Fundamental Dynamical Units for Physics-Informed Structural Inference from Perturbation Time-Series in Networked Systems

# Abstract
Source: [https://arxiv.org/html/2609.11934](https://arxiv.org/html/2609.11934)
Fundamental Dynamical Units for Physics\-Informed Structural Inference from Perturbation Time\-Series in Networked Systems

Nima Nouri

Artificial Intelligence for Science Innovation, AstraZeneca, Waltham, MA, USA

Corresponding author: nima\.nouri@astrazeneca\.com

In networked dynamical systems, the parameter of primary mechanistic interest is signed interaction structure\. Recovering this structure from perturbation time\-series data is a fundamental identification problem, compounded by three coupled obstacles: the combinatorial complexity of interaction architectures, ambiguity of causal attribution under limited interventions, and state\-dependent dynamics that confound structural inference\. Each obstacle is structural in origin and calls for a structural solution\. We address these challenges by adopting a reductionist approach, introducing Fundamental Dynamical Units \(FDUs\): signed three\-node interaction patterns as composable primitives that convert the interaction hypothesis space into a finite, constructive, and tractable representation\. We show that local interaction structure determines the perturbation conditions required to disentangle direct from relayed influence, making intervention design a structural consequence of the FDU representation\. We embed FDU\-regularized structural inference within a physics\-informed neural ordinary differential equation \(ODE\) whose governing\-equation constraint transforms structural hypotheses into verifiable dynamical predictions, enabling joint recovery of interaction structure and perturbation\-resolved trajectories\. Validated on synthetic benchmarks with known ground truth, the framework supports structural commitment, expressed through FDU primitives, motif\-prescribed intervention design, and physics\-informed learning, as a principled basis for mechanistically interpretable inference in networked dynamical systems\.

## Introduction

Recovering the interaction structure of a networked dynamical system from trajectory observations is fundamentally non\-injective: distinct causal architectures can produce responses that are indistinguishable without carefully targeted interventions[1](https://arxiv.org/html/2609.11934#bib.bib1);[2](https://arxiv.org/html/2609.11934#bib.bib2);[3](https://arxiv.org/html/2609.11934#bib.bib3), and more observations of the wrong kind deepen this ambiguity rather than resolving it\. The objective, recovering which system components regulate which others with what directionality and sign, must be pursued under three compounding constraints: measurements are partial and temporally sparse[4](https://arxiv.org/html/2609.11934#bib.bib4), interventions are limited in scope and amplitude[5](https://arxiv.org/html/2609.11934#bib.bib5), and the governing dynamics are state\-dependent, meaning the same structural interaction can produce qualitatively different observable signatures at different operating conditions[6](https://arxiv.org/html/2609.11934#bib.bib6)\. These are not merely statistical challenges; they are structural, and demand structural solutions\.

Scaling data volume and model capacity, the dominant paradigm in contemporary machine learning, cannot resolve this difficulty\. The challenge is fundamentally one of structural identification, not merely statistical estimation: more interventional data of the same design cannot in general resolve the ambiguity in the interaction hypothesis space[7](https://arxiv.org/html/2609.11934#bib.bib7), and a more expressive model can fit multiple competing structures with comparable fidelity[8](https://arxiv.org/html/2609.11934#bib.bib8)\. Existing approaches partition around this impasse without resolving it: many methods that prioritize flexible trajectory fitting provide limited causal or mechanistic semantics, while approaches that enforce interpretable structure through tightly specified ODEs can often be brittle under model misspecification and incomplete interventional coverage[9](https://arxiv.org/html/2609.11934#bib.bib9)\. Neither resolves the underlying issue: mechanistically interpretable inference requires structural principles built into the model itself, not imposed post hoc on unconstrained fits or recovered by scale alone, motivating a deliberate return to first\-principles design[10](https://arxiv.org/html/2609.11934#bib.bib10);[11](https://arxiv.org/html/2609.11934#bib.bib11)\.

We formalize this challenge as three coupled design constraints: interaction\-structure space \(constraining combinatorial complexity while preserving mechanistic expressivity\); intervention\-grounded interpretation \(disambiguating direct from relayed influence using structurally targeted interventions\); and state\-dependent nonlinear dynamics \(learning dynamics that generalize across operating regimes while maintaining mechanistic meaning\)\. Taken together, these requirements are mutually constraining: representation, causal attribution, and dynamical learning must be treated as coupled design constraints rather than independent modeling choices\.

Interaction\-structure space constitutes the first obstacle: the space of possible signed interaction architectures is combinatorially vast[12](https://arxiv.org/html/2609.11934#bib.bib12), requiring a hypothesis space grounded in the structural origin of the identification difficulty rather than in computational convenience alone\. The key insight is that multi\-path confounding has a minimal structural origin: a directed pairj→ij\\to icarries no pathway ambiguity by itself; a third node is the minimal addition that creates a relay routej→k→ij\\to k\\to iin parallel, making the triad the fundamental unit of multi\-path confounding\. We therefore represent network architectures as compositions of Fundamental Dynamical Units \(FDUs\): signed three\-node interaction patterns, each capturing a distinct local pattern of direct and relayed regulatory influence\. As reusable building blocks, FDUs provide a compositional basis for assembling network\-scale architectures from local three\-node patterns[13](https://arxiv.org/html/2609.11934#bib.bib13);[14](https://arxiv.org/html/2609.11934#bib.bib14);[15](https://arxiv.org/html/2609.11934#bib.bib15);[16](https://arxiv.org/html/2609.11934#bib.bib16), grounding the hypothesis space in the level at which confounding originates\. Each FDU goes beyond pairwise connectivity, coupling nearby edge choices and making direct\-versus\-relayed pathways explicit within each three\-node subgraph: a shift from surface to volumetric triangulation\(Fig\.[1](https://arxiv.org/html/2609.11934#Sx11.F1)a,b\), just as surface meshes encode boundary topology whereas volumetric representations impose interior structural constraints\. The resulting FDU\-compositional prior makes the interaction hypothesis space substantially more tractable and ensures every learned structure carries interpretable mechanistic meaning\.

Intervention\-grounded interpretation constitutes the second obstacle: even when a model reproduces observed trajectories, the underlying mechanism can remain ambiguous\. Apparent influence between two nodes may reflect a direct interaction, a relayed pathway through an intermediate, or a mixture of both, and these alternatives cannot generally be resolved from purely observational data[1](https://arxiv.org/html/2609.11934#bib.bib1);[17](https://arxiv.org/html/2609.11934#bib.bib17)\. We therefore treat intervention design not as an independent modeling choice but as a structural consequence: in this framework, once the FDU representation identifies which parallel pathways are present, the perturbation conditions required to disentangle direct from relayed influence are prescribed by the local interaction structure\. Concretely, interventions are treated not merely as additional training data but as structurally targeted probes[17](https://arxiv.org/html/2609.11934#bib.bib17);[3](https://arxiv.org/html/2609.11934#bib.bib3): in our framework, the local FDU type specifies which perturbation combinations differentially excite parallel pathways, enabling attribution of observed responses to specific mechanistic alternatives\. This framing elevates interpretability to a design objective: model structure and inference are organized to produce interaction hypotheses that are intervention\-consistent and amenable to empirical refutation\.

State\-dependent nonlinear dynamics constitutes the third obstacle: the observable effect of a structural interaction depends not only on whether the interaction exists, but on where in state space the system currently operates\. An interaction that is structurally present and causally active can produce near\-zero observable signal when the system’s state places it in an insensitive regime, a property of nonlinear systems that we refer to as operating\-point dependence[18](https://arxiv.org/html/2609.11934#bib.bib18)\. A model fitted at a single operating point or steady state cannot in general distinguish a genuinely absent edge from one that is present but locally insensitive[19](https://arxiv.org/html/2609.11934#bib.bib19), and the effective interaction strengths it recovers reflect local operating conditions rather than invariant structural parameters\. Addressing this requires continuous\-time learning that tracks the system through its full trajectory, including transient phases where structural interactions are most discriminable[19](https://arxiv.org/html/2609.11934#bib.bib19), with sufficient mechanistic grounding to separate structural parameters from state\-dependent response amplitudes\.

Prior work on directed structure recovery from time\-series data has developed along two broad traditions\. The first comprises data\-driven methods that recover interaction structure under parsimony constraints: these span regression\-based and Granger\-causal frameworks[20](https://arxiv.org/html/2609.11934#bib.bib20);[21](https://arxiv.org/html/2609.11934#bib.bib21), sparse system identification[22](https://arxiv.org/html/2609.11934#bib.bib22), perturbation\-response network reconstruction[23](https://arxiv.org/html/2609.11934#bib.bib23), directed acyclic graph \(DAG\) learning from time\-series under linear dynamics[24](https://arxiv.org/html/2609.11934#bib.bib24), and latent graph inference from dynamics including graph\-neural\-ODE approaches that infer cyclic directed graphs but not signed edges[25](https://arxiv.org/html/2609.11934#bib.bib25);[26](https://arxiv.org/html/2609.11934#bib.bib26);[27](https://arxiv.org/html/2609.11934#bib.bib27)\. The second comprises causal formalizations that establish identifiability conditions for structural recovery under dynamical interventions[28](https://arxiv.org/html/2609.11934#bib.bib28);[29](https://arxiv.org/html/2609.11934#bib.bib29), clarifying when causal structure can in principle be distinguished from trajectory data under specific model assumptions\. Both traditions make genuine progress on scalable recovery and causal semantics, but treat the perturbation panel as a fixed input rather than as a structural consequence of local interaction architecture, and do not organize the hypothesis space around local direct\-versus\-relay ambiguity as the primary structural unit; neither addresses operating\-point dependence, leaving recovered interaction estimates conflated with state\-dependent response sensitivity rather than separated into invariant structural parameters and operating\-point\-specific signal amplitudes\. A related body of work identifies network motifs as a post\-hoc structural vocabulary[13](https://arxiv.org/html/2609.11934#bib.bib13);[14](https://arxiv.org/html/2609.11934#bib.bib14);[15](https://arxiv.org/html/2609.11934#bib.bib15), applying motif descriptions to networks after structure has been inferred, with edge sign treated as a secondary attribute where considered at all; the present framework instead uses an exhaustive signed encoding of three\-node patterns as the hypothesis space itself, making signed interaction type a primary inferential target and motif structure the basis of inference rather than a post\-hoc descriptor\. This distinction extends to the nature of the inference output\. When the motif representation constitutes the hypothesis space, structural attribution, identifying which signed primitives generate each recovered interaction and at what compositional weight, is produced intrinsically by inference rather than by a secondary analysis decoupled from the evidence\. In networks with multiple overlapping subgraphs, post\-hoc assignment is ambiguous where intrinsic attribution is not\.

Building on this perspective, we turn to continuous\-time neural dynamical models as a substrate for hybrid inference\. Neural ordinary differential equations \(NODEs\)[30](https://arxiv.org/html/2609.11934#bib.bib30)recast deep networks as continuous\-time dynamical systems by learning a vector field and integrating it with differentiable solvers, yielding continuous trajectories and enabling end\-to\-end training\. In practice, however, standard NODE formulations do not constrain the learned vector field to correspond to a mechanistically interpretable interaction structure: different vector fields can reproduce similar trajectories without yielding a unique, interpretable decomposition into directed interactions\. Neural controlled differential equations \(Neural CDEs\)[31](https://arxiv.org/html/2609.11934#bib.bib31)address part of this gap by driving latent dynamics with an explicit control path, improving the treatment of irregular sampling and partial observability, yet the governing field is still rarely mechanistically constrained\. In parallel, physics\-informed approaches encourage dynamical consistency by penalizing violations of differential\-equation constraints during training, often improving plausibility and temporal generalization[32](https://arxiv.org/html/2609.11934#bib.bib32);[33](https://arxiv.org/html/2609.11934#bib.bib33);[34](https://arxiv.org/html/2609.11934#bib.bib34); nevertheless, such constraints alone do not specify which interaction structures generate the observed dynamics, nor do they resolve ambiguity when multiple mechanisms are compatible with the same trajectories\. Physics\-informed approaches have been extended to inverse problems, recovering continuous scalar or functional parameters from trajectory data, and sparse methods have identified dominant dynamical terms without prescribed structure[35](https://arxiv.org/html/2609.11934#bib.bib35);[22](https://arxiv.org/html/2609.11934#bib.bib22); neither addresses the combinatorial problem of inferring signed directed topology, in which edge existence, direction, and polarity must be jointly recovered from interventional observations under relay ambiguity\. Taken together, continuous\-time neural models tend to emphasize flexible data fit, whereas physics\-informed formulations tend to emphasize dynamical consistency; neither alone fully satisfies the joint requirements of interpretable structure, intervention\-grounded attribution, and state\-dependent mechanistic learning\. This motivates a physics\-informed neural ODE perspective that couples continuous\-time dynamical learning with mechanistic constraints and an explicit representation of interaction structure\.

Accordingly, we develop a FDU\-regularized physics\-informed neural ODE framework that makes multi\-path confounding explicit and operational\. Specifically, this work contributes \(i\) an exhaustive, label\-consistent encoding of signed three\-node interaction patterns into permutation\-invariant structural classes; \(ii\) a structure\-derived perturbation design principle that grounds panel selection in the identifiability requirements of local interaction architecture; and \(iii\) a FDU\-regularized physics\-informed neural ODE that embeds this structured prior within continuous\-time dynamical learning for joint trajectory prediction and signed interaction inference\(Fig\.[1](https://arxiv.org/html/2609.11934#Sx11.F1)c,d\)\. We demonstrate the framework in a proof\-of\-concept demonstration on synthetic benchmarks with known ground truth, establishing signed interaction recovery and motif\-level attribution as falsifiable expectations\. Together, these contributions establish a principled route toward compact, testable hypotheses about multi\-component dynamical systems from heterogeneous observational and interventional time\-series data\.

## Results

The following Results develop the structural theory from first principles; implementation details are provided in Methods\.

### The Signed Three\-Node Interaction Space Admits an Exhaustive and Constructive Dictionary of Fundamental Dynamical Units

The signed three\-node interaction space is the smallest structural level admitting both direct and relay pathways between the same node pair, and therefore constitutes the natural primitive space for motif\-centric signed structural inference\. We sought to characterize this space exhaustively, enumerating all admissible signed configurations and establishing a label\-consistent encoding\.

