v\)<\(1−ε\)​d​\(u,v\),u↔v\}\\left\\\{\\underline\{d\}\(u,v\)<\(1\-\\varepsilon\)d\(u,v\),\\;u\\leftrightarrow v\\right\\\}is precisely the event that the sampled pair belongs toYcY^\{c\}\. Therefore,

ℙ\(d¯\(u,v\)<\(1−ε\)d\(u,v\),u↔v\)=𝔼\[\|Yc\|nl2\]=𝔼\[\|Yc\|\|U\|⋅\|U\|nl2\]\\displaystyle\\mathbb\{P\}\\left\(\\underline\{d\}\(u,v\)<\(1\-\\varepsilon\)d\(u,v\),\\;u\\leftrightarrow v\\right\)=\\mathbb\{E\}\\left\[\\frac\{\|Y^\{c\}\|\}\{n\_\{l\}^\{2\}\}\\right\]=\\mathbb\{E\}\\left\[\\frac\{\|Y^\{c\}\|\}\{\|U\|\}\\cdot\\frac\{\|U\|\}\{n\_\{l\}^\{2\}\}\\right\]≥𝔼​\[\|Yc\|\|U\|⋅\|U\|nl2​\|\|Yc\|\|U\|\>​δ\]​ℙ​\(\|Yc\|\|U\|\>δ\)≥δ​ε′​𝔼​\[\|U\|nl2​\|\|Yc\|\|U\|\>​δ\]\\displaystyle\\geq\\mathbb\{E\}\\left\[\\frac\{\|Y^\{c\}\|\}\{\|U\|\}\\cdot\\frac\{\|U\|\}\{n\_\{l\}^\{2\}\}\\;\\middle\|\\;\\frac\{\|Y^\{c\}\|\}\{\|U\|\}\>\\delta\\right\]\\mathbb\{P\}\\left\(\\frac\{\|Y^\{c\}\|\}\{\|U\|\}\>\\delta\\right\)\\geq\\delta\\varepsilon^\{\\prime\}\\mathbb\{E\}\\left\[\\frac\{\|U\|\}\{n\_\{l\}^\{2\}\}\\;\\middle\|\\;\\frac\{\|Y^\{c\}\|\}\{\|U\|\}\>\\delta\\right\]Again since\|𝒞\(1\)\|=Θ​\(n\)\|\\mathcal\{C\}\_\{\(1\)\}\|=\\Theta\(n\)and\|𝒞\(i\)\|=O​\(log⁡n\)\|\\mathcal\{C\}\_\{\(i\)\}\|=O\(\\log n\)fori\>1i\>1w\.h\.p\.\(Bollobáset al\.[2007](https://arxiv.org/html/2607.10074#bib.bib264), Theorems 3\.1 and 3\.12\),\|U\|nl2=∑i\|𝒞\(i\)\|2nl2→ℙζ2\>0\\frac\{\|U\|\}\{n\_\{l\}^\{2\}\}=\\frac\{\\sum\_\{i\}\|\\mathcal\{C\}\_\{\(i\)\}\|^\{2\}\}\{n\_\{l\}^\{2\}\}\\xrightarrow\{\\mathbb\{P\}\}\\zeta^\{2\}\>0asnl→∞n\_\{l\}\\to\\infty\. It follows that for sufficiently largenln\_\{l\},

ℙ\\displaystyle\\mathbb\{P\}\(d¯\(u,v\)<\(1−ε\)d\(u,v\),u↔v\)≥δ​ε′​ζ22\.\\displaystyle\\left\(\\underline\{d\}\(u,v\)<\(1\-\\varepsilon\)d\(u,v\),\\;u\\leftrightarrow v\\right\)\\geq\\frac\{\\delta\\varepsilon^\{\\prime\}\\zeta^\{2\}\}\{2\}\.However, asnl→∞n\_\{l\}\\to\\infty, this strictly positive constant bound directly contradicts the polynomial upper bound established in \([15](https://arxiv.org/html/2607.10074#S7.E15)\) asδ​ε′​ζ22≫nl−4​δ\\frac\{\\delta\\varepsilon^\{\\prime\}\\zeta^\{2\}\}\{2\}\\gg n\_\{l\}^\{\-4\\delta\}, where the right\-hand side vanishes\. Thus,\|Yc\|\|U\|→ℙ0\\frac\{\|Y^\{c\}\|\}\{\|U\|\}\\xrightarrow\{\\mathbb\{P\}\}0, completing the proof\.

#### Upper\-Bound Distortion\.

Suppose that Proposition[4\.4](https://arxiv.org/html/2607.10074#S4.ThmtheoremEnv4)holds with probability at least1−n−4​δ1\-n^\{\-4\\delta\}instead of w\.h\.p\. In that case,

ℙ\(d¯​\(u,v\)d​\(u,v\)\>1\+ε\|u↔v\)≤n−4​δ\.\\mathbb\{P\}\\left\(\\frac\{\\bar\{d\}\(u,v\)\}\{d\(u,v\)\}\>1\+\\varepsilon\\;\\middle\|\\;u\\leftrightarrow v\\right\)\\leq n^\{\-4\\delta\}\.Therefore, we have:

ℙ\(d¯\(u,v\)\>\(1\+ε\)d\(u,v\),u↔v\)\\displaystyle\\mathbb\{P\}\\left\(\\bar\{d\}\(u,v\)\>\(1\+\\varepsilon\)d\(u,v\),\\;u\\leftrightarrow v\\right\)≤ℙ\(u↔v\)n−4​δ=𝔼\[∑i\|𝒞\(i\)\|2n2\]n−4​δ≤n−4​δ,\\displaystyle\\leq\\mathbb\{P\}\(u\\leftrightarrow v\)n^\{\-4\\delta\}=\\mathbb\{E\}\\left\[\\frac\{\\sum\_\{i\}\|\\mathcal\{C\}\_\{\(i\)\}\|^\{2\}\}\{n^\{2\}\}\\right\]n^\{\-4\\delta\}\\leq n^\{\-4\\delta\},\(16\)where in the final step we use𝔼​\[∑i\|𝒞\(i\)\|2n2\]≤𝔼​\[\|𝒞\(1\)\|n\]→ζ<1\\mathbb\{E\}\\left\[\\frac\{\\sum\_\{i\}\|\\mathcal\{C\}\_\{\(i\)\}\|^\{2\}\}\{n^\{2\}\}\\right\]\\leq\\mathbb\{E\}\\left\[\\frac\{\|\\mathcal\{C\}\_\{\(1\)\}\|\}\{n\}\\right\]\\to\\zeta<1as\|𝒞\(1\)\|=Θ​\(n\)\|\\mathcal\{C\}\_\{\(1\)\}\|=\\Theta\(n\)and\|𝒞\(i\)\|=O​\(log⁡n\)\|\\mathcal\{C\}\_\{\(i\)\}\|=O\(\\log n\)fori\>1i\>1w\.h\.p\.\(Bollobáset al\.[2007](https://arxiv.org/html/2607.10074#bib.bib264), Theorems 3\.1 and 3\.12\)\.

Furthermore, with probability1−n−4​δ1\-n^\{\-4\\delta\}for a sufficiently smallδ\\delta, the diameter of the graph components satisfiesdiameter​\(G\)≤C​logλ1⁡n\\text\{diameter\}\(G\)\\leq C\\log\_\{\\lambda\_\{1\}\}n\(Riordan and Wormald[2010](https://arxiv.org/html/2607.10074#bib.bib54)\)\. For any connected pairi↔ji\\leftrightarrow jand a chosen landmarkss, the upper\-bound metric satisfiesd¯​\(i,j\)≤d​\(i,s\)\+d​\(j,s\)≤2​C​logλ1⁡n\\bar\{d\}\(i,j\)\\leq d\(i,s\)\+d\(j,s\)\\leq 2C\\log\_\{\\lambda\_\{1\}\}n\. Sinced​\(i,j\)≥1d\(i,j\)\\geq 1for all distinct pairs,d¯​\(i,j\)d​\(i,j\)≤2​C​logλ1⁡n\\frac\{\\bar\{d\}\(i,j\)\}\{d\(i,j\)\}\\leq 2C\\log\_\{\\lambda\_\{1\}\}n\.

LetYYdenote the set of node pairs where the upper\-bound distance approximation succeeds\. To evaluate the global average distortion over all connected pairs\|U\|\|U\|, we decompose the sum overYYand its complementYcY^\{c\}:

1\|U\|​∑i↔jd¯​\(i,j\)d​\(i,j\)\\displaystyle\\frac\{1\}\{\|U\|\}\\sum\_\{i\\leftrightarrow j\}\\frac\{\\bar\{d\}\(i,j\)\}\{d\(i,j\)\}=1\|U\|​∑\(i,j\)∈Yd¯​\(i,j\)d​\(i,j\)\+1\|U\|​∑\(i,j\)∈Ycd¯​\(i,j\)d​\(i,j\)\\displaystyle=\\frac\{1\}\{\|U\|\}\\sum\_\{\(i,j\)\\in Y\}\\frac\{\\bar\{d\}\(i,j\)\}\{d\(i,j\)\}\+\\frac\{1\}\{\|U\|\}\\sum\_\{\(i,j\)\\in Y^\{c\}\}\\frac\{\\bar\{d\}\(i,j\)\}\{d\(i,j\)\}≤1\|U\|​∑\(i,j\)∈Y\(1\+ε\)\+1\|U\|​∑\(i,j\)∈Yc2​C​logλ1⁡n\\displaystyle\\leq\\frac\{1\}\{\|U\|\}\\sum\_\{\(i,j\)\\in Y\}\(1\+\\varepsilon\)\+\\frac\{1\}\{\|U\|\}\\sum\_\{\(i,j\)\\in Y^\{c\}\}2C\\log\_\{\\lambda\_\{1\}\}n=\(1\+ε\)​\|Y\|\|U\|\+2​C​logλ1⁡n​\|Yc\|\|U\|\\displaystyle=\(1\+\\varepsilon\)\\frac\{\|Y\|\}\{\|U\|\}\+2C\\log\_\{\\lambda\_\{1\}\}n\\frac\{\|Y^\{c\}\|\}\{\|U\|\}=1\+ε\+\(2​C​logλ1⁡n−\(1\+ε\)\)​\|Yc\|\|U\|\.\\displaystyle=1\+\\varepsilon\+\\left\(2C\\log\_\{\\lambda\_\{1\}\}n\-\(1\+\\varepsilon\)\\right\)\\frac\{\|Y^\{c\}\|\}\{\|U\|\}\.Thus, to show that1\|U\|​∑i↔jd¯​\(i,j\)d​\(i,j\)≤1\+ε\\frac\{1\}\{\|U\|\}\\sum\_\{i\\leftrightarrow j\}\\frac\{\\bar\{d\}\(i,j\)\}\{d\(i,j\)\}\\leq 1\+\\varepsilonw\.h\.p\., it suffices to establish that\|Yc\|\|U\|→ℙ0\\frac\{\|Y^\{c\}\|\}\{\|U\|\}\\xrightarrow\{\\mathbb\{P\}\}0, and we prove this by contradiction\. Supposelimn→∞ℙ​\(\|Yc\|\|U\|\>δ\)\>0\\lim\_\{n\\to\\infty\}\\mathbb\{P\}\\left\(\\frac\{\|Y^\{c\}\|\}\{\|U\|\}\>\\delta\\right\)\>0\. Then there exists a subsequence\(nl\)l≥1\(n\_\{l\}\)\_\{l\\geq 1\}such thatℙ​\(\|Yc\|\|U\|\>δ\)≥ε′\\mathbb\{P\}\\left\(\\frac\{\|Y^\{c\}\|\}\{\|U\|\}\>\\delta\\right\)\\geq\\varepsilon^\{\\prime\}for some constantε′\>0\\varepsilon^\{\\prime\}\>0\. By selecting a pair of vertices\(u,v\)\(u,v\)uniformly at random from the graph, the joint event\{d¯​\(u,v\)\>\(1\+ε\)​d​\(u,v\),u↔v\}\\left\\\{\\bar\{d\}\(u,v\)\>\(1\+\\varepsilon\)d\(u,v\),\\;u\\leftrightarrow v\\right\\\}maps precisely to sampling a pair fromYcY^\{c\}\. Hence,