We consider signed, directed three\-node systems in which each unordered pair\{1,2\}\\\{1,2\\\},\{2,3\}\\\{2,3\\\}, and\{1,3\}\\\{1,3\\\}is pairwise\-connected, meaning that at least one directed influence is present between the two nodes, the minimal condition for a relay route to coexist with a direct edge between the same pair\. For each unordered pair of nodes\{i,j\}\\\{i,j\\\}, the directed edgesi→ji\\to jandj→ij\\to ican each be absent, activating, or inhibiting, provided they are not simultaneously absent \(pairwise connectivity\)\. This yields eight admissible patterns per unordered pair: four single\-direction signed configurations \(activating or inhibiting, in either direction\) and four mutual configurations \(mutual activating, mutual inhibiting, and the two mixed\-sign cases\)\. Because the three unordered pairs can be specified independently, the total number of pairwise\-connected signed triads is83=5128^\{3\}=512\.

We represent each signed triad by a ternary matrixT∈\{0,1,2\}3×3T\\in\\\{0,1,2\\\}^\{3\\times 3\}, with rows indexing target nodes and columns indexing source nodes, so thatTi​jT\_\{ij\}encodes the influence of sourcejjon targetii\(0: no edge; 1: activationj→ij\\to i; 2: inhibitionj⊣ij\\dashv i\)\. For example,

T=\(002100010\),T=\\begin\{pmatrix\}0&0&2\\\\ 1&0&0\\\\ 0&1&0\\end\{pmatrix\},\(1\)encodes a negative\-feedback cycle1→2→3⊣11\\to 2\\to 3\\dashv 1\. FlatteningTTrow\-wise yields a 9\-character motif code \(here,002100010\), which serves as the canonical identifier for each triad in all subsequent figures and tables\.

The triad encoding admits a compact, constructive dictionary of signed three\-node primitives for structural composition: the Fundamental Dynamical Unit \(FDU\) dictionary\. Each FDU is represented as a pair of binary adjacency matrices\(φ,ψ\)∈\{0,1\}3×3×\{0,1\}3×3\(\\varphi,\\psi\)\\in\\\{0,1\\\}^\{3\\times 3\}\\times\\\{0,1\\\}^\{3\\times 3\}, with rows as targets and columns as sources, whereφi​j=1\\varphi\_\{ij\}=1denotes an activating influencej→ij\\to i,ψi​j=1\\psi\_\{ij\}=1denotes an inhibiting influencej→ij\\to i, andφi​j=ψi​j=0\\varphi\_\{ij\}=\\psi\_\{ij\}=0denotes no influence\. No directed edge is simultaneously activating and inhibiting \(φi​j=ψi​j=1\\varphi\_\{ij\}=\\psi\_\{ij\}=1is prohibited\)\.

The FDU dictionary’s triadic component is grounded in the simplest unambiguous structural primitives: signed tournaments, three\-node patterns with exactly one directed signed edge per unordered pair, no mutual connections, and no self\-regulation\. Each unordered pair admits four possibilities \(two directions, two signs\), yielding43=644^\{3\}=64triadic tournaments\. Auto\-regulation is captured by six unary self\-motifs: three auto\-activating \(one per node, withφi​i=1\\varphi\_\{ii\}=1and all off\-diagonal entries zero\) and three auto\-inhibiting \(one per node, withψi​i=1\\psi\_\{ii\}=1and all off\-diagonal entries zero\)\. The FDU dictionary therefore contains 70 motifs \(64 triadic tournaments\+\+6 unary self\-motifs\)\.

This dictionary is constructive: more general triadic architectures can be assembled by selecting a subset of FDUs and taking the edge\-wise union of their signed adjacencies\. For a set𝒦\\mathcal\{K\}of FDUs\{\(φ\(k\),ψ\(k\)\)\}k∈𝒦\\\{\(\\varphi^\{\(k\)\},\\psi^\{\(k\)\}\)\\\}\_\{k\\in\\mathcal\{K\}\}, the superposition is defined as

φ=⋁k∈𝒦φ\(k\),ψ=⋁k∈𝒦ψ\(k\)\\varphi=\\bigvee\_\{k\\in\\mathcal\{K\}\}\\varphi^\{\(k\)\},\\qquad\\psi=\\bigvee\_\{k\\in\\mathcal\{K\}\}\\psi^\{\(k\)\}\(2\)where∨\\veedenotes elementwise logical OR \(equivalently, elementwise summation followed by thresholding at\>0\>0\)\. Sign\-consistent superpositions require that no directed edge is assigned both activating and inhibiting roles across the selected FDUs\. Under this restriction, the same union operation applies directly to ternary motif codes: zeros are filled by nonzeros, and any overlapping nonzero entries must agree\.

The FDU dictionary satisfies a completeness property: any sign\-consistent, pairwise\-connected three\-node triad, including those with mutual interactions, can be realized as the union of at most two tournament FDUs \([Supplementary Material](https://arxiv.org/html/2609.11934#Sx11.SSx1)\)\. Specifically, a triad with no mutual pairs is reconstructed by one tournament FDU; otherwise two tournament FDUs suffice, resolving each mutual pair in opposite directions while agreeing on all single\-direction \(forced\) edges\. The number of distinct minimal decompositions is exactly2m−12^\{m\-1\}for a triad withm≥1m\\geq 1mutual pairs\. Consequently,m=1m=1yields a unique decomposition,m=2m=2yields exactly two, andm=3m=3yields exactly four\. When multiple minimal decompositions exist, a canonical representative is selected and additional valid superpositions treated as representational redundancy\.

Two cases illustrate the completeness property at its extremes\. The pairwise\-connected triad010100120contains a mutual positive interaction on\{1,2\}\\\{1,2\\\}and is not itself a tournament\. It admits the two\-tournament reconstruction\(Fig\.[2](https://arxiv.org/html/2609.11934#Sx11.F2)a\):

000100120∨010000120\\texttt\{000100120\}\\vee\\texttt\{010000120\}\(3\)The decomposition is unique: a single mutual pair leaves no representational choice\. As a second illustration, the triad011201110, which containsm=3m=3mutual unordered pairs, admits exactly2m−1=42^\{m\-1\}=4distinct minimal decompositions\(Fig\.[2](https://arxiv.org/html/2609.11934#Sx11.F2)b\):

000200110∨011001000\\displaystyle\\texttt\{000200110\}\\vee\\texttt\{011001000\}\(4\)000201100∨011000010\\displaystyle\\texttt\{000201100\}\\vee\\texttt\{011000010\}001200010∨010001100\\displaystyle\\texttt\{001200010\}\\vee\\texttt\{010001100\}001201000∨010000110\\displaystyle\\texttt\{001201000\}\\vee\\texttt\{010000110\}In each case, the two tournament FDUs jointly realize both directions for each mutual pair while preserving the signs and orientations of all forced edges, yielding the same target triad under sign\-consistent union\. All four decompositions realize the same target triad; the multiplicity is representational redundancy, not inferential ambiguity, and canonical selection handles it in practice \([Supplementary Material](https://arxiv.org/html/2609.11934#Sx11.SSx1)\)\.

Self\-regulation extends naturally to the same construction because unary self\-motifs operate exclusively on diagonal entries, which triadic FDUs leave at zero: the two components are structurally orthogonal, and their union cannot introduce sign conflicts\. The corresponding unary self\-motifs are incorporated by the same union operation, leaving the superposition and sign\-consistency rules unchanged\. Auto\-inhibition on component 3, for instance, converts000102100to000102102\(Fig\.[2](https://arxiv.org/html/2609.11934#Sx11.F2)c\):

000102100∨000000002\\texttt\{000102100\}\\vee\\texttt\{000000002\}\(5\)
Together, these results establish that signed triadic interaction structure is compositionally closed: every admissible three\-node configuration is exactly recoverable from a sparse combination of the simplest unambiguous signed primitives\. The FDU dictionary thus constitutes a complete and exact language for signed three\-node interaction architecture\.

### The Signed Three\-Node Interaction Space Partitions into Five Permutation\-Invariant Structural Classes

A complete enumeration characterizes the signed triad primitive space but does not reveal whether the 512 signed configurations reduce to a compact set of qualitative structural types\. Having established this space exhaustively, we partitioned its elements into permutation\-invariant, sign\-sensitive structural families\. This taxonomy assigns each triad to a permutation\-invariant structural class within two primary branches, parallel\-path \(PP\) and single\-path \(SS\), each further subdivided by coherence state or feedback polarity\.

The classification builds on two matrix representations of the canonical triad encodingT∈\{0,1,2\}3×3T\\in\\\{0,1,2\\\}^\{3\\times 3\}\(rows: targets; columns: sources\): a signed adjacencyA∈\{−1,0,\+1\}3×3A\\in\\\{\-1,0,\+1\\\}^\{3\\times 3\}and an unsigned adjacencyU∈\{0,1\}3×3U\\in\\\{0,1\\\}^\{3\\times 3\}:

Ai​j=\{\+1,Ti​j=1−1,Ti​j=20,Ti​j=0andUi​j=𝟏​\[Ti​j≠0\]\.A\_\{ij\}=\\begin\{cases\}\+1,&T\_\{ij\}=1\\\\ \-1,&T\_\{ij\}=2\\\\ 0,&T\_\{ij\}=0\\end\{cases\}\\qquad\\text\{and\}\\qquad U\_\{ij\}=\\mathbf\{1\}\[T\_\{ij\}\\neq 0\]\.\(6\)Under this convention,Ai​jA\_\{ij\}is the sign on the directed edgej→ij\\to i, andUi​jU\_\{ij\}indicates whetherj→ij\\to iexists, irrespective of sign\. The matrixU2=U​UU^\{2\}=UUcounts directed length\-2 walks from sourcejjto targetii, so that\(U2\)i​j\(U^\{2\}\)\_\{ij\}detects whether a direct influencej→ij\\to iis accompanied by a two\-step relayj→k→ij\\to k\\to i\. In a three\-node system,\(U2\)i​j∈\{0,1\}\(U^\{2\}\)\_\{ij\}\\in\\\{0,1\\\}for alli≠ji\\neq j\.

Using\(U,U2\)\(U,U^\{2\}\), parallel\-path motifs \(classPP\) are those for which at least one ordered pair \(j→ij\\to i\) admits both a direct edge and a two\-step relay in parallel:

∃i≠j​s\.t\.​Ui​j=1​and​\(U2\)i​j≥1\.\\exists\\,i\\neq j\\;\\text\{s\.t\.\}\\;U\_\{ij\}=1\\;\\text\{and\}\\;\(U^\{2\}\)\_\{ij\}\\geq 1\.\(7\)In this case, influence fromjjtoiidecomposes into a direct arm \(j→ij\\to i\) and an indirect arm through the relay nodekk\(uniquely the third node in a three\-node system,j→k→ij\\to k\\to i\)\. The feed\-forward triad000100110is a canonical example: node 1 regulates node 3 both directly\(1→3\)\(1\\to 3\)and indirectly via node 2 \(1→2→31\\to 2\\to 3\), yieldingU31=1U\_\{31\}=1and\(U2\)31=1\(U^\{2\}\)\_\{31\}=1\.

Within classPP, coherence is determined by whether the direct and relayed contributions agree in sign on each parallel ordered pair\. For a parallel ordered pairj→ij\\to iwith relay nodek≠i,jk\\neq i,j, the direct and relay signs are:

sgndirect​\(j→i\)=Ai​j,sgnrelay​\(j→i\)=Ak​j​Ai​k\.\\begin\{split\}\\mathrm\{sgn\}\_\{\\mathrm\{direct\}\}\(j\\to i\)&=A\_\{ij\},\\\\ \\mathrm\{sgn\}\_\{\\mathrm\{relay\}\}\(j\\to i\)&=A\_\{kj\}\\,A\_\{ik\}\.\\end\{split\}\(8\)wheresgnrelay\\mathrm\{sgn\}\_\{\\mathrm\{relay\}\}is the sign of the two\-step pathj→k→ij\\to k\\to i\(product of edge signs along the path\)\. A motif is coherent onj→ij\\to iifsgndirect=sgnrelay\\mathrm\{sgn\}\_\{\\mathrm\{direct\}\}=\\mathrm\{sgn\}\_\{\\mathrm\{relay\}\}, and incoherent otherwise\. The feed\-forward triad000100110is coherent, because all nonzero entries ofAAare\+1\+1\. In contrast, the related classPPmotif000100120, in which the2⊣32\\dashv 3arm is inhibiting rather than activating, is incoherent on the ordered pair1→31\\to 3: the direct arm is activating \(A31=\+1A\_\{31\}=\+1\), whereas the relayed arm is inhibiting \(A21​A32=\(\+1\)​\(−1\)=−1A\_\{21\}A\_\{32\}=\(\+1\)\(\-1\)=\-1\)\.

Collectively, classPPpartitions into three permutation\-invariant subclasses according to the coherence pattern across parallel ordered pairs:PcP\_\{c\}: coherent on all parallel ordered pairs;Pin​\-​cP\_\{\\mathrm\{in\\text\{\-\}c\}\}: incoherent on all parallel ordered pairs; andPxP\_\{x\}: mixed, containing at least one coherent and at least one incoherent parallel ordered pair\. For example,000201220is mixed: it is coherent on1→21\\to 2\(directA21=−1A\_\{21\}=\-1; relayA31​A23=\(−1\)​\(\+1\)=−1A\_\{31\}A\_\{23\}=\(\-1\)\(\+1\)=\-1\), but incoherent on1→31\\to 3\(directA31=−1A\_\{31\}=\-1; relayA21​A32=\(−1\)​\(−1\)=\+1A\_\{21\}A\_\{32\}=\(\-1\)\(\-1\)=\+1\)\.

ClassSStriads are exactly the signed directed 3\-cycles: triads in which each unordered pair carries a single directed edge and these edges together form a cycle\. Formally, classSSis defined by the absence of any parallel direct\-relay configuration:

∀i≠j:¬\(Ui​j=1​and​\(U2\)i​j≥1\)\.\\forall\\,i\\neq j:\\;\\neg\\\!\\left\(U\_\{ij\}=1\\;\\text\{and\}\\;\(U^\{2\}\)\_\{ij\}\\geq 1\\right\)\.\(9\)In these motifs, influence may propagate through chains or cycles, but no ordered pairj→ij\\to iever has both a direct edge and a two\-step routej→k→ij\\to k\\to isimultaneously, a condition that forces this tournament structure as the unique admissible architecture\.

Within classSS, triads are further stratified by feedback polarity, characterized by the signs of simple directed cycles\. For an oriented 3\-cyclei→j→k→ii\\to j\\to k\\to i, the cycle sign is the product of edge signs along the cycle:

sgn​\(i→j→k→i\)=Aj​i​Ak​j​Ai​k,\\mathrm\{sgn\}\(i\\to j\\to k\\to i\)=A\_\{ji\}\\,A\_\{kj\}\\,A\_\{ik\},\(10\)since edgei→ji\\to jcorresponds to entryAj​iA\_\{ji\}under our target\-by\-source convention\. A classSStriad isS−S\_\{\-\}if any simple directed cycle has negative sign, andS\+S\_\{\+\}otherwise\. A positive\-feedback single\-path motif is exemplified by001100010, in which the components form a directed 3\-cycle1→2→3→11\\to 2\\to 3\\to 1with all edges activating, yielding only positive simple cycles\. A negative\-feedback single\-path motif is given by002100010, which implements the cycle1→2→3⊣11\\to 2\\to 3\\dashv 1with net negative sign\.

Together, the parallel\-path versus single\-path split, the coherence state within classPP, and the feedback polarity within classSSdefine a permutation\-robust taxonomy that assigns each triad to exactly one of five structural classes\(Fig\.[2](https://arxiv.org/html/2609.11934#Sx11.F2)d,e\):

\{Pc,Pin​\-​c,Px,S\+,S−\}\.\\\{P\_\{c\},\\;P\_\{\\mathrm\{in\\text\{\-\}c\}\},\\;P\_\{x\},\\;S\_\{\+\},\\;S\_\{\-\}\\\}\.\(11\)
The distribution of all 512 admissible triads reveals a strong structural skew\(Fig\.[2](https://arxiv.org/html/2609.11934#Sx11.F2)f\)\. Only 16 triads \(3\.1%3\.1\\%\) fall into the single\-path classSS, split evenly between positive\-feedback and negative\-feedback architectures \(8 inS\+S\_\{\+\}, 8 inS−S\_\{\-\}\)\. By contrast, 496 triads \(96\.9%96\.9\\%\) are parallel\-path motifs \(classPP\), containing at least one ordered pair for which a direct edge and a two\-step route coexist in parallel\. Within classPP, coherent and incoherent motifs occur in equal numbers \(124 each inPcP\_\{c\}andPin​\-​cP\_\{\\mathrm\{in\\text\{\-\}c\}\}\), while the largest group comprises mixed motifs \(248 inPxP\_\{x\}\) that contain both coherent and incoherent parallel pairs within the same triad\.

Two structural features follow\. First, parallel direct\-relay structure is combinatorially generic: direct and relay routes coexist in all but the 16 classSSconfigurations\. Second, the predominance ofPxP\_\{x\}\(248 motifs\) indicates that structural heterogeneity, containing both coherent and incoherent parallel ordered pairs, is common within the classPPspace\. Collectively, this invariant\-based partition provides a compact, permutation\-robust summary of local three\-node architecture that explicitly resolves whether each motif contains parallel direct\-relay channels and, when present, whether those channels align or oppose in sign \([Supplementary Material](https://arxiv.org/html/2609.11934#Sx11.SSx1)\)\.

### Signed Triad Structure Dictates Minimal Perturbation Panels for Relay Cancellation

In directed signed networks, influence between two nodes can propagate simultaneously along a direct edge and a relay path through an intermediate node, so that a change in a source node may elicit a composite response at the target that conflates both contributions\. This conflation is not a modeling artifact; it is a structural property of the triad\. Separating these contributions is therefore a prerequisite for structural identifiability\. We asked, for a given signed triad, what minimal perturbation panel achieves this separation on every parallel pair\. We show that the answer is determined by the triad’s local signed structure\.

The analysis proceeds around a reference state: responses are locally linear in perturbation amplitude, and only the sign structure of couplings governs panel design\. We consider three primitive perturbation regimes that can be assembled into perturbation panels: \(i\) single\-component perturbations, in which one node is perturbed at a time,𝐮=±u0​𝐞j\\mathbf\{u\}=\\pm u\_\{0\}\\,\\mathbf\{e\}\_\{j\}; \(ii\) opposite\-sign co\-perturbations, in which two nodes are perturbed with matched magnitude and opposite polarity,𝐮=±u0​\(𝐞j−𝐞k\)\\mathbf\{u\}=\\pm u\_\{0\}\(\\mathbf\{e\}\_\{j\}\-\\mathbf\{e\}\_\{k\}\); and \(iii\) same\-sign co\-perturbations, in which two nodes are perturbed with matched magnitude and the same polarity,𝐮=±u0​\(𝐞j\+𝐞k\)\\mathbf\{u\}=\\pm u\_\{0\}\(\\mathbf\{e\}\_\{j\}\+\\mathbf\{e\}\_\{k\}\)\. Here𝐮∈ℝ3\\mathbf\{u\}\\in\\mathbb\{R\}^\{3\}is the perturbation vector with componentsuju\_\{j\}at nodejj,u0\>0u\_\{0\}\>0is the scalar perturbation amplitude, and𝐞ℓ∈ℝ3\\mathbf\{e\}\_\{\\ell\}\\in\\mathbb\{R\}^\{3\}is theℓ\\ell\-th standard basis vector \(11in coordinateℓ\\ell,0otherwise\)\. “Matched magnitude” means the two nonzero components have equal absolute value \(\|uj\|=\|uk\|\|u\_\{j\}\|=\|u\_\{k\}\|\), so that polarity \(same versus opposite sign\) is the only additional degree of freedom introduced by co\-perturbation\. Under this matched\-amplitude constraint, signed triad structure alone determines the panel assignment\.

Fix a parallel ordered pair \(j→ij\\to i\) for which influence can propagate both directly and indirectly within the triad\. In a three\-node motif, the relay nodekkis uniquely determined as the remaining node distinct fromiiandjj\. Using the signed adjacency convention \(rows are targets, columns are sources\), define the signed edge indicators along the direct and relayed arms by

a\\displaystyle a≡Ak​j∈\{±1\}\(sign of​j→k\),\\displaystyle\\equiv A\_\{kj\}\\in\\\{\\pm 1\\\}\\quad\(\\text\{sign of \}j\\to k\),\(12\)b\\displaystyle b≡Ai​k∈\{±1\}\(sign of​k→i\),\\displaystyle\\equiv A\_\{ik\}\\in\\\{\\pm 1\\\}\\quad\(\\text\{sign of \}k\\to i\),c\\displaystyle c≡Ai​j∈\{±1\}\(sign of​j→i\)\.\\displaystyle\\equiv A\_\{ij\}\\in\\\{\\pm 1\\\}\\quad\(\\text\{sign of \}j\\to i\)\.For such a parallel ordered pair, all three quantities are nonzero\. Under additive sign\-only linearization \(magnitudes absorbed into a local scaling\), the relay node responds to perturbations atjjandkkas

Δ​xk≈a​uj\+uk,\\Delta x\_\{k\}\\approx a\\,u\_\{j\}\+u\_\{k\},\(13\)and the response atiidecomposes into a direct term plus a relay term:

Δ​xi≈c​uj\+b​Δ​xk≈c​uj\+b​\(a​uj\+uk\)\.\\Delta x\_\{i\}\\approx c\\,u\_\{j\}\+b\\,\\Delta x\_\{k\}\\approx c\\,u\_\{j\}\+b\(a\\,u\_\{j\}\+u\_\{k\}\)\.\(14\)Choosinguku\_\{k\}to enforceΔ​xk≈0\\Delta x\_\{k\}\\approx 0suppresses the relayed arm, leavingΔ​xi≈c​uj\\Delta x\_\{i\}\\approx c\\,u\_\{j\}: the sign of the response atiidirectly encodes the sign of the direct edgej→ij\\to i, converting a composite response into a signed edge readout\. The relay\-cancellation condition is therefore

uk=−a​uj\.u\_\{k\}=\-a\\,u\_\{j\}\.\(15\)Under matched\-amplitude co\-perturbations \(\|uj\|=\|uk\|\|u\_\{j\}\|=\|u\_\{k\}\|\), this reduces to a polarity rule: whena=\+1a=\+1\(activatingj→kj\\to k\), cancellation requires opposite\-sign co\-perturbationuk=−uju\_\{k\}=\-u\_\{j\}, whereas whena=−1a=\-1\(inhibitingj⊣kj\\dashv k\), cancellation requires same\-sign co\-perturbationuk=\+uju\_\{k\}=\+u\_\{j\}\. Notably, the required polarity depends only on the sign of the first legj→kj\\to kand not on whether the parallel ordered pair is coherent or incoherent\. The design criterion is structurally leaner than the coherence criterion: coherence depends on the relationship among all three local edge signs \(aa,bb,cc\), whereas relay cancellation depends onaaalone\. The relay\-cancellation polarity is thus fully encoded in a single signed quantity, the first relay lega=Ak​ja=A\_\{kj\}, making panel design a direct read\-off from local signed triad structure\(Fig\.[3](https://arxiv.org/html/2609.11934#Sx11.F3)\)\.

Applied across the five structural classes, this condition yields three design rules\.

Design Rule 1\.In classS\+S\_\{\+\}andS−S\_\{\-\}motifs, no ordered pair has a direct edge coexisting with a two\-step relay\. Because relay interference is structurally absent, single\-component perturbations are sufficient to probe all signed routes within the motif \(Fig\.[3](https://arxiv.org/html/2609.11934#Sx11.F3): rows 1–2\)\.

Design Rule 2\.In classPcP\_\{c\}andPin​\-​cP\_\{\\mathrm\{in\\text\{\-\}c\}\}motifs, the co\-perturbation polarity required for relay cancellation is fully determined by the sign of the first relay legAk​jA\_\{kj\}: opposite\-sign co\-perturbation \(uk=−uju\_\{k\}=\-u\_\{j\}\) whenAk​j=\+1A\_\{kj\}=\+1; same\-sign co\-perturbation \(uk=\+uju\_\{k\}=\+u\_\{j\}\) whenAk​j=−1A\_\{kj\}=\-1\. When multiple parallel ordered pairs within the same motif carry opposing first\-leg signs, both co\-perturbation polarities must be included \(Fig\.[3](https://arxiv.org/html/2609.11934#Sx11.F3): rows 3–5\)\.

Design Rule 3\.In classPxP\_\{x\}motifs, direct and relayed contributions reinforce on coherent parallel ordered pairs and oppose on incoherent ones\. Supporting attribution across this mixed coherence structure requires both co\-perturbation polarities: together they yield a controlled relay\-off and relay\-on contrast, with each polarity selectively suppressing or amplifying the relayed arm depending on the pair, supporting robust attribution of direct and relayed contributions across all parallel ordered pairs within the motif \(Fig\.[3](https://arxiv.org/html/2609.11934#Sx11.F3): row 6\)\.

Together, these rules define four minimal panel categories for each signed triad: single\-component, opposite\-sign only, same\-sign only, or dual\-polarity\. Here “minimal” refers to the least demanding regime sufficient, under the local linear\-response and matched\-amplitude assumptions, to cancel relayed contributions on all parallel ordered pairs within the motif and to support attribution across all coherence configurations present\.

All 512 admissible signed triads are assigned to exactly one panel category\(Table[1](https://arxiv.org/html/2609.11934#Sx11.T1);[Supplementary Material](https://arxiv.org/html/2609.11934#Sx11.SSx1)\)\. All 16 single\-path motifs \(8S\+S\_\{\+\}and 8S−S\_\{\-\}\) fall into the single\-component category\. The remaining 496 motifs are parallel\-path motifs \(classPP\)\. WithinPcP\_\{c\}, 52 motifs require only opposite\-sign co\-perturbations, 51 require only same\-sign co\-perturbations, and 21 require dual\-polarity panels\. The incoherent subclassPin​\-​cP\_\{\\mathrm\{in\\text\{\-\}c\}\}exhibits the mirror pattern \(51, 52, and 21, respectively\)\. The 21 dual\-polarity motifs within each ofPcP\_\{c\}andPin​\-​cP\_\{\\mathrm\{in\\text\{\-\}c\}\}arise from the Rule 2 extension: these motifs contain multiple parallel ordered pairs whose first relay legs carry opposite signs, generating conflicting polarity requirements despite uniform coherence across all pairs\. For example,000102220\(classPcP\_\{c\}\) has two coherent parallel ordered pairs with opposing first\-leg signs:a=−1a=\-1on1→21\\to 2via33, requiring same\-sign co\-perturbation, anda=\+1a=\+1on1→31\\to 3via22, requiring opposite\-sign co\-perturbation; both polarities must therefore be included\. Mixed motifsPxP\_\{x\}comprise the most demanding regime: all 248 motifs require dual\-polarity panels\. Across all 496 classPPmotifs, 103 require only opposite\-sign co\-perturbations, 103 require only same\-sign co\-perturbations, and 290 \(≈58%\\approx 58\\%\) require dual\-polarity panels\. Dual\-polarity co\-perturbation is thus the most common requirement, applying to 290 of 512 \(57%57\\%\) of all admissible signed triads\.

The structure\-derived panel assignment converts the five\-class structural taxonomy into actionable intervention conditions required by the physics\-informed inference model\.

### FDU\-Regularized Continuous\-Time Dynamics Constitute a Constructive Physics\-Informed Framework for Signed Motif Inference

Four defining properties embed structural commitment throughout the parameterization: a constructively complete FDU bank spanning all signed structural primitives acrossNN\-node systems; an attention mechanism that concentrates structural attribution onto a compact set of FDU primitives; a topology\-magnitude factorization separating which interactions exist from how strong they are; and a physics\-informed dynamical model that maps structural parameters to sign\-interpretable, operating\-point\-aware dynamics\.

A system ofNNinteracting nodes has statex​\(t\)∈ℝNx\(t\)\\in\\mathbb\{R\}^\{N\}, wherexi​\(t\)x\_\{i\}\(t\)denotes the state of nodeiiat timett\. Signed directed influence is parameterized through two nonnegative strength matrices

α,β∈ℝ≥0N×N,\\alpha,\\beta\\in\\mathbb\{R\}\_\{\\geq 0\}^\{N\\times N\},\(16\)whereαi​j\\alpha\_\{ij\}encodes the strength of an activating influencej→ij\\to iandβi​j\\beta\_\{ij\}encodes the strength of an inhibiting influencej⊣ij\\dashv i\. Under the target\-source convention \(rows are targets, columns are sources\), the signed interaction pattern implied by\(α,β\)\(\\alpha,\\beta\)is directly comparable to the triadic encodings \(activating channel supported whereαi​j\>0\\alpha\_\{ij\}\>0, inhibiting channel supported whereβi​j\>0\\beta\_\{ij\}\>0\)\.