ℙ\(d¯\(u,v\)\>\(1\+ε\)d\(u,v\),u↔v\)=𝔼\[\|Yc\|nl2\]=𝔼\[\|Yc\|\|U\|⋅\|U\|nl2\]\\displaystyle\\mathbb\{P\}\\left\(\\bar\{d\}\(u,v\)\>\(1\+\\varepsilon\)d\(u,v\),\\;u\\leftrightarrow v\\right\)=\\mathbb\{E\}\\left\[\\frac\{\|Y^\{c\}\|\}\{n\_\{l\}^\{2\}\}\\right\]=\\mathbb\{E\}\\left\[\\frac\{\|Y^\{c\}\|\}\{\|U\|\}\\cdot\\frac\{\|U\|\}\{n\_\{l\}^\{2\}\}\\right\]≥𝔼​\[\|Yc\|\|U\|⋅\|U\|nl2​\|\|Yc\|\|U\|\>​δ\]​ℙ​\(\|Yc\|\|U\|\>δ\)≥δ​ε′​𝔼​\[\|U\|nl2​\|\|Yc\|\|U\|\>​δ\]\.\\displaystyle\\geq\\mathbb\{E\}\\left\[\\frac\{\|Y^\{c\}\|\}\{\|U\|\}\\cdot\\frac\{\|U\|\}\{n\_\{l\}^\{2\}\}\\;\\middle\|\\;\\frac\{\|Y^\{c\}\|\}\{\|U\|\}\>\\delta\\right\]\\mathbb\{P\}\\left\(\\frac\{\|Y^\{c\}\|\}\{\|U\|\}\>\\delta\\right\)\\geq\\delta\\varepsilon^\{\\prime\}\\mathbb\{E\}\\left\[\\frac\{\|U\|\}\{n\_\{l\}^\{2\}\}\\;\\middle\|\\;\\frac\{\|Y^\{c\}\|\}\{\|U\|\}\>\\delta\\right\]\.Again since\|𝒞\(1\)\|=Θ​\(n\)\|\\mathcal\{C\}\_\{\(1\)\}\|=\\Theta\(n\)and\|𝒞\(i\)\|=O​\(log⁡n\)\|\\mathcal\{C\}\_\{\(i\)\}\|=O\(\\log n\)fori\>1i\>1w\.h\.p\.\(Bollobáset al\.[2007](https://arxiv.org/html/2607.10074#bib.bib264), Theorems 3\.1 and 3\.12\),\|U\|nl2=∑i\|𝒞\(i\)\|2nl2→ℙζ2\>0\\frac\{\|U\|\}\{n\_\{l\}^\{2\}\}=\\frac\{\\sum\_\{i\}\|\\mathcal\{C\}\_\{\(i\)\}\|^\{2\}\}\{n\_\{l\}^\{2\}\}\\xrightarrow\{\\mathbb\{P\}\}\\zeta^\{2\}\>0asnl→∞n\_\{l\}\\to\\infty\. It follows that for sufficiently largenln\_\{l\},

ℙ\(d¯\(u,v\)\>\(1\+ε\)d\(u,v\),u↔v\)\\displaystyle\\mathbb\{P\}\\left\(\\bar\{d\}\(u,v\)\>\(1\+\\varepsilon\)d\(u,v\),\\;u\\leftrightarrow v\\right\)≥δ​ε′​ζ22\.\\displaystyle\\geq\\frac\{\\delta\\varepsilon^\{\\prime\}\\zeta^\{2\}\}\{2\}\.However, asnl→∞n\_\{l\}\\to\\infty, this logarithmic decay rate strictly dominates the polynomial upper bound established in \([16](https://arxiv.org/html/2607.10074#S7.E16)\) asδ​ε′​ζ22​logλ1⁡nl≫nl−4​δ\\frac\{\\delta\\varepsilon^\{\\prime\}\\zeta^\{2\}\}\{2\\log\_\{\\lambda\_\{1\}\}n\_\{l\}\}\\gg n\_\{l\}^\{\-4\\delta\}, yielding a clear contradiction\. Thus,\|Yc\|\|U\|​logλ1⁡n→ℙ0\\frac\{\|Y^\{c\}\|\}\{\|U\|\}\\log\_\{\\lambda\_\{1\}\}n\\xrightarrow\{\\mathbb\{P\}\}0, completing the proof\.

### 7\.9Proof of Theorem[5\.1](https://arxiv.org/html/2607.10074#S5.ThmtheoremEnv1)

#### Step 1\. Construction and Coupling\.

For anyδ\>0\\delta\>0, sinceκ∈L2​\(𝒳2,μ×μ\)\\kappa\\in L^\{2\}\(\\mathcal\{X\}^\{2\},\\mu\\times\\mu\)may be unbounded, we use a truncation thresholdMn\>0M\_\{n\}\>0to isolate the singularities withlimn→∞nMn=0\\lim\_\{n\\to\\infty\}\\frac\{n\}\{M\_\{n\}\}=0\. LetκMn​\(x,y\)=min⁡\(κ​\(x,y\),Mn\)\\kappa\_\{M\_\{n\}\}\(x,y\)=\\min\(\\kappa\(x,y\),M\_\{n\}\)be the truncated kernel\. We then construct a finite measurable partition𝒫δ=\{𝒳1,…,𝒳T\}\\mathcal\{P\}\_\{\\delta\}=\\\{\\mathcal\{X\}\_\{1\},\\dots,\\mathcal\{X\}\_\{T\}\\\}on the bounded domain to define the step functions:

κδ−​\(x,y\)=∑i,j=1T\(inf\(x,y\)∈Xi×Xj​κMn​\(x,y\)\)​𝟏Xi×Xj​\(x,y\)\\kappa\_\{\\delta\}^\{\-\}\(x,y\)=\\sum\_\{i,j=1\}^\{T\}\\left\(\\mathrm\{inf\}\_\{\(x,y\)\\in X\_\{i\}\\times X\_\{j\}\}\\kappa\_\{M\_\{n\}\}\(x,y\)\\right\)\\mathbf\{1\}\_\{X\_\{i\}\\times X\_\{j\}\}\(x,y\)κδ\+​\(x,y\)=∑i,j=1T\(sup\(x,y\)∈Xi×Xj​κMn​\(x,y\)\)​𝟏Xi×Xj​\(x,y\)\\kappa\_\{\\delta\}^\{\+\}\(x,y\)=\\sum\_\{i,j=1\}^\{T\}\\left\(\\mathrm\{sup\}\_\{\(x,y\)\\in X\_\{i\}\\times X\_\{j\}\}\\kappa\_\{M\_\{n\}\}\(x,y\)\\right\)\\mathbf\{1\}\_\{X\_\{i\}\\times X\_\{j\}\}\(x,y\)such that

‖κMn−κδ±‖2≤δ2\.\\\|\\kappa\_\{M\_\{n\}\}\-\\kappa\_\{\\delta\}^\{\\pm\}\\\|\_\{2\}\\leq\\frac\{\\delta\}\{2\}\.Consequently,κδ−​\(x,y\)≤κMn​\(x,y\)≤κδ\+​\(x,y\)\\kappa\_\{\\delta\}^\{\-\}\(x,y\)\\leq\\kappa\_\{M\_\{n\}\}\(x,y\)\\leq\\kappa\_\{\\delta\}^\{\+\}\(x,y\)\. On the other hand, the lower bound remains globally valid onκ\\kappa, i\.e\.κδ−​\(x,y\)≤κ​\(x,y\)\\kappa\_\{\\delta\}^\{\-\}\(x,y\)\\leq\\kappa\(x,y\)μ×μ\\mu\\times\\mu\-a\.e\., while the upper bound holds everywhere except on the singular tail setΩn=\{\(x,y\):κ​\(x,y\)\>Mn\}\\Omega\_\{n\}=\\\{\(x,y\):\\kappa\(x,y\)\>M\_\{n\}\\\}whereκδ\+​\(x,y\)≤κ​\(x,y\)\\kappa\_\{\\delta\}^\{\+\}\(x,y\)\\leq\\kappa\(x,y\)\. Sinceκ∈L2\\kappa\\in L^\{2\}, the global energy of the kernel is finite \(∬κ2​𝑑μ2<∞\\iint\\kappa^\{2\}\\,d\\mu^\{2\}<\\infty\), so

‖κ−κMn‖22=∬Ωn\|κ​\(x,y\)−Mn\|2​𝑑μ​\(x\)​𝑑μ​\(y\)\\\|\\kappa\-\\kappa\_\{M\_\{n\}\}\\\|\_\{2\}^\{2\}=\\iint\_\{\\Omega\_\{n\}\}\|\\kappa\(x,y\)\-M\_\{n\}\|^\{2\}\\,d\\mu\(x\)\\,d\\mu\(y\)is finite and vanishes asMn→∞M\_\{n\}\\to\\infty\. Sincelimn→∞nMn=0\\lim\_\{n\\to\\infty\}\\frac\{n\}\{M\_\{n\}\}=0, there exists anNδ\>0N\_\{\\delta\}\>0such that for alln≥Nδn\\geq N\_\{\\delta\}the tail mass satisfies‖κ−κMn‖2≤δ2\\\|\\kappa\-\\kappa\_\{M\_\{n\}\}\\\|\_\{2\}\\leq\\frac\{\\delta\}\{2\}\. Then by triangle inequality,

‖κ−κδ±‖2≤‖κ−κMn‖2\+‖κMn−κδ±‖2≤δ2\+δ2=δ\.\\\|\\kappa\-\\kappa\_\{\\delta\}^\{\\pm\}\\\|\_\{2\}\\leq\\\|\\kappa\-\\kappa\_\{M\_\{n\}\}\\\|\_\{2\}\+\\\|\\kappa\_\{M\_\{n\}\}\-\\kappa\_\{\\delta\}^\{\\pm\}\\\|\_\{2\}\\leq\\frac\{\\delta\}\{2\}\+\\frac\{\\delta\}\{2\}=\\delta\.
We define a coupling of the graphsGκδ−,GκMn,Gκδ\+G\_\{\\kappa\_\{\\delta\}^\{\-\}\},G\_\{\\kappa\_\{M\_\{n\}\}\},G\_\{\\kappa\_\{\\delta\}^\{\+\}\}, andGκG\_\{\\kappa\}by sampling latent positionsx1,…,xn∼μx\_\{1\},\\dots,x\_\{n\}\\sim\\muand independent edge variablesUi​j∼Uniform​\(0,1\)U\_\{ij\}\\sim\\mathrm\{Uniform\}\(0,1\)\. An edge\(i,j\)\(i,j\)exists inGfG\_\{f\}ifUi​j≤f​\(xi,xj\)/nU\_\{ij\}\\leq\{f\(x\_\{i\},x\_\{j\}\)\}/\{n\}\. Sinceκδ−≤κMn≤κδ\+\\kappa\_\{\\delta\}^\{\-\}\\leq\\kappa\_\{M\_\{n\}\}\\leq\\kappa\_\{\\delta\}^\{\+\}holds everywhere, we preserve the deterministic inclusionE​\(Gκδ−\)⊆E​\(GκMn\)⊆E​\(Gκδ\+\)E\(G\_\{\\kappa\_\{\\delta\}^\{\-\}\}\)\\subseteq E\(G\_\{\\kappa\_\{M\_\{n\}\}\}\)\\subseteq E\(G\_\{\\kappa\_\{\\delta\}^\{\+\}\}\)\. Let

Xn=∑1≤i<j≤n𝟏\{\(i,j\)∈E​\(Gκ\)∖E​\(GκMn\)\}\.X\_\{n\}=\\sum\_\{1\\leq i<j\\leq n\}\\mathbf\{1\}\_\{\\\{\(i,j\)\\in E\(G\_\{\\kappa\}\)\\setminus E\(G\_\{\\kappa\_\{M\_\{n\}\}\}\)\\\}\}\.It follows that

𝔼​\[Xn\]=∑1≤i<j≤nℙ​\(\(i,j\)∈E​\(Gκ\)∖E​\(GκMn\)\)\\mathbb\{E\}\[X\_\{n\}\]=\\sum\_\{1\\leq i<j\\leq n\}\\mathbb\{P\}\\big\(\(i,j\)\\in E\(G\_\{\\kappa\}\)\\setminus E\(G\_\{\\kappa\_\{M\_\{n\}\}\}\)\\big\)An edge\(i,j\)\(i,j\)lands in the singular discrepancy set if and only if its uniform variable satisfiesκMn​\(xi,xj\)n<Ui​j≤κ​\(xi,xj\)n\\frac\{\\kappa\_\{M\_\{n\}\}\(x\_\{i\},x\_\{j\}\)\}\{n\}<U\_\{ij\}\\leq\\frac\{\\kappa\(x\_\{i\},x\_\{j\}\)\}\{n\}\. Conditioning on the sampled latent positionsxi,xj∼μx\_\{i\},x\_\{j\}\\sim\\mu, the width of this window is:

ℙ​\(\(i,j\)∈E​\(Gκ\)∖E​\(GκMn\)∣xi,xj\)\\displaystyle\\mathbb\{P\}\\big\(\(i,j\)\\in E\(G\_\{\\kappa\}\)\\setminus E\(G\_\{\\kappa\_\{M\_\{n\}\}\}\)\\mid x\_\{i\},x\_\{j\}\\big\)=κ​\(xi,xj\)−κMn​\(xi,xj\)n\\displaystyle=\\frac\{\\kappa\(x\_\{i\},x\_\{j\}\)\-\\kappa\_\{M\_\{n\}\}\(x\_\{i\},x\_\{j\}\)\}\{n\}=κ​\(xi,xj\)−Mnn​𝟏Ωn​\(xi,xj\)\\displaystyle=\\frac\{\\kappa\(x\_\{i\},x\_\{j\}\)\-M\_\{n\}\}\{n\}\\mathbf\{1\}\_\{\\Omega\_\{n\}\}\(x\_\{i\},x\_\{j\}\)Taking the expectation over the latent space positions and noting that there are exactly\(n2\)=n​\(n−1\)2\\binom\{n\}\{2\}=\\frac\{n\(n\-1\)\}\{2\}identically distributed pairs, we get:

𝔼​\[Xn\]\\displaystyle\\mathbb\{E\}\[X\_\{n\}\]=n​\(n−1\)2⋅1n​∬Ωn\(κ​\(x,y\)−Mn\)​𝑑μ​\(x\)​𝑑μ​\(y\)\\displaystyle=\\frac\{n\(n\-1\)\}\{2\}\\cdot\\frac\{1\}\{n\}\\iint\_\{\\Omega\_\{n\}\}\\big\(\\kappa\(x,y\)\-M\_\{n\}\\big\)\\,d\\mu\(x\)\\,d\\mu\(y\)≤n2​∬Ωnκ​\(x,y\)​𝑑μ​\(x\)​𝑑μ​\(y\)\\displaystyle\\leq\\frac\{n\}\{2\}\\iint\_\{\\Omega\_\{n\}\}\\kappa\(x,y\)\\,d\\mu\(x\)\\,d\\mu\(y\)
On the domainΩn\\Omega\_\{n\}, the inequalityκ​\(x,y\)\>Mn\\kappa\(x,y\)\>M\_\{n\}implies1≤κ​\(x,y\)Mn1\\leq\\frac\{\\kappa\(x,y\)\}\{M\_\{n\}\}\. Substituting this into the integrand allows us to pull out the truncation ceiling:

𝔼​\[Xn\]\\displaystyle\\mathbb\{E\}\[X\_\{n\}\]≤n2​∬Ωnκ​\(x,y\)​𝑑μ​\(x\)​𝑑μ​\(y\)≤n2​∬Ωnκ​\(x,y\)​\(κ​\(x,y\)Mn\)​𝑑μ​\(x\)​𝑑μ​\(y\)\\displaystyle\\leq\\frac\{n\}\{2\}\\iint\_\{\\Omega\_\{n\}\}\\kappa\(x,y\)\\,d\\mu\(x\)\\,d\\mu\(y\)\\leq\\frac\{n\}\{2\}\\iint\_\{\\Omega\_\{n\}\}\\kappa\(x,y\)\\left\(\\frac\{\\kappa\(x,y\)\}\{M\_\{n\}\}\\right\)\\,d\\mu\(x\)\\,d\\mu\(y\)≤n2​1Mn​∬𝒳2κ2​\(x,y\)​𝑑μ​\(x\)​𝑑μ​\(y\)=n​‖κ‖222​Mn\.\\displaystyle\\leq\\frac\{n\}\{2\}\\frac\{1\}\{M\_\{n\}\}\\iint\_\{\\mathcal\{X\}^\{2\}\}\\kappa^\{2\}\(x,y\)\\,d\\mu\(x\)\\,d\\mu\(y\)=\\frac\{n\\\|\\kappa\\\|\_\{2\}^\{2\}\}\{2M\_\{n\}\}\.Applying Markov’s inequality,

ℙ​\(E​\(Gκ\)≠E​\(GκMn\)\)≤n​‖κ‖222​Mn⟶0as​n→∞,\\mathbb\{P\}\\big\(E\(G\_\{\\kappa\}\)\\neq E\(G\_\{\\kappa\_\{M\_\{n\}\}\}\)\\big\)\\leq\\frac\{n\\\|\\kappa\\\|\_\{2\}^\{2\}\}\{2M\_\{n\}\}\\longrightarrow 0\\quad\\text\{as \}n\\to\\infty,soE​\(Gκ\)=E​\(GκMn\)E\(G\_\{\\kappa\}\)=E\(G\_\{\\kappa\_\{M\_\{n\}\}\}\)w\.h\.p\., which yields the stochastic edge set inclusion:

E​\(Gκδ−\)⊆E​\(Gκ\)⊆E​\(Gκδ\+\)w\.h\.p\.E\(G\_\{\\kappa\_\{\\delta\}^\{\-\}\}\)\\subseteq E\(G\_\{\\kappa\}\)\\subseteq E\(G\_\{\\kappa\_\{\\delta\}^\{\+\}\}\)\\quad\\text\{w\.h\.p\.\}Since the shortest\-path distanced​\(u1,u2\)d\(u\_\{1\},u\_\{2\}\)is a monotonically non\-increasing function of the edge set, we obtain:

dκδ\+​\(u1,u2\)≤dκ​\(u1,u2\)≤dκδ−​\(u1,u2\)w\.h\.p\.d\_\{\\kappa\_\{\\delta\}^\{\+\}\}\(u\_\{1\},u\_\{2\}\)\\leq d\_\{\\kappa\}\(u\_\{1\},u\_\{2\}\)\\leq d\_\{\\kappa\_\{\\delta\}^\{\-\}\}\(u\_\{1\},u\_\{2\}\)\\quad\\text\{w\.h\.p\.\}

#### Step 2\. Spectral Convergence\.

Let𝒯κ,𝒯κδ−,𝒯κδ\+\\mathcal\{T\}\_\{\\kappa\},\\mathcal\{T\}\_\{\\kappa\_\{\\delta\}^\{\-\}\},\\mathcal\{T\}\_\{\\kappa\_\{\\delta\}^\{\+\}\}denote the integral operators associated with their respective kernels, acting onL2​\(𝒳,μ\)L^\{2\}\(\\mathcal\{X\},\\mu\)via\(Tf​ϕ\)​\(x\)=∫𝒳f​\(x,y\)​ϕ​\(y\)​𝑑μ​\(y\)\.\(T\_\{f\}\\phi\)\(x\)=\\int\_\{\\mathcal\{X\}\}f\(x,y\)\\phi\(y\)\\,d\\mu\(y\)\.\. Sinceκ∈L2​\(𝒳2,μ×μ\)\\kappa\\in L^\{2\}\(\\mathcal\{X\}^\{2\},\\mu\\times\\mu\), these are Hilbert\-Schmidt operators whose operator norms are strictly bounded by theL2L^\{2\}norm of their kernels\(Reed[1980](https://arxiv.org/html/2607.10074#bib.bib282), Theorem VI\.22\):