Motif regularization of\(α,β\)\(\\alpha,\\beta\)is achieved through a finite bank ofN×NN\\times Nbinary masks, constructed by lifting the primitive FDUs into theNN\-node setting: each triadic FDU is embedded onto node triplets \(placing its3×33\\times 3pattern into theN×NN\\times Nmask at the selected rows and columns, zeros elsewhere\), and each unary self\-motif onto individual nodes, yielding\{\(M↑\(ℓ\),M↓\(ℓ\)\)\}ℓ=1L\\\{\(M\_\{\\uparrow\}^\{\(\\ell\)\},M\_\{\\downarrow\}^\{\(\\ell\)\}\)\\\}\_\{\\ell=1\}^\{L\}, where↑\\uparrowindexes activating and↓\\downarrowindexes inhibiting edge templates, andL=L​\(N\)L=L\(N\)is finite for any fixedNN\(Fig\.[4](https://arxiv.org/html/2609.11934#Sx11.F4)a\)\. Because the FDU dictionary is constructively complete, the bank provides a sufficient generative vocabulary: any signed triadic architecture in the target network is expressible as an attention\-weighted superposition of bank entries\. The bank size is given by:

L​\(N\)=64⋅\(N3\)\+2​N,L\(N\)=64\\cdot\\binom\{N\}\{3\}\+2N,\(17\)where the first term counts placements of the 64 tournament FDUs across all 3\-node subsets and the second accounts for self\-regulation at each of theNNnodes \(activating and inhibiting\)\. AtN=3N=3,L​\(3\)=70L\(3\)=70: exactly the FDU dictionary\. AtN=4N=4,L​\(4\)=264L\(4\)=264; atN=20N=20,L​\(20\)=73,000L\(20\)=73\{,\}000, reflecting the polynomialO​\(N3\)O\(N^\{3\}\)growth of the hypothesis space with network size\. Each mask pair specifies which directed edges are admissible as activating versus inhibiting under motifℓ\\ell\. In networks containing singletons, the bank is augmented with a null entry carrying zero masks\.

Attention over this bank is induced by the entmax transform[36](https://arxiv.org/html/2609.11934#bib.bib36), a sparsity\-promoting normalizationω​\(⋅\)\\omega\(\\cdot\)that maps learnable logitsz∈ℝLz\\in\\mathbb\{R\}^\{L\}to simplex\-constrained FDU attention weightse∈ℝLe\\in\\mathbb\{R\}^\{L\}:

e=ω​\(z\),eℓ≥0,∑ℓ=1Leℓ=1\.e=\\omega\(z\),\\qquad e\_\{\\ell\}\\geq 0,\\qquad\\sum\_\{\\ell=1\}^\{L\}e\_\{\\ell\}=1\.\(18\)This induces motif\-level sparsity that \(i\) discourages diffuse superpositions over many motifs, thereby regularizing the combinatorial hypothesis class, and \(ii\) yields a parsimonious interpretation of the inferred structure as a small set of dominant FDU primitives whose edge\-wise union supports the effective signed interaction pattern\(Fig\.[4](https://arxiv.org/html/2609.11934#Sx11.F4)b\)\.

These weights define motif\-derived structural support masks

σ↑=∑ℓeℓ​M↑\(ℓ\),σ↓=∑ℓeℓ​M↓\(ℓ\),\\sigma\_\{\\uparrow\}=\\sum\_\{\\ell\}e\_\{\\ell\}\\,M\_\{\\uparrow\}^\{\(\\ell\)\},\\qquad\\sigma\_\{\\downarrow\}=\\sum\_\{\\ell\}e\_\{\\ell\}\\,M\_\{\\downarrow\}^\{\(\\ell\)\},\(19\)where\(σ↑\)i​j\(\\sigma\_\{\\uparrow\}\)\_\{ij\}and\(σ↓\)i​j\(\\sigma\_\{\\downarrow\}\)\_\{ij\}quantify how strongly the current FDU mixture supports activating or inhibiting influence on edgej→ij\\to i\. Entry\-wise gating by the support masks yields the effective signed interaction strengths used by the dynamics:

αeff=α⊙σ↑,βeff=β⊙σ↓,\\alpha^\{\\mathrm\{eff\}\}=\\alpha\\odot\\sigma\_\{\\uparrow\},\\qquad\\beta^\{\\mathrm\{eff\}\}=\\beta\\odot\\sigma\_\{\\downarrow\},\(20\)with⊙\\odotdenoting Hadamard multiplication\. In this factorization, FDUs act as structural priors that constrain signed topology through\(σ↑,σ↓\)\(\\sigma\_\{\\uparrow\},\\sigma\_\{\\downarrow\}\), while\(α,β\)\(\\alpha,\\beta\)encode magnitudes for the admissible interactions\. This factorization separates two logically distinct inference problems: which interactions exist \(topology, determined by FDU attention\) and how strong they are \(magnitude, encoded inα,β\\alpha,\\beta\), allowing each to be learned through its natural parameterization\(Fig\.[4](https://arxiv.org/html/2609.11934#Sx11.F4)c\)\. Because\(σ↑,σ↓\)\(\\sigma\_\{\\uparrow\},\\sigma\_\{\\downarrow\}\)arise from superposition over a finite FDU dictionary, the parameterization can express composite architectures not represented by any single motif, while remaining anchored to interpretable building blocks\.

This parameterization acquires continuous\-time semantics through an ODE governing the statexi​\(t\)x\_\{i\}\(t\)of each node,i=1,…,Ni=1,\\ldots,N:

d​xi​\(t\)d​t=Fi​\(x​\(t\)\)−γi​xi​\(t\)\+ui​\(t\),\\frac\{dx\_\{i\}\(t\)\}\{dt\}=F\_\{i\}\(x\(t\)\)\-\\gamma\_\{i\}\\,x\_\{i\}\(t\)\+u\_\{i\}\(t\),\(21\)whereFiF\_\{i\}is the net signed interaction drive,γi\>0\\gamma\_\{i\}\>0is a first\-order decay rate, andui​\(t\)u\_\{i\}\(t\)is an exogenous input representing a controlled perturbation applied to nodeii\. The controlled inputsui​\(t\)u\_\{i\}\(t\)are instantiated as the relay\-cancellation panel prescribed by design rules 1–3, guaranteeing that training data spans the relay\-off and relay\-on conditions necessary for signed edge attribution under the structural class of the target motif\. The physics\-informed constraint thatx​\(t\)x\(t\)satisfies the ODE at every observed time point reduces structural degeneracy: where multiple interaction structures may fit isolated observations, only a subset generate ODE\-consistent trajectories across the full perturbation panel\.

FiF\_\{i\}is an additive superposition of activating and inhibiting channels from all sources, capturing saturating nonlinear effects while preserving sign interpretability:

Fi​\(x\)=∑j=1N\(αi​jeff​H↑​\(xj;κj,Kj\)\+βi​jeff​H↓​\(xj;κj,Kj\)\),F\_\{i\}\(x\)=\\sum\_\{j=1\}^\{N\}\\\!\\left\(\\alpha\_\{ij\}^\{\\mathrm\{eff\}\}\\,H\_\{\\uparrow\}\(x\_\{j\};\\kappa\_\{j\},K\_\{j\}\)\+\\beta\_\{ij\}^\{\\mathrm\{eff\}\}\\,H\_\{\\downarrow\}\(x\_\{j\};\\kappa\_\{j\},K\_\{j\}\)\\right\),\(22\)whereαi​jeff≥0\\alpha\_\{ij\}^\{\\mathrm\{eff\}\}\\geq 0andβi​jeff≥0\\beta\_\{ij\}^\{\\mathrm\{eff\}\}\\geq 0are the FDU\-gated effective strengths for activating and inhibiting influence on the directed edgej→ij\\to i\. HereH↑H\_\{\\uparrow\}is monotone increasing and saturating, whereasH↓H\_\{\\downarrow\}is monotone decreasing and saturating\. For systems with nonnegative states, a standard choice[37](https://arxiv.org/html/2609.11934#bib.bib37)is Hill\-type forms:

H↑​\(x;κ,K\)=xκK\+xκ,H↓​\(x;κ,K\)=KK\+xκ,\\begin\{split\}H\_\{\\uparrow\}\(x;\\kappa,K\)&=\\frac\{x^\{\\kappa\}\}\{K\+x^\{\\kappa\}\},\\\\ H\_\{\\downarrow\}\(x;\\kappa,K\)&=\\frac\{K\}\{K\+x^\{\\kappa\}\},\\end\{split\}\(23\)withκj≥1\\kappa\_\{j\}\\geq 1controlling cooperativity andKj\>0K\_\{j\}\>0setting the half\-saturation scale for source nodejj\. The saturating Hill forms directly address operating\-point dependence: when a source node’s state lies far from the half\-saturation pointKK, the Hill response is near\-maximal or near\-zero, and even a structurally present interaction contributes negligible drive toFiF\_\{i\}\. Trajectory\-spanning training across perturbation conditions is therefore the operative mechanism: transient phases in which the source traverses the sensitive regime nearKKcarry the discriminative signal for structural recovery\. Under this parameterization,∂H↑/∂x\>0\\partial H\_\{\\uparrow\}/\\partial x\>0and∂H↓/∂x<0\\partial H\_\{\\downarrow\}/\\partial x<0forx\>0x\>0, so edges withαi​jeff\>0\\alpha\_\{ij\}^\{\\mathrm\{eff\}\}\>0andβi​jeff=0\\beta\_\{ij\}^\{\\mathrm\{eff\}\}=0induce∂Fi/∂xj\>0\\partial F\_\{i\}/\\partial x\_\{j\}\>0\(small\-signal activation\), whereas edges withβi​jeff\>0\\beta\_\{ij\}^\{\\mathrm\{eff\}\}\>0andαi​jeff=0\\alpha\_\{ij\}^\{\\mathrm\{eff\}\}=0induce∂Fi/∂xj<0\\partial F\_\{i\}/\\partial x\_\{j\}<0\(small\-signal inhibition\)\(Fig\.[4](https://arxiv.org/html/2609.11934#Sx11.F4)d\)\. The parameterization is trained to satisfy channel exclusivity: each directed edge operates in at most one channel\. Under channel exclusivity, the sign of∂Fi/∂xj\\partial F\_\{i\}/\\partial x\_\{j\}directly encodes the signed interactionj→ij\\to i, grounding the learned dynamics in the motif representation\.

This framework resolves all three coupled inferential challenges: a tractable interaction\-structure hypothesis space, a structure\-derived perturbation design, and a physics\-informed, state\-dependent mechanistic dynamical model\.

### FDU\-Regularized Physics\-Informed Inference Recovers Signed Structure, Motif Identity, and Dynamics Without Per\-Edge Structural Labels

The FDU\-regularized parameterization provides a tractable and constructively complete hypothesis class for signed motif inference; its empirical adequacy is assessed against representative motifs spanning all five structural classes\. We asked whether the inferred\(α,β\)\(\\alpha,\\beta\)channels recover edge presence, source\-to\-target directionality, and activating or inhibiting polarity, whether the learned dynamics faithfully regenerate perturbation\-resolved trajectories, and whether the FDU attention mechanism localises onto the generating motif’s support set\.

The six representative motifs\(Fig\.[3](https://arxiv.org/html/2609.11934#Sx11.F3)\)span all five structural classes and all three perturbation panel types prescribed by design rules 1–3:002100010\(classS−S\_\{\-\}\),001100010\(classS\+S\_\{\+\}\),000100110\(classPcP\_\{c\}\),000200210\(classPcP\_\{c\}\),000100120\(classPin​\-​cP\_\{\\mathrm\{in\\text\{\-\}c\}\}\), and000201220\(classPxP\_\{x\}\)\. The two classPcP\_\{c\}representatives cover both the opposite\-sign and same\-sign co\-perturbation sub\-cases, spanning the full range of panel types within that class\. For each motif, ground\-truth trajectories were generated by simulation from the FDU\-regularized continuous\-time system with known structure, under the unperturbed reference condition and the class\-dependent perturbation panel: classSSmotifs \(002100010,001100010\) received single\-component panels; the two classPcP\_\{c\}motifs received opposite\-sign \(000100110\) and same\-sign \(000200210\) co\-perturbation panels respectively; the classPin​\-​cP\_\{\\mathrm\{in\\text\{\-\}c\}\}motif \(000100120\) received an opposite\-sign co\-perturbation panel; and the classPxP\_\{x\}motif \(000201220\) received a dual\-polarity panel\. Training and validation used sparse observations distributed across distinct dynamical phases \(initialization, early response, mid\-to\-late transient, and steady state; 3:1 training\-to\-validation split\)\. Each motif was trained jointly across all perturbation conditions in its class\-specific panel under a fixed protocol with identical hyperparameters across experiments \(see Methods\)\.

Structural recovery is assessed from the inferred activating and inhibiting channels \(Fig\.[5](https://arxiv.org/html/2609.11934#Sx11.F5)a–f: top row of each panel\)\. Across the six motifs, dominant interactions concentrate on the ground\-truth ordered edges: activating couplings inα\\alphaand inhibiting couplings inβ\\beta, with non\-edges remaining suppressed in both channels\. This channel separation establishes recovery of edge presence, source\-to\-target directionality, and activating\-or\-inhibiting polarity directly from\(α,β\)\(\\alpha,\\beta\), without additional sign assignment beyond the two\-channel parameterization\. This signed partition emerges from training rather than explicit supervision: the assignment of interactions to activating or inhibiting channels is not prescribed per edge, but arises through the physics\-informed dynamics and soft channel exclusivity constraint across all six motifs\.

Trajectory regeneration is evaluated across all six motifs and their class\-specific perturbation panels\([Supplementary Fig\. 1](https://arxiv.org/html/2609.11934#Sx11.SSx1)\)\. Predicted trajectories closely track ground\-truth simulations over the full simulated time horizon, capturing transient responses and late\-time stabilization under the reference condition, single\-component perturbations, and class\-specific co\-perturbations\. Reconstruction error is quantified on the held\-out test set spanning the full temporal horizon, on time points not used in training or validation, per condition and per state as the range\-normalised root\-mean\-square error \(Fig\.[5](https://arxiv.org/html/2609.11934#Sx11.F5)a–f: row 2 left\): the median lies in the range0\.120\.12to0\.210\.21across motifs, indicating that predicted dynamics remain faithful to ground\-truth variability across all conditions and states\. Training loss curves decrease and stabilize across all experiments \([Supplementary Fig\. 1](https://arxiv.org/html/2609.11934#Sx11.SSx1): bottom row\), supporting convergence of the joint trajectory\-structure inference\. These results confirm that the inferred continuous\-time model generalises across the full temporal horizon from sparse multi\-phase supervision, rather than fitting isolated dynamical phases\. The simultaneous recovery of accurate trajectories and correct signed structure indicates that the structural constraints imposed by FDU regularization do not compromise predictive fidelity\.