‖𝒯κ−𝒯κδ±‖\\displaystyle\\\|\\mathcal\{T\}\_\{\\kappa\}\-\\mathcal\{T\}\_\{\\kappa\_\{\\delta\}^\{\\pm\}\}\\\|≤‖𝒯κ−𝒯κδ±‖HS=‖κ−κδ±‖2≤δ\.\\displaystyle\\leq\\\|\\mathcal\{T\}\_\{\\kappa\}\-\\mathcal\{T\}\_\{\\kappa\_\{\\delta\}^\{\\pm\}\}\\\|\_\{\\mathrm\{HS\}\}=\\\|\\kappa\-\\kappa\_\{\\delta\}^\{\\pm\}\\\|\_\{2\}\\leq\\delta\.\(17\)
The primitivity ofκ\\kappaimplies that the principal eigenvalueλ1​\(𝒯κ\)\\lambda\_\{1\}\(\\mathcal\{T\}\_\{\\kappa\}\)is algebraically simple and isolated from the rest of the spectrum by a positive spectral gap\. By analytic perturbation theory for linear operators\(Kato and Katåo[1966](https://arxiv.org/html/2607.10074#bib.bib277), Theorem 3\.16\), the mapT↦λ1​\(T\)T\\mapsto\\lambda\_\{1\}\(T\)is locally Lipschitz continuous with respect to the operator norm topology\. Combining this directly with ourL2L^\{2\}\-driven operator norm bound‖𝒯κ−𝒯κδ±‖≤δ\\\|\\mathcal\{T\}\_\{\\kappa\}\-\\mathcal\{T\}\_\{\\kappa\_\{\\delta\}^\{\\pm\}\}\\\|\\leq\\delta, the bilateral spectral deviations scale linearly withδ\\delta:

λ1​\(𝒯κδ±\)=λ1​\(𝒯κ\)\+O​\(δ\)\.\\lambda\_\{1\}\(\\mathcal\{T\}\_\{\\kappa\_\{\\delta\}^\{\\pm\}\}\)=\\lambda\_\{1\}\(\\mathcal\{T\}\_\{\\kappa\}\)\+O\(\\delta\)\.
Since the true kernel is assumed to be supercritical \(λ1​\(𝒯κ\)\>1\\lambda\_\{1\}\(\\mathcal\{T\}\_\{\\kappa\}\)\>1\), choosingδ\\deltasufficiently small ensures thatλ1​\(𝒯κδ−\)\>1\\lambda\_\{1\}\(\\mathcal\{T\}\_\{\\kappa\_\{\\delta\}^\{\-\}\}\)\>1\. Furthermore, by construction,κδ−​\(x,y\)≤κδ\+​\(x,y\)\\kappa\_\{\\delta\}^\{\-\}\(x,y\)\\leq\\kappa\_\{\\delta\}^\{\+\}\(x,y\)andκδ−​\(x,y\)≤κ​\(x,y\)\\kappa\_\{\\delta\}^\{\-\}\(x,y\)\\leq\\kappa\(x,y\)holdμ×μ\\mu\\times\\mu\-a\.e\. globally across the entire domain\. By the monotonicity property of the spectral radius for positive operators\(Schaefer[1974](https://arxiv.org/html/2607.10074#bib.bib278), Proposition 4\.1\), this entrywise operator dominance yieldsλ1​\(𝒯κδ−\)≤min⁡\{λ1​\(𝒯κδ\+\),λ1​\(𝒯κ\)\}\\lambda\_\{1\}\(\\mathcal\{T\}\_\{\\kappa\_\{\\delta\}^\{\-\}\}\)\\leq\\min\\big\\\{\\lambda\_\{1\}\(\\mathcal\{T\}\_\{\\kappa\_\{\\delta\}^\{\+\}\}\),\\lambda\_\{1\}\(\\mathcal\{T\}\_\{\\kappa\}\)\\big\\\}\.

#### Step 3\. Primitivity ofκδ±\\kappa\_\{\\delta\}^\{\\pm\}\.

Since the truncationκMn​\(x,y\)=min⁡\(κ​\(x,y\),Mn\)\\kappa\_\{M\_\{n\}\}\(x,y\)=\\min\(\\kappa\(x,y\),M\_\{n\}\)only dampens large values without altering the kernel’s support,κMn\\kappa\_\{M\_\{n\}\}natively inherits primitivity fromκ\\kappa\. The pointwise dominanceκδ\+​\(x,y\)≥κMn​\(x,y\)\\kappa\_\{\\delta\}^\{\+\}\(x,y\)\\geq\\kappa\_\{M\_\{n\}\}\(x,y\)everywhere on𝒳2\\mathcal\{X\}^\{2\}ensures thatκδ\+\\kappa\_\{\\delta\}^\{\+\}preserves these positive\-measure pathways, thereby inheriting primitivity as well\.

Conversely, while blockwise infima can introduce zeros, the primitivity ofκδ−\\kappa\_\{\\delta\}^\{\-\}is preserved for a sufficiently smallδ\>0\\delta\>0\. Becauseκ\\kappais primitive,𝒯κ\\mathcal\{T\}\_\{\\kappa\}possesses an isolated, simple dominant eigenvalueλ1​\(𝒯κ\)\\lambda\_\{1\}\(\\mathcal\{T\}\_\{\\kappa\}\)and a strictly positive principal eigenfunctionϕ1≥c\>0\\phi\_\{1\}\\geq c\>0μ\\mu\-a\.e\.\(Schaefer[1974](https://arxiv.org/html/2607.10074#bib.bib278)\)\. By the spectral stability of simple eigenvalues and projections\(Kato and Katåo[1966](https://arxiv.org/html/2607.10074#bib.bib277)\), under the operator norm perturbation‖𝒯κ−𝒯κδ−‖≤δ\\\|\\mathcal\{T\}\_\{\\kappa\}\-\\mathcal\{T\}\_\{\\kappa\_\{\\delta\}^\{\-\}\}\\\|\\leq\\deltain \([17](https://arxiv.org/html/2607.10074#S7.E17)\), the operator𝒯κδ−\\mathcal\{T\}\_\{\\kappa\_\{\\delta\}^\{\-\}\}retains a unique dominant eigenvalueλ1​\(𝒯κδ−\)\\lambda\_\{1\}\(\\mathcal\{T\}\_\{\\kappa\_\{\\delta\}^\{\-\}\}\), and its principal eigenfunction satisfies‖ϕ1−−ϕ1‖2→0\\\|\\phi\_\{1\}^\{\-\}\-\\phi\_\{1\}\\\|\_\{2\}\\to 0asδ→0\\delta\\to 0\. For a sufficiently smallδ\\delta, thisL2L^\{2\}convergence guarantees thatϕ1−\>0\\phi\_\{1\}^\{\-\}\>0μ\\mu\-a\.e\., thereby ruling out both periodicity and reducibility \(via Theorems 3\.16 and 3\.5\)\. Consequently,κδ−\\kappa\_\{\\delta\}^\{\-\}inherits primitivity\.

### 7\.10Proof of Theorem[5\.2](https://arxiv.org/html/2607.10074#S5.ThmtheoremEnv2)

#### Step 1: Kernel Approximation\.

Letε′∈\(0,ε\)\\varepsilon^\{\\prime\}\\in\(0,\\varepsilon\)such that\(1\+ε′\)2<1\+ε\(1\+\\varepsilon^\{\\prime\}\)^\{2\}<1\+\\varepsilon\. By Theorem[5\.1](https://arxiv.org/html/2607.10074#S5.ThmtheoremEnv1), there exists aδ\>0\\delta\>0with primitive step\-function kernelsκδ−\\kappa\_\{\\delta\}^\{\-\}andκδ\+\\kappa\_\{\\delta\}^\{\+\}such that‖κ−κδ±‖2≤δ\\\|\\kappa\-\\kappa\_\{\\delta\}^\{\\pm\}\\\|\_\{2\}\\leq\\delta\. As shown in the proof of Theorem[5\.1](https://arxiv.org/html/2607.10074#S5.ThmtheoremEnv1), we can construct from these step\-function kernels finite\-type inhomogeneous random graphsGκδ−G\_\{\\kappa\_\{\\delta\}^\{\-\}\}andGκδ\+G\_\{\\kappa\_\{\\delta\}^\{\+\}\}onTTtypes via a monotone coupling such that

E​\(Gκδ−\)⊆E​\(Gκ\)⊆E​\(Gκδ\+\)w\.h\.p\.E\(G\_\{\\kappa\_\{\\delta\}^\{\-\}\}\)\\subseteq E\(G\_\{\\kappa\}\)\\subseteq E\(G\_\{\\kappa\_\{\\delta\}^\{\+\}\}\)\\quad\\text\{w\.h\.p\.\}\(18\)and

1<λ1​\(𝒯κδ−\)≤min⁡\{λ1​\(𝒯κδ\+\),λ1​\(𝒯κ\)\}withλ1​\(𝒯κδ±\)=λ1​\(𝒯κ\)±O​\(δ\)\.\\displaystyle 1<\\lambda\_\{1\}\(\\mathcal\{T\}\_\{\\kappa\_\{\\delta\}^\{\-\}\}\)\\leq\\min\\\{\\lambda\_\{1\}\(\\mathcal\{T\}\_\{\\kappa\_\{\\delta\}^\{\+\}\}\),\\lambda\_\{1\}\(\\mathcal\{T\}\_\{\\kappa\}\)\\\}\\quad\\text\{with\}\\quad\\lambda\_\{1\}\(\\mathcal\{T\}\_\{\\kappa\_\{\\delta\}^\{\\pm\}\}\)=\\lambda\_\{1\}\(\\mathcal\{T\}\_\{\\kappa\}\)\\pm O\(\\delta\)\.\(19\)

#### Step 2: Landmark Estimation on Finite Types\.

SinceGκδ−G\_\{\\kappa\_\{\\delta\}^\{\-\}\}andGκδ\+G\_\{\\kappa\_\{\\delta\}^\{\+\}\}are finite\-type graphs with primitive step functions, they satisfy Assumptions[3\.1](https://arxiv.org/html/2607.10074#S3.ThmtheoremEnv1)–[3\.3](https://arxiv.org/html/2607.10074#S3.ThmtheoremEnv3)\. Applying Theorems[4\.1](https://arxiv.org/html/2607.10074#S4.ThmtheoremEnv1)and[4\.2](https://arxiv.org/html/2607.10074#S4.ThmtheoremEnv2)under the required conditions forθ,r,R\\theta,r,R, the estimatorsd¯\\underline\{d\}andd¯\\overline\{d\}constructed onGκG\_\{\\kappa\}satisfy:

\(1−ε′\)​dκδ\+​\(u1,u2\)≤d¯κδ\+​\(u1,u2\)andd¯κδ−​\(u1,u2\)≤\(1\+ε′\)​dκδ−​\(u1,u2\)w\.h\.p\.\\displaystyle\(1\-\\varepsilon^\{\\prime\}\)d\_\{\\kappa\_\{\\delta\}^\{\+\}\}\(u\_\{1\},u\_\{2\}\)\\leq\\underline\{d\}\_\{\\kappa\_\{\\delta\}^\{\+\}\}\(u\_\{1\},u\_\{2\}\)\\quad\\text\{and\}\\quad\\overline\{d\}\_\{\\kappa\_\{\\delta\}^\{\-\}\}\(u\_\{1\},u\_\{2\}\)\\leq\(1\+\\varepsilon^\{\\prime\}\)d\_\{\\kappa\_\{\\delta\}^\{\-\}\}\(u\_\{1\},u\_\{2\}\)\\quad\\text\{w\.h\.p\.\}\(20\)

#### Step 3: Upper\-Bound Distortion\.

As explained in Section[2\.2](https://arxiv.org/html/2607.10074#S2.SS2),

d¯​\(u1,u2\)=mini,j∈\{0,…,\(r\+1\)​R−1\}⁡\{d​\(u1,Si\)\+d​\(u2,Sj\):arg⁡mins∈Si⁡d​\(u1,s\)=arg⁡mins∈Sj⁡d​\(u2,s\)\},\\bar\{d\}\(u\_\{1\},u\_\{2\}\)=\\min\_\{i,j\\in\\\{0,\\dots,\(r\+1\)R\-1\\\}\}\\left\\\{d\(u\_\{1\},S\_\{i\}\)\+d\(u\_\{2\},S\_\{j\}\):\\arg\\min\_\{s\\in S\_\{i\}\}d\(u\_\{1\},s\)=\\arg\\min\_\{s\\in S\_\{j\}\}d\(u\_\{2\},s\)\\right\\\},whered​\(u,S\)=mins∈S⁡d​\(u,s\)d\(u,S\)=\\min\_\{s\\in S\}d\(u,s\)denotes the distance betweenuuand the closest landmark inSS\. Sinced¯\\bar\{d\}concerns common landmarks, without loss of generality we write

d¯κδ−​\(u1,u2\)=dκδ−​\(u1,s∗\)\+dκδ−​\(u2,s∗\)\\overline\{d\}\_\{\\kappa\_\{\\delta\}^\{\-\}\}\(u\_\{1\},u\_\{2\}\)=d\_\{\\kappa\_\{\\delta\}^\{\-\}\}\(u\_\{1\},s^\{\*\}\)\+d\_\{\\kappa\_\{\\delta\}^\{\-\}\}\(u\_\{2\},s^\{\*\}\)for some landmarks∗s^\{\*\}\. Since shortest\-path distances are monotone with respect to edge addition, \([18](https://arxiv.org/html/2607.10074#S7.E18)\) implies

d¯κδ−​\(u1,u2\)=dκδ−​\(u1,s∗\)\+dκδ−​\(u2,s∗\)≥dκ​\(u1,s∗\)\+dκ​\(u2,s∗\)≥d¯κ​\(u1,u2\)w\.h\.p\.\\overline\{d\}\_\{\\kappa\_\{\\delta\}^\{\-\}\}\(u\_\{1\},u\_\{2\}\)=d\_\{\\kappa\_\{\\delta\}^\{\-\}\}\(u\_\{1\},s^\{\*\}\)\+d\_\{\\kappa\_\{\\delta\}^\{\-\}\}\(u\_\{2\},s^\{\*\}\)\\geq d\_\{\\kappa\}\(u\_\{1\},s^\{\*\}\)\+d\_\{\\kappa\}\(u\_\{2\},s^\{\*\}\)\\geq\\overline\{d\}\_\{\\kappa\}\(u\_\{1\},u\_\{2\}\)\\quad\\text\{w\.h\.p\.\}Combining this with \([20](https://arxiv.org/html/2607.10074#S7.E20)\), we obtain

d¯κ​\(u1,u2\)≤d¯κδ−​\(u1,u2\)≤\(1\+ε′\)​dκδ−​\(u1,u2\)w\.h\.p\.\\overline\{d\}\_\{\\kappa\}\(u\_\{1\},u\_\{2\}\)\\leq\\overline\{d\}\_\{\\kappa\_\{\\delta\}^\{\-\}\}\(u\_\{1\},u\_\{2\}\)\\leq\(1\+\\varepsilon^\{\\prime\}\)d\_\{\\kappa\_\{\\delta\}^\{\-\}\}\(u\_\{1\},u\_\{2\}\)\\quad\\text\{w\.h\.p\.\}
To complete the proof for the upper\-bound distortion, it suffices to show thatdκδ−​\(u1,u2\)≤\(1\+ε′\)​dκ​\(u1,u2\)d\_\{\\kappa\_\{\\delta\}^\{\-\}\}\(u\_\{1\},u\_\{2\}\)\\leq\(1\+\\varepsilon^\{\\prime\}\)d\_\{\\kappa\}\(u\_\{1\},u\_\{2\}\)w\.h\.p\. Conditionally onu1,u2u\_\{1\},u\_\{2\}being in the same connected component, Theorem 6\.2 from\\swapHofstadvan der \([2024b](https://arxiv.org/html/2607.10074#bib.bib10)\)implies thatd​\(u1,u2\)/logλ1⁡n→ℙ1d\(u\_\{1\},u\_\{2\}\)/\\log\_\{\\lambda\_\{1\}\}n\\xrightarrow\{\\scriptscriptstyle\\mathbb\{P\}\}1\. Thus, for any fixedϵ\>0\\epsilon\>0, the typical distances satisfy

\(1−ϵ\)​logλ1​\(𝒯κ\)⁡n≤dκ​\(u1,u2\)≤\(1\+ϵ\)​logλ1​\(𝒯κ\)⁡nw\.h\.p\.\(1\-\\epsilon\)\\log\_\{\\lambda\_\{1\}\(\\mathcal\{T\}\_\{\\kappa\}\)\}n\\leq d\_\{\\kappa\}\(u\_\{1\},u\_\{2\}\)\\leq\(1\+\\epsilon\)\\log\_\{\\lambda\_\{1\}\(\\mathcal\{T\}\_\{\\kappa\}\)\}n\\quad\\text\{w\.h\.p\.\}and similarly fordκδ−​\(u1,u2\)d\_\{\\kappa\_\{\\delta\}^\{\-\}\}\(u\_\{1\},u\_\{2\}\)with respect toλ1​\(𝒯κδ−\)\\lambda\_\{1\}\(\\mathcal\{T\}\_\{\\kappa\_\{\\delta\}^\{\-\}\}\)\. It follows that for anyϵ\>0\\epsilon\>0and sufficiently largenn, the distance ratio satisfies

dκδ−​\(u1,u2\)dκ​\(u1,u2\)≤\(1\+ϵ\)​log⁡λ1​\(𝒯κ\)\(1−ϵ\)​log⁡λ1​\(𝒯κδ−\)w\.h\.p\.\\frac\{d\_\{\\kappa\_\{\\delta\}^\{\-\}\}\(u\_\{1\},u\_\{2\}\)\}\{d\_\{\\kappa\}\(u\_\{1\},u\_\{2\}\)\}\\leq\\frac\{\(1\+\\epsilon\)\\log\\lambda\_\{1\}\(\\mathcal\{T\}\_\{\\kappa\}\)\}\{\(1\-\\epsilon\)\\log\\lambda\_\{1\}\(\\mathcal\{T\}\_\{\\kappa\_\{\\delta\}^\{\-\}\}\)\}\\quad\\text\{w\.h\.p\.\}Since the functiong​\(λ\)=1log⁡λg\(\\lambda\)=\\frac\{1\}\{\\log\\lambda\}is continuous and differentiable forλ\>1\\lambda\>1, the Mean Value Theorem paired with the spectral proximityλ1​\(𝒯κδ−\)=λ1​\(𝒯κ\)−O​\(δ\)\\lambda\_\{1\}\(\\mathcal\{T\}\_\{\\kappa\_\{\\delta\}^\{\-\}\}\)=\\lambda\_\{1\}\(\\mathcal\{T\}\_\{\\kappa\}\)\-O\(\\delta\)from \([19](https://arxiv.org/html/2607.10074#S7.E19)\) implies

0≤1log⁡λ1​\(𝒯κδ−\)−1log⁡λ1​\(𝒯κ\)≤C​δ0\\leq\\frac\{1\}\{\\log\\lambda\_\{1\}\(\\mathcal\{T\}\_\{\\kappa\_\{\\delta\}^\{\-\}\}\)\}\-\\frac\{1\}\{\\log\\lambda\_\{1\}\(\\mathcal\{T\}\_\{\\kappa\}\)\}\\leq C\\deltawhereC=c0​supλ∈\[λ1​\(𝒯κ\)−δ,λ1​\(𝒯κ\)\]\|g′​\(λ\)\|C=c\_\{0\}\\sup\_\{\\lambda\\in\[\\lambda\_\{1\}\(\\mathcal\{T\}\_\{\\kappa\}\)\-\\delta,\\,\\lambda\_\{1\}\(\\mathcal\{T\}\_\{\\kappa\}\)\]\}\\left\|g^\{\\prime\}\(\\lambda\)\\right\|for a structural constantc0\>0c\_\{0\}\>0matching the error bound in \([19](https://arxiv.org/html/2607.10074#S7.E19)\)\. This supreme remains strictly bounded becauseg′​\(λ\)=−1λ​\(log⁡λ\)2g^\{\\prime\}\(\\lambda\)=\\frac\{\-1\}\{\\lambda\(\\log\\lambda\)^\{2\}\}is non\-zero for allλ\>1\\lambda\>1\. By selectingδ\\deltaandϵ\\epsilonsufficiently small, we guarantee that