Motif identification is assessed from the FDU attention distributions over the bank \(Fig\.[5](https://arxiv.org/html/2609.11934#Sx11.F5)a–f: row 2 right\)\. For the five motifs with a unique single\-FDU representation, attention mass concentrates on the generating triad encoding, with markedly smaller weights on alternatives\. For the classPxP\_\{x\}motif000201220, attention follows the anticipated compositional pattern: dominant weight distributes across the two FDUs000201200and000200220, consistent with the unique minimal two\-FDU decomposition for a motif with a single mutual pair \([Supplementary Material](https://arxiv.org/html/2609.11934#Sx11.SSx1)\)\. Together, these patterns confirm that the attention mechanism localises onto the FDU support set consistent with the generating architecture, stable across class\-dependent perturbation panels\. The FDU attention profile thus provides a structural basis for the recovered interaction magnitudes: beyond localising onto the correct motif, it identifies the structural primitive underlying the inferred interaction pattern, grounding structural recovery in the motif representation rather than in interaction magnitudes alone\.

Element\-wise convergence ofαi​j\\alpha\_\{ij\}andβi​j\\beta\_\{ij\}across training epochs provides a mechanistic audit of identifiability \(Fig\.[5](https://arxiv.org/html/2609.11934#Sx11.F5)a–f: row 3\)\. Entries corresponding to true directed interactions increase and stabilize in the appropriate channel, whereas coefficients for non\-edges remain near baseline or decay toward zero\. In motifs containing inhibiting interactions, growth concentrates inβ\\betafor the relevant ordered pairs whileα\\alpharemains suppressed, yielding explicit polarity separation at the parameter level\. Sparse signed structure thus emerges through selective parameter stabilization rather than diffuse weight sharing across candidate edges\.

Beyond the six triad\-only experiments, compositional structure recovery is tested on motif000102102\([Supplementary Fig\. 2](https://arxiv.org/html/2609.11934#Sx11.SSx1)\), which comprises the triadic architecture000102100\(classPin​\-​cP\_\{\\mathrm\{in\\text\{\-\}c\}\}\) and a unary auto\-inhibiting self\-edge \(000000002\)\. Trajectories are generated under the reference condition and the opposite\-sign co\-perturbation panel of itsPin​\-​cP\_\{\\mathrm\{in\\text\{\-\}c\}\}base triad, under the same fixed protocol as the triad\-only experiments\. The inferred activating and inhibiting channels recover the self\-edge in the appropriate channel while maintaining sparsity on unrelated ordered pairs\. Attention localises to two dominant entries: the triadic component000102100and the unary self\-motif000000002\(both present in the bank\), autonomously discovering the two\-component FDU decomposition from trajectory data without per\-edge structural labels\. Element\-wise parameter trajectories show selective stabilization of the self\-coupling alongside the triadic interactions\. Trajectory regeneration remains accurate across all perturbation conditions with a median range\-normalised root\-mean\-square error of0\.190\.19and no evidence of condition\-specific degradation, and training loss decreases and stabilizes consistently\. The compositional FDU representation thus extends seamlessly to composite triadic\-unary architectures: structural complexity is absorbed without sacrificing predictive fidelity or channel interpretability\.

Collectively, across all five structural classes and a compositional architecture extending FDU coverage to a unary auto\-inhibiting self\-edge, the FDU\-regularized physics\-informed neural ODE jointly recovers signed interaction structure and perturbation\-resolved dynamics from sparse time\-course data\. Three forms of structural recovery emerge without per\-edge structural labels: assignment of activating and inhibiting interactions to their respective channels, localisation of FDU attention onto the generating motif support set, and autonomous discovery of compositional FDU decompositions\. Structural constraints and predictive fidelity are not in tension: the jointly inferred solutions remain accurate across the full perturbation panel and consistent with the ground\-truth dynamical system\. These results establish that structural inductive biases constitute the operative inferential mechanism for mechanistically interpretable signed motif inference from perturbation time courses\.

### FDU\-Regularized Inference Recovers Signed Interaction Structure of a Network\-Embedded Triad Under Open\-System Confounding

Structural inference within a self\-contained three\-node system rests on a closure assumption: the perturbation panel covers all nodes, and the local dynamics are fully determined by intra\-block interactions alone\. The relay\-cancellation argument exploits this closure directly: for any ordered pair\(i→j\)\(i\\to j\)in a three\-node motif, the only possible two\-step relay is mediated by the third node, making relay cancellation a unique, structurally determined operation\. In networks withN\>3N\>3, this closure fails: even when inference is restricted to a local motif, its observed dynamics can be strongly shaped by influences from nodes outside the local block that provide alternative relay routes and exogenous drives with no representation in the closed local model\. When this external contribution is unmodeled, it is generically projected onto intra\-block edges, producing spurious couplings indistinguishable from true local regulation under standard fitting\. The open\-system partition of the network state makes this confounding structurally explicit and yields a decomposition of the local dynamics that separates the FDU\-representable structure from the environmental contribution\.

The full network state separates into a local blockxB​\(t\)∈ℝ\|B\|x\_\{B\}\(t\)\\in\\mathbb\{R\}^\{\|B\|\}, comprising the\|B\|\|B\|nodes under inference \(B⊂\{1,…,N\}B\\subset\\\{1,\\ldots,N\\\}, with\|B\|=3\|B\|=3for an embedded triad\), and an environment blockxE​\(t\)∈ℝN−\|B\|x\_\{E\}\(t\)\\in\\mathbb\{R\}^\{N\-\|B\|\}, consisting of all remaining nodes \(environmentEE, the set of all nodes outsideBB\):

x​\(t\)=\(xB​\(t\),xE​\(t\)\)\.x\(t\)=\\bigl\(x\_\{B\}\(t\),\\,x\_\{E\}\(t\)\\bigr\)\.\(24\)The key consequence of this partition is that the true dynamics of each local nodei∈Bi\\in Bdepend on both blocks:

d​xi​\(t\)d​t=Fi​\(xB​\(t\),xE​\(t\)\)−γi​xi​\(t\)\+ui​\(t\),i∈B,\\frac\{dx\_\{i\}\(t\)\}\{dt\}=F\_\{i\}\(x\_\{B\}\(t\),x\_\{E\}\(t\)\)\-\\gamma\_\{i\}\\,x\_\{i\}\(t\)\+u\_\{i\}\(t\),\\qquad i\\in B,\(25\)whereFi​\(xB,xE\)F\_\{i\}\(x\_\{B\},x\_\{E\}\)is the net signed interaction drive on nodeiifrom the full network,γi\>0\\gamma\_\{i\}\>0is a first\-order decay rate, andui​\(t\)u\_\{i\}\(t\)is a controlled perturbation input\. ThexE​\(t\)x\_\{E\}\(t\)dependence inFiF\_\{i\}is the structural source of open\-system confounding: environment nodes contribute to the drive on local nodes, and this contribution is not representable by a closed model of the local block alone\.

ThexE​\(t\)x\_\{E\}\(t\)dependence across all local nodesi∈Bi\\in Bis captured collectively in the vector\-form dynamics of the local block:

d​xB​\(t\)d​t=FB​\(xB​\(t\),xE​\(t\)\)−γB​xB​\(t\)\+uB​\(t\),\\frac\{dx\_\{B\}\(t\)\}\{dt\}=F\_\{B\}\(x\_\{B\}\(t\),x\_\{E\}\(t\)\)\-\\gamma\_\{B\}\\,x\_\{B\}\(t\)\+u\_\{B\}\(t\),\(26\)whereFB​\(xB,xE\)∈ℝ\|B\|F\_\{B\}\(x\_\{B\},x\_\{E\}\)\\in\\mathbb\{R\}^\{\|B\|\}is the vector of net signed interaction drives withii\-th entryFi​\(xB,xE\)F\_\{i\}\(x\_\{B\},x\_\{E\}\)for eachi∈Bi\\in B,γB∈ℝ\|B\|\\gamma\_\{B\}\\in\\mathbb\{R\}^\{\|B\|\}is the vector of first\-order decay rates withii\-th entryγi\\gamma\_\{i\}, anduB​\(t\)∈ℝ\|B\|u\_\{B\}\(t\)\\in\\mathbb\{R\}^\{\|B\|\}is the vector of controlled perturbation inputs withii\-th entryui​\(t\)u\_\{i\}\(t\)\. Even with the full system state available,FBF\_\{B\}contains contributions from the environmentEEnot representable by the FDU structural prior\.FBF\_\{B\}therefore admits a decomposition:

FB​\(xB,xE\)=F~B​\(x\)\+w​\(t\),F\_\{B\}\(x\_\{B\},x\_\{E\}\)=\\widetilde\{F\}\_\{B\}\(x\)\+w\(t\),\(27\)whereF~B​\(x\)∈ℝ\|B\|\\widetilde\{F\}\_\{B\}\(x\)\\in\\mathbb\{R\}^\{\|B\|\}captures all signed directed interactions representable by the FDU prior across the full system statexx, andw​\(t\)∈ℝ\|B\|w\(t\)\\in\\mathbb\{R\}^\{\|B\|\}\(the latent forcing term\) is the exact remainder between the true interaction field and the FDU\-regularized approximation\. The local block dynamics then become:

d​xB​\(t\)d​t=F~B​\(x​\(t\)\)−γB​xB​\(t\)\+uB​\(t\)\+w​\(t\)\.\\frac\{dx\_\{B\}\(t\)\}\{dt\}=\\widetilde\{F\}\_\{B\}\(x\(t\)\)\-\\gamma\_\{B\}\\,x\_\{B\}\(t\)\+u\_\{B\}\(t\)\+w\(t\)\.\(28\)The termw​\(t\)w\(t\)is therefore not an ad hoc modification; it is the unavoidable remainder between the true local interaction field and any structured approximation operating in an open dynamical system[38](https://arxiv.org/html/2609.11934#bib.bib38);[39](https://arxiv.org/html/2609.11934#bib.bib39)\. Because this remainder is defined over the full interaction field without reference to the structural class of the local block, the open\-system decomposition is class\-agnostic: the triad’s class determines the perturbation panel required for relay cancellation \(design rules 1–3\), but leaves the latent forcing requirement unchanged\.

Sincew​\(t\)w\(t\)is defined as the residual between the true interaction field and the FDU approximation, it is not directly observable; estimating it from the perturbation panel jointly withF~B\\widetilde\{F\}\_\{B\}is therefore necessary\. While the decomposition holds independently of the triad’s structural class, the realisedw​\(t\)w\(t\)is specific to each perturbation condition\. Two structural requirements constrain its form\. First,w​\(t\)w\(t\)is inherently time\-varying: sincexE​\(t\)x\_\{E\}\(t\)changes throughout a perturbation experiment, the environmental contribution to local dynamics cannot be captured by a static offset\. Second,w​\(t\)w\(t\)must be condition\-conditioned: each perturbation condition drives a distinctxE​\(t\)x\_\{E\}\(t\)trajectory, so the environmental drive differs systematically across the perturbation panel\. A finite Fourier time basisΦ​\(t\)\\Phi\(t\)[22](https://arxiv.org/html/2609.11934#bib.bib22)with condition\-dependent learned coefficients provides one estimable parameterization that satisfies both requirements:

Φ​\(t\)=\[1,sin⁡\(2​π​j​t/τ\),cos⁡\(2​π​j​t/τ\)\]j=1J∈ℝ1\+2​J,\\Phi\(t\)=\\bigl\[1,\\;\\sin\(2\\pi j\\,t/\\tau\),\\;\\cos\(2\\pi j\\,t/\\tau\)\\bigr\]\_\{j=1\}^\{J\}\\in\\mathbb\{R\}^\{1\+2J\},\(29\)whereτ\>0\\tau\>0normalizes the time scale andJJsets the frequency resolution\. The basisΦ​\(t\)\\Phi\(t\)is smooth and globally defined, boundingw​\(t\)w\(t\)to a finite\-dimensional function space; condition dependence is encoded in learned coefficients over this basis, allowingw​\(t\)w\(t\)to track the environmental trajectory specific to each perturbation condition\. This parameterization maintains separability from structural inference: intra\-block interactions encoded inF~B\\widetilde\{F\}\_\{B\}are time\-persistent, structurally organized features that hold across perturbation conditions;w​\(t\)w\(t\)is smooth, condition\-specific, and carries no directed graph structure\. The two components therefore impose distinct functional constraints on the observed local dynamics, making joint estimation of signed interaction structure and environmental forcing tractable from the perturbation panel alone\. Four penalties constrainw​\(t\)w\(t\)during training to maintain this separation:ℒenergy\\mathcal\{L\}\_\{\\mathrm\{energy\}\}keeping the forcing minimal,ℒsmooth\\mathcal\{L\}\_\{\\mathrm\{smooth\}\}enforcing temporal regularity,ℒrow\\mathcal\{L\}\_\{\\mathrm\{row\}\}localising environmental drive to a sparse node subset, andℒalign\\mathcal\{L\}\_\{\\mathrm\{align\}\}preventingw​\(t\)w\(t\)from absorbing perturbation\-driven structural signal \(see Methods\)\.

We asked whether the framework recovers the signed interaction structure of a target triad embedded within a larger open system\. One architecture commonly observed in empirical directed networks combines global sparsity with dense local interaction motifs, where individual nodes participate in multiple overlapping interaction subgraphs spanning distinct structural classes\. The proof\-of\-concept network encodes this architecture\(Fig\.[6](https://arxiv.org/html/2609.11934#Sx11.F6)a\): a 20\-node directed signed network with 10 overlapping signed triad subnetworks, comprising 43 directed signed edges across four structural classes \(PxP\_\{x\}: 4;Pin​\-​cP\_\{\\mathrm\{in\\text\{\-\}c\}\}: 3;S−S\_\{\-\}: 2;PcP\_\{c\}: 1\)\. The network is globally sparse: no node participates in more than two triad subnetworks and no node carries more than three signed edges per direction, precluding interaction hubs\. Node 12 \(singleton\) participates in no triad subnetwork; the FDU bank is augmented with a null entry to cover such cases, bringing the total to 73,001 candidate placements \(L​\(20\)\+1L\(20\)\+1\)\. The inference target is the triad of nodes 1, 16, and 18, a classPin​\-​cP\_\{\\mathrm\{in\\text\{\-\}c\}\}triad, with seven directed edges sourced from the three target nodes, five spanning intra\-triad interactions and two outgoing to the broader network\. The triad constitutes an upstream interaction core whose internal dynamics are self\-contained for two of its three nodes: nodes 1 and 18 receive no inputs from outside the triad\. Node 16 is the triad’s sole external interface, receiving one activating input from node 4 and propagating inhibiting signals back to nodes 4 and 5\.