\(1\+ϵ\)​log⁡λ1​\(𝒯κ\)\(1−ϵ\)​log⁡λ1​\(𝒯κδ−\)≤1\+ϵ1−ϵ​\(1\+C​δ​log⁡λ1​\(𝒯κ\)\)≤1\+ε′\\frac\{\(1\+\\epsilon\)\\log\\lambda\_\{1\}\(\\mathcal\{T\}\_\{\\kappa\}\)\}\{\(1\-\\epsilon\)\\log\\lambda\_\{1\}\(\\mathcal\{T\}\_\{\\kappa\_\{\\delta\}^\{\-\}\}\)\}\\leq\\frac\{1\+\\epsilon\}\{1\-\\epsilon\}\(1\+C\\delta\\log\\lambda\_\{1\}\(\\mathcal\{T\}\_\{\\kappa\}\)\)\\leq 1\+\\varepsilon^\{\\prime\}which yieldsdκδ−​\(u1,u2\)≤\(1\+ε′\)​dκ​\(u1,u2\)d\_\{\\kappa\_\{\\delta\}^\{\-\}\}\(u\_\{1\},u\_\{2\}\)\\leq\(1\+\\varepsilon^\{\\prime\}\)d\_\{\\kappa\}\(u\_\{1\},u\_\{2\}\)w\.h\.p\. Compounding these bounds into our original chain yields:

d¯κ​\(u1,u2\)≤\(1\+ε′\)​dκδ−​\(u1,u2\)≤\(1\+ε′\)2​dκ​\(u1,u2\)≤\(1\+ε\)​dκ​\(u1,u2\)w\.h\.p\.\\overline\{d\}\_\{\\kappa\}\(u\_\{1\},u\_\{2\}\)\\leq\(1\+\\varepsilon^\{\\prime\}\)d\_\{\\kappa\_\{\\delta\}^\{\-\}\}\(u\_\{1\},u\_\{2\}\)\\leq\(1\+\\varepsilon^\{\\prime\}\)^\{2\}d\_\{\\kappa\}\(u\_\{1\},u\_\{2\}\)\\leq\(1\+\\varepsilon\)d\_\{\\kappa\}\(u\_\{1\},u\_\{2\}\)\\quad\\text\{w\.h\.p\.\}

#### Step 4: Lower\-Bound Distortion\.

To proved¯κ​\(u1,u2\)≥\(1−ε\)​dκ​\(u1,u2\)\\underline\{d\}\_\{\\kappa\}\(u\_\{1\},u\_\{2\}\)\\geq\(1\-\\varepsilon\)d\_\{\\kappa\}\(u\_\{1\},u\_\{2\}\)w\.h\.p\., we use proof techniques similar to those in the proof of Theorem[4\.1](https://arxiv.org/html/2607.10074#S4.ThmtheoremEnv1)\. LetZZbe defined as in the proof of Theorem[4\.1](https://arxiv.org/html/2607.10074#S4.ThmtheoremEnv1)andXXbe the event thatu1,u2u\_\{1\},u\_\{2\}are in the giant component for the graphGκG\_\{\\kappa\}\. Let\(Z\+,X\+\)\(Z^\{\+\},X^\{\+\}\)and\(Z−,X−\)\(Z^\{\-\},X^\{\-\}\)denote the corresponding events for the graphsGκδ\+G\_\{\\kappa\_\{\\delta\}^\{\+\}\}andGκδ−G\_\{\\kappa\_\{\\delta\}^\{\-\}\}, respectively\. From \([18](https://arxiv.org/html/2607.10074#S7.E18)\), we haveX−⊆X⊆X\+X^\{\-\}\\subseteq X\\subseteq X^\{\+\}w\.h\.p\., so

ℙ​\(Z\|X\)\\displaystyle\\mathbb\{P\}\(Z\|X\)≥ℙ​\(Z∩X−\|X\)=ℙ​\(Z∩X−∩X\)ℙ​\(X\)=ℙ​\(Z∩X−\)ℙ​\(X−\)​ℙ​\(X−\)ℙ​\(X\)=ℙ​\(Z\|X−\)​ℙ​\(X−\)ℙ​\(X\)\.\\displaystyle\\geq\\mathbb\{P\}\(Z\\cap X^\{\-\}\|X\)=\\frac\{\\mathbb\{P\}\(Z\\cap X^\{\-\}\\cap X\)\}\{\\mathbb\{P\}\(X\)\}=\\frac\{\\mathbb\{P\}\(Z\\cap X^\{\-\}\)\}\{\\mathbb\{P\}\(X^\{\-\}\)\}\\frac\{\\mathbb\{P\}\(X^\{\-\}\)\}\{\\mathbb\{P\}\(X\)\}=\\mathbb\{P\}\(Z\|X^\{\-\}\)\\frac\{\\mathbb\{P\}\(X^\{\-\}\)\}\{\\mathbb\{P\}\(X\)\}\.
ByBollobáset al\.\([2007](https://arxiv.org/html/2607.10074#bib.bib264), Theorem 3\.1\), supercriticality from \([19](https://arxiv.org/html/2607.10074#S7.E19)\) ensures that the giant component probabilities match the integrated branching survival profiles∫𝒳ζκ​𝑑μ\\int\_\{\\mathcal\{X\}\}\\zeta\_\{\\kappa\}\\,d\\muand∫𝒳ζκδ−​𝑑μ\\int\_\{\\mathcal\{X\}\}\\zeta\_\{\\kappa\_\{\\delta\}^\{\-\}\}\\,d\\mu\. Sinceμ​\(𝒳\)=1\\mu\(\\mathcal\{X\}\)=1, the Cauchy–Schwarz inequality bridges the norms via‖κ−κδ−‖1≤‖κ−κδ−‖2≤δ\\\|\\kappa\-\\kappa\_\{\\delta\}^\{\-\}\\\|\_\{1\}\\leq\\\|\\kappa\-\\kappa\_\{\\delta\}^\{\-\}\\\|\_\{2\}\\leq\\delta, so‖ζκ−ζκδ−‖1=O​\(δ\)\\\|\\zeta\_\{\\kappa\}\-\\zeta\_\{\\kappa\_\{\\delta\}^\{\-\}\}\\\|\_\{1\}=O\(\\delta\)follows fromBollobáset al\.\([2007](https://arxiv.org/html/2607.10074#bib.bib264), Theorem 6\.4\)which establishes the qualitative continuity of these profiles underL1L^\{1\}kernel perturbations\. Consequently, using the algebraic identitya2−b2=\(a\+b\)​\(a−b\)a^\{2\}\-b^\{2\}=\(a\+b\)\(a\-b\)and noting that both survival functions are bounded above by11, we obtain

ℙ​\(X\)−ℙ​\(X−\)≤2​∫𝒳\|ζκ​\(x\)−ζκδ−​\(x\)\|​𝑑μ​\(x\)=2​‖ζκ−ζκδ−‖1=O​\(δ\)\.\\mathbb\{P\}\(X\)\-\\mathbb\{P\}\(X^\{\-\}\)\\leq 2\\int\_\{\\mathcal\{X\}\}\\left\|\\zeta\_\{\\kappa\}\(x\)\-\\zeta\_\{\\kappa\_\{\\delta\}^\{\-\}\}\(x\)\\right\|d\\mu\(x\)=2\\\|\\zeta\_\{\\kappa\}\-\\zeta\_\{\\kappa\_\{\\delta\}^\{\-\}\}\\\|\_\{1\}=O\(\\delta\)\.Given thatℙ​\(x∈𝒞\(1\)\|κ\)\\mathbb\{P\}\(x\\in\\mathcal\{C\}\_\{\(1\)\}\|\\kappa\)is strictly bounded away from zero by the supercriticality ofκ\\kappa\(Bollobáset al\.[2007](https://arxiv.org/html/2607.10074#bib.bib264), Theorem 3\.1\), we obtainℙ​\(X\)\>0\\mathbb\{P\}\(X\)\>0, which implies

ℙ​\(X−\)ℙ​\(X\)=1−ℙ​\(X\)−ℙ​\(X−\)ℙ​\(X\)≥1−2​‖ζκ−ζκδ−‖2ℙ​\(X\)=1−O​\(δ\)\.\\frac\{\\mathbb\{P\}\(X^\{\-\}\)\}\{\\mathbb\{P\}\(X\)\}=1\-\\frac\{\\mathbb\{P\}\(X\)\-\\mathbb\{P\}\(X^\{\-\}\)\}\{\\mathbb\{P\}\(X\)\}\\geq 1\-\\frac\{2\\\|\\zeta\_\{\\kappa\}\-\\zeta\_\{\\kappa\_\{\\delta\}^\{\-\}\}\\\|\_\{2\}\}\{\\mathbb\{P\}\(X\)\}=1\-O\(\\delta\)\.Takingδ→0\\delta\\to 0yieldsℙ​\(X−\)ℙ​\(X\)→1\\frac\{\\mathbb\{P\}\(X^\{\-\}\)\}\{\\mathbb\{P\}\(X\)\}\\to 1, which implies

ℙ​\(Z\|X\)\\displaystyle\\mathbb\{P\}\(Z\|X\)≥ℙ​\(Z\|X−\)​ℙ​\(X−\)ℙ​\(X\)→ℙ​\(Z\|X−\)\\displaystyle\\geq\\mathbb\{P\}\(Z\|X^\{\-\}\)\\frac\{\\mathbb\{P\}\(X^\{\-\}\)\}\{\\mathbb\{P\}\(X\)\}\\to\\mathbb\{P\}\(Z\|X^\{\-\}\)
Next, we lower boundℙ​\(Z∣X−\)\\mathbb\{P\}\(Z\\mid X^\{\-\}\)by upper boundingℙ​\(Zc∣X−\)\\mathbb\{P\}\(Z^\{c\}\\mid X^\{\-\}\)\. Recall the definitions of the neighborhoodNk​\(u\)N\_\{k\}\(u\)and the boundary∂Nk​\(u\)\\partial N\_\{k\}\(u\)as formulated in the proof of Theorem[4\.1](https://arxiv.org/html/2607.10074#S4.ThmtheoremEnv1)for the graphGκG\_\{\\kappa\}\. Analogously, we define\(Nk\+​\(u\),∂Nk\+​\(u\)\)\(N^\{\+\}\_\{k\}\(u\),\\partial N^\{\+\}\_\{k\}\(u\)\)and\(Nk−​\(u\),∂Nk−​\(u\)\)\(N^\{\-\}\_\{k\}\(u\),\\partial N^\{\-\}\_\{k\}\(u\)\)to be the corresponding node sets in the random graphsGκδ\+G\_\{\\kappa\_\{\\delta\}^\{\+\}\}andGκδ−G\_\{\\kappa\_\{\\delta\}^\{\-\}\}, respectively\. We also definek1=ε′​dκ​\(u1,u2\)k\_\{1\}=\\varepsilon^\{\\prime\}d\_\{\\kappa\}\(u\_\{1\},u\_\{2\}\)andk2=\(1−ε\+ε′\)​dκ​\(u1,u2\)k\_\{2\}=\(1\-\\varepsilon\+\\varepsilon^\{\\prime\}\)d\_\{\\kappa\}\(u\_\{1\},u\_\{2\}\)with

0<ε′=min⁡\{ε2,ε−θ\}−ε′′<min⁡\{ε2,ε−θ\}\\displaystyle 0<\\varepsilon^\{\\prime\}=\\min\\left\\\{\\frac\{\\varepsilon\}\{2\},\\varepsilon\-\\theta\\right\\\}\-\\varepsilon^\{\\prime\\prime\}<\\min\\left\\\{\\frac\{\\varepsilon\}\{2\},\\varepsilon\-\\theta\\right\\\}for someε′′∈\(0,min⁡\{ε2,ε−θ\}\)\\varepsilon^\{\\prime\\prime\}\\in\\left\(0,\\min\\left\\\{\\frac\{\\varepsilon\}\{2\},\\varepsilon\-\\theta\\right\\\}\\right\)chosen so thatNk1​\(u1\)∩Nk2​\(u2\)=∅N\_\{k\_\{1\}\}\(u\_\{1\}\)\\cap N\_\{k\_\{2\}\}\(u\_\{2\}\)=\\varnothing\. Conditioning throughout on the eventX−X^\{\-\}, we have similar to \([12](https://arxiv.org/html/2607.10074#S7.E12)\) that

ℙ​\(Zc∣X−\)​𝟏X−\\displaystyle\\mathbb\{P\}\(Z^\{c\}\\mid X^\{\-\}\)\\mathbf\{1\}\_\{X^\{\-\}\}≤exp⁡\(−R​\|∂Nk1​\(u1\)\|n​Mr​\(1−\|Nk1​\(u1\)\|\+\|Nk2​\(u2\)\|n\)Mr−1\)​𝟏X−\\displaystyle\\leq\\exp\\bigg\(\-R\\frac\{\|\\partial N\_\{k\_\{1\}\}\(u\_\{1\}\)\|\}\{n\}M^\{r\}\\bigg\(1\-\\frac\{\|N\_\{k\_\{1\}\}\(u\_\{1\}\)\|\+\|N\_\{k\_\{2\}\}\(u\_\{2\}\)\|\}\{n\}\\bigg\)^\{M^\{r\}\-1\}\\bigg\)\\mathbf\{1\}\_\{X^\{\-\}\}≤exp⁡\(−R​\|∂Nk1−​\(u1\)\|n​Mr​\(1−\|Nk1\+​\(u1\)\|\+\|Nk2\+​\(u2\)\|n\)Mr−1\)​𝟏X−,\\displaystyle\\leq\\exp\\bigg\(\-R\\frac\{\|\\partial N^\{\-\}\_\{k\_\{1\}\}\(u\_\{1\}\)\|\}\{n\}M^\{r\}\\bigg\(1\-\\frac\{\|N^\{\+\}\_\{k\_\{1\}\}\(u\_\{1\}\)\|\+\|N^\{\+\}\_\{k\_\{2\}\}\(u\_\{2\}\)\|\}\{n\}\\bigg\)^\{M^\{r\}\-1\}\\bigg\)\\mathbf\{1\}\_\{X^\{\-\}\},where ”≤\\leq” follows from the monotone edge coupling in \([18](https://arxiv.org/html/2607.10074#S7.E18)\)\.

Conditioned onX−X^\{\-\}withX−⊆X⊆\{u1,u2​are in the same connected component of​Gκ\}X^\{\-\}\\subseteq X\\subseteq\\\{u\_\{1\},u\_\{2\}\\text\{ are in the same connected component of \}G\_\{\\kappa\}\\\}, Theorem 6\.2 from\\swapHofstadvan der \([2024b](https://arxiv.org/html/2607.10074#bib.bib10)\)implies thatdκ​\(u1,u2\)/logλ1​\(𝒯κ\)⁡n→ℙ1d\_\{\\kappa\}\(u\_\{1\},u\_\{2\}\)/\\log\_\{\\lambda\_\{1\}\(\\mathcal\{T\}\_\{\\kappa\}\)\}n\\xrightarrow\{\\mathbb\{P\}\}1\. In other words,

\(1−ϵ\)​logλ1​\(𝒯κ\)⁡n≤dκ​\(u1,u2\)≤\(1\+ϵ\)​logλ1​\(𝒯κ\)⁡nw\.h\.p\.\(1\-\\epsilon\)\\log\_\{\\lambda\_\{1\}\(\\mathcal\{T\}\_\{\\kappa\}\)\}n\\leq d\_\{\\kappa\}\(u\_\{1\},u\_\{2\}\)\\leq\(1\+\\epsilon\)\\log\_\{\\lambda\_\{1\}\(\\mathcal\{T\}\_\{\\kappa\}\)\}n\\quad\\text\{w\.h\.p\.\}for any fixedϵ\>0\\epsilon\>0\. Choosingϵ\\epsilonsmall enough so thatε′​ϵ<1−ε′\\varepsilon^\{\\prime\}\\epsilon<1\-\\varepsilon^\{\\prime\}, we obtain

k1≤ε′​\(1\+ϵ\)​logλ1​\(𝒯κ\)⁡n<log⁡λ1​\(𝒯κδ−\)log⁡λ1​\(𝒯κ\)​logλ1​\(𝒯κδ−\)⁡n≤logλ1​\(𝒯κδ−\)⁡nw\.h\.p\.,k\_\{1\}\\leq\\varepsilon^\{\\prime\}\(1\+\\epsilon\)\\log\_\{\\lambda\_\{1\}\(\\mathcal\{T\}\_\{\\kappa\}\)\}n<\\frac\{\\log\{\\lambda\_\{1\}\(\\mathcal\{T\}\_\{\\kappa\_\{\\delta\}^\{\-\}\}\)\}\}\{\\log\{\\lambda\_\{1\}\(\\mathcal\{T\}\_\{\\kappa\}\)\}\}\\log\_\{\\lambda\_\{1\}\(\\mathcal\{T\}\_\{\\kappa\_\{\\delta\}^\{\-\}\}\)\}n\\leq\\log\_\{\\lambda\_\{1\}\(\\mathcal\{T\}\_\{\\kappa\_\{\\delta\}^\{\-\}\}\)\}n\\quad\\text\{w\.h\.p\.\},which allows us to apply Proposition[4\.3](https://arxiv.org/html/2607.10074#S4.ThmtheoremEnv3)to\|∂Nk1−​\(u1\)\|\|\\partial N^\{\-\}\_\{k\_\{1\}\}\(u\_\{1\}\)\|\. By Proposition[4\.3](https://arxiv.org/html/2607.10074#S4.ThmtheoremEnv3)and the fact thatκδ−\\kappa\_\{\\delta\}^\{\-\}is primitive, there exists a constantc\>0c\>0such that

\|∂Nk1−​\(u1\)\|≥∑t=1T\\displaystyle\|\\partial N^\{\-\}\_\{k\_\{1\}\}\(u\_\{1\}\)\|\\geq\\sum\_\{t=1\}^\{T\}\(1−ε\)​c​λ1​\(𝒯κδ−\)k1≥T​\(1−ε\)​c⋅nε′​\(1−ϵ\)​logλ1​\(𝒯κ\)⁡λ1​\(𝒯κδ−\)w\.h\.p\.\\displaystyle\(1\-\\varepsilon\)c\\lambda\_\{1\}\(\\mathcal\{T\}\_\{\\kappa\_\{\\delta\}^\{\-\}\}\)^\{k\_\{1\}\}\\geq T\(1\-\\varepsilon\)c\\cdot n^\{\\varepsilon^\{\\prime\}\(1\-\\epsilon\)\\log\_\{\\lambda\_\{1\}\(\\mathcal\{T\}\_\{\\kappa\}\)\}\\lambda\_\{1\}\(\\mathcal\{T\}\_\{\\kappa\_\{\\delta\}^\{\-\}\}\)\}\\quad\\text\{w\.h\.p\.\}\(21\)
Next, recall from \([19](https://arxiv.org/html/2607.10074#S7.E19)\) thatλ1​\(𝒯κδ\+\)=λ1​\(𝒯κ\)\+O​\(δ\)\\lambda\_\{1\}\(\\mathcal\{T\}\_\{\\kappa\_\{\\delta\}^\{\+\}\}\)=\\lambda\_\{1\}\(\\mathcal\{T\}\_\{\\kappa\}\)\+O\(\\delta\)\. Choosingϵ\\epsilonandδ\\deltasmall enough so thatϵ​\(1−ε\+ε′\)<\(1−θ\)​logλ1​\(𝒯κδ\+\)⁡λ1​\(𝒯κ\)−\(1−ε\+ε′\)\\epsilon\(1\-\\varepsilon\+\\varepsilon^\{\\prime\}\)<\(1\-\\theta\)\\log\_\{\\lambda\_\{1\}\(\\mathcal\{T\}\_\{\\kappa\_\{\\delta\}^\{\+\}\}\)\}\\lambda\_\{1\}\(\\mathcal\{T\}\_\{\\kappa\}\)\-\(1\-\\varepsilon\+\\varepsilon^\{\\prime\}\), we obtain

k2\\displaystyle k\_\{2\}≤\(1−ε\+ε′\)​\(1\+ϵ\)​logλ1​\(𝒯κ\)⁡n<\(1−θ\)​logλ1​\(𝒯κδ\+\)⁡nw\.h\.p\.,\\displaystyle\\leq\(1\-\\varepsilon\+\\varepsilon^\{\\prime\}\)\(1\+\\epsilon\)\\log\_\{\\lambda\_\{1\}\(\\mathcal\{T\}\_\{\\kappa\}\)\}n<\\left\(1\-\\theta\\right\)\\log\_\{\\lambda\_\{1\}\(\\mathcal\{T\}\_\{\\kappa\_\{\\delta\}^\{\+\}\}\)\}n\\quad\\text\{w\.h\.p\.\},and so there existsγ∈\(0,1−θ\)\\gamma\\in\\left\(0,1\-\\theta\\right\)such thatk1<k2<γ​logλ1​\(𝒯κδ\+\)⁡nk\_\{1\}<k\_\{2\}<\\gamma\\log\_\{\\lambda\_\{1\}\(\\mathcal\{T\}\_\{\\kappa\_\{\\delta\}^\{\+\}\}\)\}n\. By Markov’s inequality and Lemma[4\.6](https://arxiv.org/html/2607.10074#S4.ThmtheoremEnv6), there existsδ′\>0\\delta^\{\\prime\}\>0such that