The triad class and its external connectivity jointly determine the perturbation panel, which addresses two structural requirements\. Relay cancellation within thePin​\-​cP\_\{\\mathrm\{in\\text\{\-\}c\}\}triad requires opposite\-sign co\-perturbation on the parallel ordered pairs \(design rule 2\), generating six conditions across all three target node pairs in both directions\. Single\-component perturbations in both directions on each target node isolate each node’s direct downstream effects, making the outgoing connections to the broader network observable in the perturbation response; together with a reference condition, the panel comprises 13 conditions, where comprehensive coverage is motivated by structural ignorance prior to inference\. The perturbation effect profile across all 20 nodes characterizes node 16’s role as external interface: nodes 4 and 5, the direct recipients of node 16’s inhibiting output, show the highest median effect sizes among non\-target nodes; nodes 7 and 15, two steps removed through node 5’s outgoing inhibiting connections, show intermediate effects; the remaining 13 non\-target nodes exhibit near\-zero effects \(median<0\.01<0\.01\), reflecting the triad’s limited external connectivity\(Fig\.[6](https://arxiv.org/html/2609.11934#Sx11.F6)b\)\.

Structural recovery at the 20\-node scale is assessed from the inferred\(α,β\)\(\\alpha,\\beta\)channels\(Fig\.[6](https://arxiv.org/html/2609.11934#Sx11.F6)c\)\. All seven directed edges from target nodes 1, 16, and 18 are recovered: the two edges from node 1 concentrate in the activating channel, the three edges from node 16 concentrate exclusively in the inhibiting channel, and node 18 contributes one activating and one inhibiting edge\. Convergence trajectories for the target source columns confirm the channel separation\(Fig\.[6](https://arxiv.org/html/2609.11934#Sx11.F6)d\): inhibiting\-channel entries separate cleanly between recovered edges and near\-zero non\-edge entries; the activating channel shows the same separation, with non\-edge entries at distinctly lower magnitudes than the recovered edges\. Polarity is correctly assigned across all seven edges, and trajectory fidelity remains low across all perturbation conditions and target nodes, with a median range\-normalised root\-mean\-square error of0\.120\.12\(Fig\.[6](https://arxiv.org/html/2609.11934#Sx11.F6)e\), demonstrating that open\-system confounding, when explicitly modeled via latent forcing, does not degrade inferential precision\.

The FDU attention profile\(Fig\.[6](https://arxiv.org/html/2609.11934#Sx11.F6)f\)provides the structural basis for the recovered interaction magnitudes: each non\-zero entry in the\(α,β\)\(\\alpha,\\beta\)channels is grounded in specific structural primitives whose attention weights quantify their contribution, linking recovered edges to a compact set of contributing structural primitives\. Of the 73,001 candidate placements, 72,995 are suppressed to zero or near\-zero weight \(individual weights<0\.05<0\.05\): this concentration is not merely a sparsity statistic but evidence that a hypothesis space of 73,001 structural candidates was correctly navigated, with a specific structural claim emerging from inference\. The weights within this support are distributed: the classPin​\-​cP\_\{\\mathrm\{in\\text\{\-\}c\}\}target triad has a minimal two\-FDU decomposition, precluding concentration on a single entry, and the surrounding network structure broadens the effective support further\. Both decompositions, \(000100120,001102000\) and \(000102100,001100020\), are represented within the top ten FDUs, with all four corresponding entries appearing at ranks 3, 4, 6, and 8, confirming that attention includes the structural primitives from which the target triad is built \([Supplementary Material](https://arxiv.org/html/2609.11934#Sx11.SSx1)\)\. The three highest\-weight FDUs outside the target triad’s decomposition \(ranks 1, 2, and 5:020002100and010000220, the latter reflecting two distinct node\-set embeddings of the same FDU type\) all involve nodes 4 and 5, the most dynamically responsive non\-target nodes in the perturbation effect profile\(Fig\.[6](https://arxiv.org/html/2609.11934#Sx11.F6)b\)and node 16’s direct downstream targets\(Fig\.[6](https://arxiv.org/html/2609.11934#Sx11.F6)c\)\. Each carries confirmed edges from the full ground\-truth network, establishing that the attention beyond the target triad’s decomposition reflects the network’s real interaction neighborhood rather than residual noise\. The remaining support FDUs each anchor on a single confirmed ground\-truth inhibiting edge: one on the inhibiting edge from node 5 to node 7, and two on the inhibiting edge from node 16 to node 4; each carries two additional edges absent from the ground truth, topologically forced by the 3\-node primitive structure and remaining sub\-threshold\. The attention profile therefore encodes two nested levels of structurally grounded inference: the exact FDU composition of the target triad at the core, and the network’s local interaction structure in the surrounding support\. This structural attribution derives from the FDU hypothesis space itself: because every recovered interaction is expressed as a weighted combination of structural primitives, grounding is produced intrinsically by inference rather than by secondary analysis of interaction magnitudes\.

The generality of structural recovery is assessed by extending inference independently to all ten embedded triads, each with its structure\-derived perturbation panel, spanning all four structural classes and all four panel types present in the network\. The area under the receiver operating characteristic curve \(AUROC\), pooled across all ten triad scoring matrices covering both signed channels for the three source nodes against all 20 targets, is0\.9980\.998; 70 of 75 ground\-truth\-positive channel entries are detected, with 5 false positives \(F1=0\.93=0\.93;[Supplementary Table 1](https://arxiv.org/html/2609.11934#Sx11.SSx1)\)\. The five false negatives are all intra\-triad edges; four inhibiting interactions and one activating edge fall just below the detection threshold, reflecting weak signals at the boundary of detectability rather than structural misidentification\. Outgoing sensitivity is perfect across all ten triads and all four panel types prescribed by the design rules: every directed edge from a target node to a non\-target node is recovered with correct polarity, confirming that the latent forcing strategy successfully decouples environmental confounding from outgoing structural parameters\.ℒalign\\mathcal\{L\}\_\{\\mathrm\{align\}\}follows the same peak\-then\-decline trajectory across all runs, regardless of structural class or panel type \([Supplementary Fig\. 3](https://arxiv.org/html/2609.11934#Sx11.SSx1)\), confirming empirically that the open\-system decomposition and the forcing penalty system are class\-agnostic; the peak magnitude scales with panel complexity, reflecting the larger perturbation\-correlated signal available in richer panels, but all runs converge to a stable low\-loss regime before structural inference is complete\. The remaining three penalties \(ℒenergy\\mathcal\{L\}\_\{\\mathrm\{energy\}\},ℒrow\\mathcal\{L\}\_\{\\mathrm\{row\}\},ℒsmooth\\mathcal\{L\}\_\{\\mathrm\{smooth\}\}\) exhibit uniform behaviour across all structural classes, confirming that the penalty design is robust and that no class or panel configuration places unusual demands on the forcing parameterization\. Disabling the latent forcing module under matched conditions collapses structural recovery to near chance \(AUROC=0\.60=0\.60; F1=0\.27=0\.27\), confirming that the module is necessary for signed structural inference under open\-system confounding\.

Together, these results demonstrate that open\-system confounding is not a fundamental barrier to signed structural inference: it is a structurally separable sub\-problem whose resolution is achievable through parameterization that exploits the functional contrast between the FDU interaction field and the environmental contribution\.

## Discussion

Signed interaction inference in networked dynamical systems has a well\-known failure mode: the quantity the data identify and the quantity the model parameterizes live at different structural levels\. Edge\-level sign assignments are under\-constrained by trajectory data, but the sign pattern of a triad is not, and it is the triad, not the edge, that carries the mechanistic content of the interaction\. Grounding the parameterization at this level resolves the mismatch: the structural constraints that determine what the data can identify and the inferential constraints that determine what the model can recover become the same constraints\.

The FDU framework rests on a specific reductionist commitment: the atomic level of representation is determined by the structure of the inference problem itself\. The triad is that level: the minimal closed structure in which direct and relay pathways coexist between the same node pair\. The triad’s structural class determines the perturbation conditions required for signed edge identifiability, making it the natural primitive for networked dynamical systems in which multi\-path confounding is structurally present\. The FDU dictionary is constructively complete at this level, ensuring that nothing in the signed interaction space falls outside the representation\.

This completeness is the first of four properties that jointly characterize the FDU dictionary as a well\-defined hypothesis space for signed structural inference\. It is complete in the representational sense: every admissible signed triad is exactly representable as a sign\-consistent superposition of at most two tournament FDU primitives\. It is interpretable: each primitive belongs to a defined structural class with explicit pathway and perturbation\-design consequences\. It is self\-specifying: the structural class of the target triad directly prescribes the perturbation conditions sufficient for signed edge identifiability\. It is tractable: theNN\-node bank grows asO​\(N3\)O\(N^\{3\}\)\. Together, these properties define the FDU dictionary as a well\-formed hypothesis space\.

The interpretability of the FDU hypothesis space follows from a specific shift in representational granularity\. Where a conventional directed graph specifies edges independently as pairwise adjacency entries, FDU superposition couples edge choices within each three\-node subgraph, making pathway composition and coherence intrinsic to the representation rather than inferences drawn from independent entries; this is the shift from surface to volumetric triangulation\. The constructive property of the FDU dictionary makes this concrete: complex interaction architectures are not special cases requiring bespoke representation, but natural compositions of the same primitive set\.

Every non\-zero entry in the learned interaction matrices is structurally mediated through FDU support: each recovered activating or inhibiting edge is explained by specific FDU primitives whose attention weights quantify their contribution to the observed dynamics\. This attribution is intrinsic to the inference: it is not derived by comparing recovered parameter values to an external structural vocabulary after the fact, but by the FDU attention mechanism, which constrains every learned interaction to remain expressible as a superposition of interpretable structural units\. In networked systems with multiple overlapping subgraphs, this makes the result structurally interpretable beyond edge recovery: the FDU attention profile identifies the structural primitives consistent with each recovered edge and their relative contributions, grounding inference in the motif representation rather than in interaction magnitudes alone\. The separation of relay\-mediated from direct contributions, and the attribution of edge sign at the structural level, are properties that follow from the choice of hypothesis space rather than from the learning procedure \(cf\.[40](https://arxiv.org/html/2609.11934#bib.bib40)\)\.

Beyond structural attribution, the structure\-derived perturbation panel resolves a specific class of inference errors\. When perturbation design is decoupled from the triad\-level structural class of the target interaction, relay\-mediated and direct contributions remain conflated, parallel\-path triads cannot be resolved into their coherent or incoherent configurations, and inferred networks accumulate spurious direct edges: relay\-mediated connections misattributed to direct influence\. The design rules demonstrate that this decoupling is not inevitable\. The signed structural class of a target triad prescribes the minimal perturbation conditions required for relay cancellation and direct\-arm isolation, making perturbation design a derived consequence of the structural hypothesis rather than an independent choice\. By construction, the prescribed panel probes the identifiable directions in parameter space along which structurally competing hypotheses diverge\. This establishes the forward direction: class\-to\-panel, not panel\-to\-class\. When the true local structure is unknown, a conservative panel combining single\-component perturbations with dual\-polarity co\-perturbations provides coverage across all admissible signed triads, preserving the framework’s applicability without requiring prior class knowledge\.

The present validation is conducted on synthetic data, a deliberate design choice that isolates modeling contributions from the challenges inherent in real\-world settings; the framework’s robustness to measurement noise, irregular temporal acquisition, model misspecification, and preprocessing variation remains to be established\. The model also holds the Hill kinetic parameters \(cooperativity, half\-saturation, and decay rate\) at ground\-truth values, isolating structural recovery from the harder problem of joint parameter estimation; in practice these are unknown and must be inferred jointly with interaction structure, substantially expanding the effective parameter space\. Another consideration is computational complexity: theO​\(N3\)O\(N^\{3\}\)bank grows polynomially withNN, and computational costs for evaluating attention over the full bank become non\-trivial at larger scales; entmax sparsification mitigates this by concentrating structural attribution onto a compact active support, reducing the effective inferential complexity regardless of bank size\. These considerations reflect deliberate scope boundaries of the present proof\-of\-concept rather than fundamental barriers, and together constitute a well\-defined agenda for extending the framework toward real\-world settings\.

Directed signed interactions, targetable perturbations, and time\-resolved observations are the entry conditions for the FDU framework\. They are most directly satisfied in molecular biological systems\. Transcriptional regulatory networks[41](https://arxiv.org/html/2609.11934#bib.bib41);[42](https://arxiv.org/html/2609.11934#bib.bib42)meet all three: genetic knockouts and overexpression provide targeted node perturbations, and transcriptomic time courses yield the required observations\. Independently, the coherent and incoherent feed\-forward loops that appear as identifiable subclasses in the structural taxonomy are the same motifs previously identified as enriched above random expectation in biological regulatory networks[13](https://arxiv.org/html/2609.11934#bib.bib13): a correspondence between what the framework deems inferentially distinguishable and what biology has selected for\. The entry conditions may also be approximated outside experimental settings, when structurally localized natural events \(social disruptions with a known locus[43](https://arxiv.org/html/2609.11934#bib.bib43), exogenous economic shocks to identifiable sectors[44](https://arxiv.org/html/2609.11934#bib.bib44);[45](https://arxiv.org/html/2609.11934#bib.bib45)\) play the role of targeted perturbations, and time\-resolved observations are available\. Testing these settings is beyond the scope of this work, but the entry conditions and design rules provide a principled basis for domain\-specific adaptation\.

The difficulty of signed structural inference has long been read as a data problem\. FDU reframes it as a representational one\. Once the hypothesis space, the perturbation design, and the dynamical model are indexed by the same primitive, signed structure is not something to be extracted from the data but something the data and the model agree on by construction\.

## Methods

Signed structural inference from perturbation time\-series places three distinct computational demands on the framework: predicting trajectories across the full perturbation panel, recovering the sparse FDU interaction structure underlying them, and, in the open\-system setting, absorbing contributions that fall outside the FDU representation\. Three modules address these demands: a reference\-conditioned trajectory predictor, a motif\-regularized structural parameterizer, and a latent forcing absorber, trained jointly under a composite objective and a progressive optimization schedule that preserves the perturbation\-response signal on which structural inference depends\. A complete list of mathematical symbols is provided in the[Supplementary Note](https://arxiv.org/html/2609.11934#Sx11.SSx1)\.