ℙ​\(\|Nki\+​\(ui\)t\|≥nγ\)≤O​\(λ1​\(𝒯κδ\+\)ki\)nγ≤n−δ′\\mathbb\{P\}\(\|N^\{\+\}\_\{k\_\{i\}\}\(u\_\{i\}\)\_\{t\}\|\\geq n^\{\\gamma\}\)\\leq\\frac\{O\(\\lambda\_\{1\}\(\\mathcal\{T\}\_\{\\kappa\_\{\\delta\}^\{\+\}\}\)^\{k\_\{i\}\}\)\}\{n^\{\\gamma\}\}\\leq n^\{\-\\delta^\{\\prime\}\}fori=1,2i=1,2andt∈\[T\]t\\in\[T\]with sufficiently largenn\. Therefore,

ℙ\(\|Nki\+\(ui\)t\|\\displaystyle\\mathbb\{P\}\(\|N^\{\+\}\_\{k\_\{i\}\}\(u\_\{i\}\)\_\{t\}\|≤nγfori=1,2andt∈\[T\]\)\\displaystyle\\leq n^\{\\gamma\}\\text\{ for \}i=1,2\\text\{ and \}t\\in\[T\]\)=1−ℙ​\(∃i∈\{1,2\},t∈\[T\]​s\.t\.​\|Nki​\(ui\)t\|≥nγ\)\\displaystyle=1\-\\mathbb\{P\}\(\\exists i\\in\\\{1,2\\\},t\\in\[T\]\\text\{ s\.t\. \}\\left\|N\_\{k\_\{i\}\}\(u\_\{i\}\)\_\{t\}\\right\|\\geq n^\{\\gamma\}\)≥1−∑i=1,2∑t=1Tℙ​\(\|Nki​\(ui\)t\|≥nγ\)≥1−2​T​n−δ′,\\displaystyle\\geq 1\-\\sum\_\{i=1,2\}\\sum\_\{t=1\}^\{T\}\\mathbb\{P\}\(\|N\_\{k\_\{i\}\}\(u\_\{i\}\)\_\{t\}\|\\geq n^\{\\gamma\}\)\\geq 1\-2Tn^\{\-\\delta^\{\\prime\}\},and so

\|Nki\+​\(ui\)\|≤T​nγfor​i=1,2​w\.h\.p\.\\displaystyle\|N^\{\+\}\_\{k\_\{i\}\}\(u\_\{i\}\)\|\\leq Tn^\{\\gamma\}\\quad\\text\{for \}i=1,2\\text\{ w\.h\.p\.\}\(22\)
From \([21](https://arxiv.org/html/2607.10074#S7.E21)\) and \([22](https://arxiv.org/html/2607.10074#S7.E22)\),

ℙ\\displaystyle\\mathbb\{P\}\(Zc∣X−\)​𝟏X−\\displaystyle\(Z^\{c\}\\mid X^\{\-\}\)\\mathbf\{1\}\_\{X^\{\-\}\}≤exp⁡\(−R​T​\(1−ε\)​c⋅nε′​\(1−ϵ\)​logλ1​\(𝒯κ\)⁡λ1​\(𝒯κδ−\)n​Mθlog⁡M​log⁡n−1​\(1−2​T​nγn\)Mθlog⁡M​log⁡n\)\\displaystyle\\leq\\exp\\bigg\(\-R\\frac\{T\(1\-\\varepsilon\)c\\cdot n^\{\\varepsilon^\{\\prime\}\(1\-\\epsilon\)\\log\_\{\\lambda\_\{1\}\(\\mathcal\{T\}\_\{\\kappa\}\)\}\\lambda\_\{1\}\(\\mathcal\{T\}\_\{\\kappa\_\{\\delta\}^\{\-\}\}\)\}\}\{n\}M^\{\\frac\{\\theta\}\{\\log M\}\\log n\-1\}\\bigg\(1\-\\frac\{2Tn^\{\\gamma\}\}\{n\}\\bigg\)^\{M^\{\\frac\{\\theta\}\{\\log M\}\\log n\}\}\\bigg\)=exp⁡\(−R​T​\(1−ε\)​c⋅n\(min⁡\{ε2,ε−θ\}−ε′′\)​\(1−ϵ\)​logλ1​\(𝒯κ\)⁡λ1​\(𝒯κδ−\)n​M​nθ​\(1−2​T​nγn\)nθ\)\.\\displaystyle=\\exp\\bigg\(\-R\\frac\{T\(1\-\\varepsilon\)c\\cdot n^\{\\left\(\\min\\left\\\{\\frac\{\\varepsilon\}\{2\},\\varepsilon\-\\theta\\right\\\}\-\\varepsilon^\{\\prime\\prime\}\\right\)\(1\-\\epsilon\)\\log\_\{\\lambda\_\{1\}\(\\mathcal\{T\}\_\{\\kappa\}\)\}\\lambda\_\{1\}\(\\mathcal\{T\}\_\{\\kappa\_\{\\delta\}^\{\-\}\}\)\}\}\{nM\}n^\{\\theta\}\\bigg\(1\-\\frac\{2Tn^\{\\gamma\}\}\{n\}\\bigg\)^\{n^\{\\theta\}\}\\bigg\)\.Sinceγ\+θ<1\\gamma\+\\theta<1,\(1−2​T​nγn\)nθ≥1−2​T​nγ\+θn→1\\bigg\(1\-\\frac\{2Tn^\{\\gamma\}\}\{n\}\\bigg\)^\{n^\{\\theta\}\}\\geq 1\-\\frac\{2Tn^\{\\gamma\+\\theta\}\}\{n\}\\to 1asn→∞n\\to\\infty\. Sinceϵ\\epsiloncan be chosen small enough so that\(−ε′′​ϵ\+ε′′\+ϵ​min⁡\{ε2,ε−θ\}\)​logλ1​\(𝒯κ\)⁡λ1​\(𝒯κδ−\)<ς\(\-\\varepsilon^\{\\prime\\prime\}\\epsilon\+\\varepsilon^\{\\prime\\prime\}\+\\epsilon\\min\\left\\\{\\frac\{\\varepsilon\}\{2\},\\varepsilon\-\\theta\\right\\\}\)\\log\_\{\\lambda\_\{1\}\(\\mathcal\{T\}\_\{\\kappa\}\)\}\\lambda\_\{1\}\(\\mathcal\{T\}\_\{\\kappa\_\{\\delta\}^\{\-\}\}\)<\\varsigmafor anyς\>0\\varsigma\>0,

R=Ω​\(n1−θ−min⁡\{ε2,ε−θ\}​logλ1​\(𝒯κ\)⁡λ1​\(𝒯κδ−\)\+ς\)R=\\Omega\\left\(n^\{1\-\\theta\-\\min\\left\\\{\\frac\{\\varepsilon\}\{2\},\\varepsilon\-\\theta\\right\\\}\\log\_\{\\lambda\_\{1\}\(\\mathcal\{T\}\_\{\\kappa\}\)\}\\lambda\_\{1\}\(\\mathcal\{T\}\_\{\\kappa\_\{\\delta\}^\{\-\}\}\)\+\\varsigma\}\\right\)is sufficient for the final bound to tend to 0\. By the supercriticality ofκδ−\\kappa\_\{\\delta\}^\{\-\}from \([19](https://arxiv.org/html/2607.10074#S7.E19)\),ℙ​\(x∈𝒞\(1\)\|κδ−\)\\mathbb\{P\}\(x\\in\\mathcal\{C\}\_\{\(1\)\}\|\\kappa\_\{\\delta\}^\{\-\}\)is strictly bounded away from zero\(Bollobáset al\.[2007](https://arxiv.org/html/2607.10074#bib.bib264), Theorem 3\.1\), soℙ​\(X−\)\\mathbb\{P\}\(X^\{\-\}\)does not vanish\. Then sinceℙ​\(Zc∣X−\)​𝟏X−\\mathbb\{P\}\(Z^\{c\}\\mid X^\{\-\}\)\\mathbf\{1\}\_\{X^\{\-\}\}vanishes,ℙ​\(Zc∣X−\)\\mathbb\{P\}\(Z^\{c\}\\mid X^\{\-\}\)vanishes\. Takingδ→0\\delta\\to 0yields

ℙ​\(Z\|X\)\\displaystyle\\mathbb\{P\}\(Z\|X\)≥ℙ​\(Z\|X−\)​ℙ​\(X−\)ℙ​\(X\)→ℙ​\(Z\|X−\)=1−ℙ​\(Zc\|X−\)\\displaystyle\\geq\\mathbb\{P\}\(Z\|X^\{\-\}\)\\frac\{\\mathbb\{P\}\(X^\{\-\}\)\}\{\\mathbb\{P\}\(X\)\}\\to\\mathbb\{P\}\(Z\|X^\{\-\}\)=1\-\\mathbb\{P\}\(Z^\{c\}\|X^\{\-\}\)which converges to 1 when

R=Ω​\(n1−θ−min⁡\{ε2,ε−θ\}\+ς\),R=\\Omega\\left\(n^\{1\-\\theta\-\\min\\left\\\{\\frac\{\\varepsilon\}\{2\},\\varepsilon\-\\theta\\right\\\}\+\\varsigma\}\\right\),consistent with the result in Theorem[4\.1](https://arxiv.org/html/2607.10074#S4.ThmtheoremEnv1)\.

## 8Conclusion

In this paper, we established sharp lower and upper\(1±ε\)\(1\\pm\\varepsilon\)\-distortion guarantees for landmark\-based distance\-preserving embeddings in inhomogeneous random graphs\. Our analysis demonstrates that the key mechanism underlying accurate distance\-preserving embeddings is exponential neighborhood growth and sufficiently rapid intersection behavior, as governed by the spectral properties of the affinity matrixDDor its continuous integral operator counterpart𝒯κ\\mathcal\{T\}\_\{\\kappa\}\.

By expanding our analysis from point\-wise high\-probability bounds to global empirical averages, we demonstrated that the hierarchical landmark framework successfully stabilizes metric distortion across whole topologies, proving that localized structural bottlenecks are statistically negligible\. Furthermore, via a novel metric sandwiching coupling, we successfully transferred these guarantees from finite block models to infinite\-dimensional, continuous latent\-space kernels\. This mathematical bridge establishes universal distortion bounds that easily accommodate non\-parametric, scale\-free, and heavy\-tailed network configurations like Chung–Lu power\-law models\.

Our results provide a unified spectral perspective linking affinity structure, neighborhood expansion, and metric distortion\. By demonstrating that a small set of landmarks can accurately bound distance stretch globally, this framework bridges random graph theory with the design of probabilistic graph spanners for heterogeneous spaces\. They offer theoretical justification for scalable distance\-preserving node embeddings in large\-scale networks exhibiting community structure, cyclic connectivity, or block decomposition\. By characterizing distortion behavior up to radiusO​\(logλ1⁡n\)O\(\\log\_\{\\lambda\_\{1\}\}n\), the analysis also clarifies the precise regime in which landmark\-based methods operate effectively in sparse, locally tree\-like graphs\.

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