### Model Architecture

The trajectory predictor maps an input triplet\(t,xref​\(t\),u\)\(t,\\,x\_\{\\mathrm\{ref\}\}\(t\),\\,u\)to the predicted statex^​\(t\)∈ℝN\\hat\{x\}\(t\)\\in\\mathbb\{R\}^\{N\}, wherettis the query time,xref​\(t\)∈ℝNx\_\{\\mathrm\{ref\}\}\(t\)\\in\\mathbb\{R\}^\{N\}is the unperturbed reference state at that same time, andu∈ℝNu\\in\\mathbb\{R\}^\{N\}is the perturbation vector\.xref​\(t\)x\_\{\\mathrm\{ref\}\}\(t\)is obtained by direct evaluation of the unperturbed ODE solution at each query time\. Conditioning onxref​\(t\)x\_\{\\mathrm\{ref\}\}\(t\)rather than a fixed initial condition att=0t=0removes the need to learn baseline dynamics from scratch: perturbation response is a differential quantity, and providing the reference state as temporal context directs representational capacity toward the perturbation\-driven deviation that carries structural information\. Becauseuuis an explicit input, a single predictor handles all perturbation conditions simultaneously without per\-condition retraining\.

The network is a multilayer perceptron withtanh\\tanhactivations throughout: their everywhere\-differentiable smoothness is required for exact computation ofd​x^/d​td\\hat\{x\}/dtthrough the network\. The exact time derivatived​x^/d​td\\hat\{x\}/dtis obtained at any queried time point without finite\-difference approximation and on arbitrarily irregular time grids; it is computed via forward\-mode automatic differentiation, specifically a Jacobian\-vector product with unit direction in the time axis, in a single forward pass through the network\. Layer normalization[46](https://arxiv.org/html/2609.11934#bib.bib46)and dropout[47](https://arxiv.org/html/2609.11934#bib.bib47)stabilize optimization under the composite physics\-informed objective; weights are initialized from a Xavier uniform distribution, which preserves signal variance across layers fortanh\\tanhnonlinearities[48](https://arxiv.org/html/2609.11934#bib.bib48)\.

The structural parameterizer separates the inference of interaction topology from the inference of interaction magnitude\. Topology, the set of directed interactions that are structurally supported, is determined by entmax\-regularized[36](https://arxiv.org/html/2609.11934#bib.bib36)attention over the FDU bank: learnable logits overLLcandidate placements are mean\-centered and transformed to a sparse simplex\-constrained weight vectore∈ℝLe\\in\\mathbb\{R\}^\{L\}, inducing the support masks\(σ↑,σ↓\)\(\\sigma\_\{\\uparrow\},\\,\\sigma\_\{\\downarrow\}\)that encode the structural support over directed edges under the inferred motif mixture\. Magnitude is encoded separately through nonnegative strength matricesα\\alphaandβ\\beta, enforced non\-negative via softplus applied to unconstrained raw parameters; their Hadamard products with the topology masks yield the effective interaction strengthsαeff=α⊙σ↑\\alpha^\{\\mathrm\{eff\}\}=\\alpha\\odot\\sigma\_\{\\uparrow\}andβeff=β⊙σ↓\\beta^\{\\mathrm\{eff\}\}=\\beta\\odot\\sigma\_\{\\downarrow\}that enter the dynamics, where⊙\\odotdenotes element\-wise multiplication\. This factorization separates topology from magnitude estimation: FDU attention resolves structural support through a sparse selection over the motif dictionary, while\(α,β\)\(\\alpha,\\beta\)scale interaction strengths continuously from data\. Two implementation choices support stable operation over the large FDU bank: logits are mean\-centered before transformation, exploiting the shift\-invariance of the entmax operator for numerical stability; and the weighted sum over FDU masks is evaluated over the active entmax support \(entries where the attention weight is strictly positive\), keeping bank traversal tractable asLLscales with network size\. The entmax operator reduces to softmax when its concentration parameter equals one, enabling smooth initialization; motif attention logits are initialized to zero, placing a uniform prior over allLLcandidate placements at the start of training so that the sparse structural support emerges entirely through optimization with no initial preference among FDUs\. The schedule governing concentration across training is described in the optimization subsection\. The Hill parametersκ\\kappa,KKand decay ratesγ\\gammaare fixed rather than learned, restricting inference to the interaction structure encoded in\(αeff,βeff\)\(\\alpha^\{\\mathrm\{eff\}\},\\beta^\{\\mathrm\{eff\}\}\)\.

A latent forcing module introduces a condition\-dependent signalw​\(t,u\)w\(t,u\)that absorbs contributions to the local dynamics that the FDU\-regularized interaction structure cannot represent, including influences from nodes outside the modeled subnetwork\. The forcing signal is expressed asw​\(t,u\)=V​C​\(u\)​Φ​\(t\)w\(t,u\)=V\\,C\(u\)\\,\\Phi\(t\), whereΦ​\(t\)∈ℝ1\+2​J\\Phi\(t\)\\in\\mathbb\{R\}^\{1\+2J\}is a deterministic Fourier time basis evaluated analytically at each query time,C​\(u\)C\(u\)is a single\-hidden\-layer network with hidden width 32 andtanh\\tanhactivation mapping the perturbation vectoruuto a matrix of basis coefficients of shaper×\(1\+2​J\)r\\times\(1\+2J\), andV∈ℝN×rV\\in\\mathbb\{R\}^\{N\\times r\}is a learnable node mixing matrix\. Using an analytic Fourier basis withJJfrequency components at time scaleτ\\tauanchorsw​\(t,u\)w\(t,u\)to a smooth, finite\-dimensional function space without learned temporal parameters; condition dependence is encoded entirely throughC​\(u\)C\(u\), so each perturbation condition drives a distinct environmental trajectory while sharing the same temporal basis structure\. The rankrrofVVcontrols the intrinsic complexity of the environmental drive: a low rank reflects the assumption that a small number of latent programs account for contributions from outside the modeled subnetwork\. Numerical values for all architectural parameters are specified in the Simulation and Experimental Setup subsection\.

### Training Objective

The training objective is applied to the output of a differentiable chain: attention weights over the FDU bank induce structural support masks that gate the interaction strengths, which determine the ODE right\-hand side against which the physics residual is evaluated\. The composite loss differentiates through this entire chain, driving motif attention toward the sparse FDU support consistent with the observed perturbation responses\. The resulting objective encodes a three\-way tension between ODE consistency, data fidelity, and structural regularization, with terms operating at different levels of the inference problem\. The central term is the physics residual, which enforces that the predicted trajectory satisfies the governing ODE at every training time point:

ℒphysics=‖d​x^d​t−\(F​\(x^\)−γ​x^\+u\+w​\(t,u\)\)‖2,\\mathcal\{L\}\_\{\\mathrm\{physics\}\}=\\left\\\|\\frac\{d\\hat\{x\}\}\{dt\}\-\\Bigl\(F\(\\hat\{x\}\)\-\\gamma\\hat\{x\}\+u\+w\(t,u\)\\Bigr\)\\right\\\|^\{2\},\(30\)whereF​\(x^\)F\(\\hat\{x\}\)is the net signed interaction drive computed from the Hill\-type FDU parameterization \(αeff\\alpha^\{\\mathrm\{eff\}\},βeff\\beta^\{\\mathrm\{eff\}\},κ\\kappa,KK\),γ​x^\\gamma\\hat\{x\}is the first\-order decay term,uuis the perturbation input, andw​\(t,u\)w\(t,u\)is the latent forcing signal\. Here∥⋅∥2\\\|\\cdot\\\|^\{2\}denotes the mean squared residual across the training batch and allNNstate dimensions\.

Four additional terms complete the training objective\. Data fidelity \(ℒdata\\mathcal\{L\}\_\{\\mathrm\{data\}\}\) is enforced as a point\-wise mean squared error between the predicted state and the observed trajectory, ensuring agreement with ground\-truth measurements at each queried time point\. Initial\-condition anchoring \(ℒinit\\mathcal\{L\}\_\{\\mathrm\{init\}\}\) is enforced through a separate forward pass on a dedicated batch of initial\-time samples, giving explicit control over the initial state independently of the physics residual\. Non\-negativity \(ℒneg\\mathcal\{L\}\_\{\\mathrm\{neg\}\}\) is encouraged through a soft quadratic penalty on the negative part of the predicted state, ensuring consistency with the nonnegative domain required by the Hill\-type interaction model\. Channel exclusivity \(ℒactinh\\mathcal\{L\}\_\{\\mathrm\{actinh\}\}\) penalizes the mean of the elementwise productαeff⊙βeff\\alpha^\{\\mathrm\{eff\}\}\\odot\\beta^\{\\mathrm\{eff\}\}, discouraging simultaneous activating and inhibiting influence on the same directed edge; the penalty operates on the post\-mask effective strengths, so that only structurally supported interactions are subject to exclusivity pressure\.

These five terms are combined through an uncertainty\-based adaptive weighting scheme[49](https://arxiv.org/html/2609.11934#bib.bib49)in which each term carries a learnable log\-variance parameterscs\_\{c\}\. The uncertainty\-weighted contribution of termccis:

ℒcuw=12​\(e−sc​ℒc\+sc\),\\mathcal\{L\}\_\{c\}^\{\\mathrm\{uw\}\}=\\tfrac\{1\}\{2\}\\\!\\left\(e^\{\-s\_\{c\}\}\\,\\mathcal\{L\}\_\{c\}\+s\_\{c\}\\right\),\(31\)wheree−sce^\{\-s\_\{c\}\}is the effective weight and the additivescs\_\{c\}prevents trivial solutions in which all variances grow without bound\. The total loss is:

ℒtotal=∑cℒcuw\+∑kλk​ℒkforcing,\\mathcal\{L\}\_\{\\mathrm\{total\}\}=\\sum\_\{c\}\\mathcal\{L\}\_\{c\}^\{\\mathrm\{uw\}\}\+\\sum\_\{k\}\\lambda\_\{k\}\\,\\mathcal\{L\}\_\{k\}^\{\\mathrm\{forcing\}\},\(32\)where the first sum is over the five adaptive terms and the second over the forcing penalties described below\. Log\-variance parameters are initialized to establish a learning hierarchy:sphysics=sdata=−2s\_\{\\mathrm\{physics\}\}=s\_\{\\mathrm\{data\}\}=\-2\(effective weighte2≈7\.4e^\{2\}\\approx 7\.4\), whilesinit=sneg=sactinh=0s\_\{\\mathrm\{init\}\}=s\_\{\\mathrm\{neg\}\}=s\_\{\\mathrm\{actinh\}\}=0\(effective weight11\)\. This initialization ensures ODE consistency and trajectory fidelity dominate early in training, preventing convergence to solutions that satisfy the structural priors before the dynamics are correctly established\.

The five adaptive terms correspond to objectives whose relative importance is genuinely uncertain and data\-dependent; the uncertainty\-weighting mechanism allows their contributions to adjust without manual tuning\. Four penalties governing the latent forcing signal are instead applied with pre\-specified weightsλk\\lambda\_\{k\}outside the adaptive tier\. These penalties are not imposed as first\-principles laws of the environmental drive; they function as identifiability\-oriented regularizers that encode the intended role ofw​\(t,u\)w\(t,u\)as a minimal, smooth residual with sparse node support that does not duplicate the perturbation channel: an energy penalty \(ℒenergy\\mathcal\{L\}\_\{\\mathrm\{energy\}\}\) controls the overall magnitude ofw​\(t,u\)w\(t,u\); a temporal smoothness penalty \(ℒsmooth\\mathcal\{L\}\_\{\\mathrm\{smooth\}\}\), computed via the same Jacobian\-vector product mechanism, enforces thatw​\(t,u\)w\(t,u\)varies smoothly in time; a row\-wise L2 penalty onVV\(ℒrow\\mathcal\{L\}\_\{\\mathrm\{row\}\}\) encodes an identifiability prior, restricting exogenous drive to a subset of nodes to prevent confounding of external influence with endogenous regulation; and an anti\-alignment penalty \(ℒalign\\mathcal\{L\}\_\{\\mathrm\{align\}\}\) enforces a separation\-of\-roles constraint:uuis the perturbation channel carrying discriminative structural information, andw​\(t,u\)w\(t,u\)must not reproduce the perturbation signal\. These penalties are placed outside the adaptive tier because the adaptive mechanism could legitimately relax structural regularizers when the data warrants it, but unconstrained forcing growth is not a valid optimization, since it would silently absorb structural signal while appearing correct to the loss\.

### Optimization Protocol

The trajectory predictor, structural parameterizer, and latent forcing module are optimized jointly using a single Adam optimizer with a shared learning rate; the forcing penalties and structural regularizers govern how the residual is partitioned between them\.

The concentration parameter of the entmax operator is annealed from the softmax limit, placing a uniform prior over the FDU bank, to a near\-sparsemax regime via a cosine schedule: slow initially to allow broad exploration of the structural hypothesis space, accelerating as the perturbation\-response signal disambiguates the correct FDU support, and decelerating again as concentration approaches its target to ease commitment rather than impose it abruptly\.

Log\-variance parameters are clamped to a symmetric interval after each gradient step, bounding the effective loss weights and preventing any single term from dominating or vanishing entirely\. A learning rate scheduler reduces the learning rate when validation loss improvement falls below a relative threshold over consecutive epochs, providing finer convergence after the initial learning phase\. Gradient clipping bounds the gradient norm per update step, stabilizing the backward pass through the physics residual\. Training terminates when validation loss shows no improvement for a fixed patience window; the model checkpoint from the best\-performing validation epoch is restored at convergence\.

### Simulation and Experimental Setup

Ground\-truth trajectories are generated by numerical integration of the FDU\-regularized ODE with known interaction structure using the RK45 \(Runge\-Kutta 45\) solver overt∈\[0,100\]t\\in\[0,\\,100\]\. All node states are initialized uniformly toxi​\(0\)=5x\_\{i\}\(0\)=5\. Hill function parameters are set toκ=2\\kappa=2andK=20K=20, uniform across all source nodes, and first\-order decay rates toγ=0\.1\\gamma=0\.1for all nodes\. Ground\-truth interaction strengths are binary: each directed edge in the FDU\-constructed network carries unit strength \(αi​j=1\\alpha\_\{ij\}=1orβi​j=1\\beta\_\{ij\}=1\)\. Withκ=2\\kappa=2, the Hill half\-activation threshold isK1/κ=20≈4\.5K^\{1/\\kappa\}=\\sqrt\{20\}\\approx 4\.5; the uniform initial statexi​\(0\)=5x\_\{i\}\(0\)=5lies near this threshold, placing the system in the sensitive dynamical regime where source activity changes produce the largest fractional response and perturbation\-driven deviations carry maximal structural information\.

The 20\-node ground\-truth network was constructed by placing 10 signed triad subnetworks, drawn uniformly at random from the 512 admissible signed triads, onto randomly selected node triplets under a fixed random seed\. Construction enforces three per\-node sparsity constraints: signed in\-degree and out\-degree are each bounded at 3, each node participates in at most 2 embedded triads, and no directed edge is reused across triads\. The resulting network comprises 43 directed signed edges spanning 15 unidirectional node pairs and 14 mutual node pairs\.

Observations are sampled sparsely across four temporally disjoint windows designed to capture distinct dynamical phases: initialization \(t∈\[0,0\.1\]t\\in\[0,\\,0\.1\]\), early\-to\-mid transient \(t∈\[1,30\]t\\in\[1,\\,30\]\), mid\-to\-late transient \(t∈\[40,60\]t\\in\[40,\\,60\]\), and late steady state \(t∈\[70,90\]t\\in\[70,\\,90\]\)\. Spanning these phases distributes structural information across the full trajectory, and the boundaries are fixed across all experiments rather than tuned to individual motif trajectories\. Each window contributes 75 training and 25 validation time points, drawn randomly from a uniform distribution within the window\. A held\-out test set of 100 time points is drawn uniformly from the full horizont∈\[0,100\]t\\in\[0,\\,100\], spanning both within\-window and gap regions; approximately31%31\\%of the horizon lies outside any training window, providing evaluation in temporally unobserved regions\.

Perturbation panels are assigned per the structural class of the target motif\. Each condition is specified by a signed amplitude vectoru∈ℝNu\\in\\mathbb\{R\}^\{N\}: non\-zero entries carry a signed amplitude of fixed magnitude1\.01\.0, uniform across nodes and equal in absolute value for positive and negative directions, and all other entries are zero; the unperturbed reference setsu=𝟎u=\\mathbf\{0\}\. Each non\-zerouiu\_\{i\}is applied as a smoothly gated step with sigmoidal onset att=0t=0and deactivation at the end of the simulation horizon, with a transition timescale of 1 time unit; the sigmoidal profile avoids discontinuities in the ODE right\-hand side\. Single\-component conditions perturb one node at a time in both directions; co\-perturbation conditions perturb two nodes simultaneously with same\-sign or opposite\-sign combinations as specified by the structural class\. Conditions cover all nodes and node pairs in the target subset exhaustively\.

The trajectory predictor uses depth 6 and hidden width 512, with layer normalization and dropout rate0\.10\.1\. The entmax concentration is annealed from1\.01\.0to1\.51\.5via a cosine schedule:

c​\(δ\)=ci\+\(cf−ci\)⋅1−cos⁡\(π​δ/Tmax\)2,c\(\\delta\)=c\_\{i\}\+\(c\_\{f\}\-c\_\{i\}\)\\cdot\\frac\{1\-\\cos\(\\pi\\delta/T\_\{\\mathrm\{max\}\}\)\}\{2\},\(33\)wherecic\_\{i\}andcfc\_\{f\}are the initial and final concentration values,δ\\deltais the current epoch, andTmaxT\_\{\\mathrm\{max\}\}the total number of epochs\. The latent forcing module, active in the 20\-node experiment, usesJ=4J=4Fourier frequency components, time scaleτ=100\\tau=100, and rankr=1r=1; the latent forcing module is disabled in the triad\-only experiments, where the closed\-system assumption renders environmental contributions structurally absent\. Forcing penalty weights follow the same cosine schedule form as the entmax concentration, decaying from10−310^\{\-3\}to10−410^\{\-4\}over training for all four penalties \(ℒenergy\\mathcal\{L\}\_\{\\mathrm\{energy\}\},ℒsmooth\\mathcal\{L\}\_\{\\mathrm\{smooth\}\},ℒrow\\mathcal\{L\}\_\{\\mathrm\{row\}\},ℒalign\\mathcal\{L\}\_\{\\mathrm\{align\}\}\); the decay reflects that strong early regularization prevents forcing from absorbing structural signal during the critical early phase, while relaxing constraints later allows it to absorb genuine residuals\. All parameters are optimized with Adam at learning rate1×10−41\\times 10^\{\-4\}with batch size 50, for a maximum of 1000 epochs\. The learning rate scheduler applies a reduction factor of0\.50\.5when validation loss improvement falls below10−310^\{\-3\}relative over 10 consecutive epochs, with a minimum learning rate of10−610^\{\-6\}\. Gradient norms are clipped to a maximum of5\.05\.0per update step\. Log\-variance parameters are clamped within\[−log⁡\(150\),log⁡\(150\)\]\[\-\\log\(150\),\\,\\log\(150\)\]after each gradient step, bounding effective loss weights to the interval\[1/150,150\]\[1/150,\\,150\]\. Early stopping is applied with a patience of 50 epochs\.

### Computational Resources

All experiments were run on a CPU\-only platform \(2\.50 GHz, 128 GiB RAM\)\. Each 3\-node experiment required approximately2\.742\.74s per epoch \(wall\-clock, including training, validation, and checkpointing\), corresponding to approximately 46 min for 1000 epochs\. The 20\-node experiment required approximately5\.625\.62s per epoch, corresponding to approximately 1 h 34 min\. Although the 20\-node FDU bank is approximately 1000\-fold larger than the 3\-node bank \(73,000 vs 70 candidate placements\), training time increased by less than twofold, a direct consequence of entmax sparsification concentrating computation onto a small active support regardless of bank size\. Experiments were implemented in Python 3\.12 using PyTorch 2\.10 for model training and automatic differentiation, SciPy 1\.17 \(RK45 solver\) for ground\-truth ODE integration, and NumPy 1\.26 for numerical operations\.

## Data Availability

All datasets used in this study are synthetic and were generated computationally using the code available at[https://github\.com/AstraZeneca/fdu\-pisi](https://github.com/AstraZeneca/fdu-pisi)\. No empirical data were collected or used\. The datasets generated and/or analysed during the current study are available in the Zenodo repository,[https://doi\.org/10\.5281/zenodo\.20377559](https://doi.org/10.5281/zenodo.20377559)\.

## Code Availability

The source code, documentation, and all materials required to reproduce the experiments are publicly available on GitHub at[https://github\.com/AstraZeneca/fdu\-pisi](https://github.com/AstraZeneca/fdu-pisi)\.

## Acknowledgments

The author is grateful to Dr\. Virginia Savova for supervision, strategic direction, and critical reading of the manuscript, and AstraZeneca Research and Development for institutional support\.

## Funding

This work was supported by AstraZeneca US\.

## Author Contributions

N\.N\. conducted all research and authored the paper\.

## Competing Interests

The author is an employee of AstraZeneca US\.

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![[Uncaptioned image]](https://arxiv.org/html/2609.11934v1/x1.png)

Fig\. 1:Motif\-centric volumetric triangulation and FDU\-regularized physics\-informed neural ODE framework for joint trajectory prediction and structural inference\.\(a\)Surface triangulation encodes geometry at the boundary: \(top\) a scalar field on a flat triangulated mesh \(finite\-element heat diffusion\); \(middle\) a three\-dimensional terrain represented as a height field on a surface mesh; \(bottom\) a complex surface mesh capturing only boundary topology\. In each case, triangulation defines connectivity among surface elements without imposing constraints on the interior\. In network terms, a conventional directed graph is the surface analog: pairwise connectivity encoded without constraining how edges compose into multi\-path structures\.\(b\)Volumetric triangulation of an interaction network\. Each Fundamental Dynamical Unit \(FDU\) is a signed three\-node subgraph \(inset: nodes A, B, C with signed directed edges\) that couples direct and relayed interaction pathways within a single primitive\. A network is represented as a sparse superposition of FDUs, visualised here as interior\-filling triangular primitives within a spherical interaction network \(purple\), so that every inferred structure carries explicit constraints on interior pathway composition, not only on pairwise connectivity\.\(c\)Joint training procedure\. Time\- and condition\-aware observations \(timett, system statexx, and perturbation input𝐮\\mathbf\{u\}\) are passed to a neural network that produces the predicted state and jointly infers network parametersϕ\\boldsymbol\{\\phi\}, ODE parameters𝜽\\boldsymbol\{\\theta\}, and FDU composition𝐝\\mathbf\{d\}\. A physics\-informed constraint layer penalises violations of the governing ODE during backpropagation, coupling data fidelity with dynamical consistency throughout training\.\(d\)Inference outputs of the trained model\. Top: the inferred FDU composition is extracted as a sparse attention\-weighted superposition over the FDU bank and reported as an interaction network \(green\) for mechanistic interpretability\. Middle and Bottom: the trained model integrates the learned ODE forward in time to produce continuous\-time trajectory predictions \(orange, dashed\) under arbitrary perturbation conditions; predicted trajectories are shown against observed state variables \(blue, solid\)\.![[Uncaptioned image]](https://arxiv.org/html/2609.11934v1/x2.png)

Fig\. 2:FDU dictionary construction and permutation\-invariant structural classification of signed three\-node triads\.\(a\)FDU decomposition of a triad with one mutual pair \(m=1m=1\)\. The pairwise\-connected triad010100120, containing a mutual activating interaction on\{A,B\}\\\{A,B\\\}, is not a tournament and admits a unique two\-FDU reconstruction via superposition \(∨\\vee\):000100120∨\\vee010000120\.\(b\)FDU decomposition of a triad with three mutual pairs \(m=3m=3\)\. The triad011201110admits exactly2m−1=42^\{m\-1\}=4distinct minimal two\-FDU decompositions; each pair jointly realizes all three mutual interactions in opposite directions while preserving forced edge orientations and signs\.\(c\)Extension to self\-regulation\. The triad000102102, combining a triadic architecture with an auto\-inhibiting self\-edge on node C, is expressed as the superposition of the triadic FDU000102100and the unary auto\-inhibiting self\-motif000000002\.\(d\)Taxonomy decision tree\. Each signed triad is tested for the presence of at least one parallel ordered pair \(classPP: parallel\-path; classSS: single\-path\), then refined by coherence state within classPP\(PcP\_\{c\}: coherent;Pin​\-​cP\_\{\\mathrm\{in\\text\{\-\}c\}\}: incoherent;PxP\_\{x\}: mixed\) or by cycle polarity within classSS\(S\+S\_\{\+\}: positive;S−S\_\{\-\}: negative\)\.\(e\)One canonical representative per structural class:000100110\(PcP\_\{c\}\),000100120\(Pin​\-​cP\_\{\\mathrm\{in\\text\{\-\}c\}\}\),000201220\(PxP\_\{x\}\),001100010\(S\+S\_\{\+\}\),002100010\(S−S\_\{\-\}\)\.\(f\)Distribution of all 512 pairwise\-connected signed triads across the five structural classes; count and percentage of total shown above each bar\. Bar colors correspond to the structural class color bands in panel \(d\)\. In panels a–c and e, arrows denote activating edges, filled circles denote inhibiting edges, and nodes are labeled A, B, C \(equivalently, nodes 1, 2, 3 in the main text\)\.![[Uncaptioned image]](https://arxiv.org/html/2609.11934v1/x3.png)

Fig\. 3:Relay\-cancellation panel design: worked examples across structural classes\.Worked examples of the relay\-cancellation derivation across structural classes, illustrating all distinct relay\-cancellation polarity cases\. Each row reports: motif barcode; structural class; signed edge indicators\(a,b,c\)\(a,b,c\)for the parallel ordered pairA→CA\\to C, whereaa\(sign of first relay legA→BA\\to B\),bb\(sign of second relay legB→CB\\to C\), andcc\(sign of direct edgeA→CA\\to C;c=0c=0for classSSmotifs, which carry no directA→CA\\to Cedge\); the minimal co\-perturbation panel\(uA,uB\)\(u\_\{A\},u\_\{B\}\)applied to nodesAAandBB; the relay\-node responseΔ​xB=a​uA\+uB\\Delta x\_\{B\}=a\\,u\_\{A\}\+u\_\{B\}; the target responseΔ​xC=c​uA\+b​\(a​uA\+uB\)\\Delta x\_\{C\}=c\\,u\_\{A\}\+b\(a\\,u\_\{A\}\+u\_\{B\}\); and the mechanistic interpretation of the result at nodeCC\. For classSSmotifs \(rows 1–2; Rule 1\), single\-component perturbations produce relay\-only transmission atCC, with no direct arm \(c=0c=0\)\. For classPcP\_\{c\}andPin​\-​cP\_\{\\mathrm\{in\\text\{\-\}c\}\}motifs \(rows 3–5; Rule 2\), the co\-perturbation polarity is matched to the first relay leg sign: opposite\-sign \(uB=−uAu\_\{B\}=\-u\_\{A\}\) whena=\+1a=\+1\(rows 3 and 5\), same\-sign \(uB=\+uAu\_\{B\}=\+u\_\{A\}\) whena=−1a=\-1\(row 4\), enforcingΔ​xB=0\\Delta x\_\{B\}=0and isolating the direct armc​uAc\\,u\_\{A\}at nodeCC\. For the classPxP\_\{x\}motif \(row 6; Rule 3\), the same\-sign panel cancels the relay onA→CA\\to C\(a=−1a=\-1\); both co\-perturbation polarities are required to yield the relay\-off and relay\-on attribution contrast across the mixed coherence structure\. In the Motif column, arrows denote activating edges, filled circles denote inhibiting edges, and nodes are labeled A, B, C \(equivalently, nodes 1, 2, 3 in the main text\)\.![[Uncaptioned image]](https://arxiv.org/html/2609.11934v1/x4.png)

Fig\. 4:FDU\-regularized parameterization: embedding, attention mechanism, and signed channel dynamics\.\(a\)FDU embedding into theNN\-node setting\. A signed three\-node primitive \(3×33\\times 3FDU, left\) is embedded onto node triplets within anNN\-node system to yield anN×NN\\times Nbinary mask pair\(M↑\(ℓ\),M↓\(ℓ\)\)\(M\_\{\\uparrow\}^\{\(\\ell\)\},M\_\{\\downarrow\}^\{\(\\ell\)\}\)\(illustrated forN=5N=5, right\), specifying which directed edges are activating versus inhibiting under motifℓ\\ell\.\(b\)Entmax attention sparsity\. Learnable logitsz∈ℝLz\\in\\mathbb\{R\}^\{L\}are mapped to simplex\-constrained attention weightse=ω​\(z\)e=\\omega\(z\)via the entmax normalization\. Increasing the entmax concentration produces progressively more decisive \(sparse\) attention, concentrating weight on a small number of dominant FDU primitives; the softmax and sparsemax limits are shown for reference\.\(c\)The three independent components of the FDU\-regularized parameterization \(all values schematic and for illustration purposes only\)\. Left: sparse entmax attention weightseℓe\_\{\\ell\}over the FDU bank; three dominant motifs \(hereℓ1\\ell\_\{1\},ℓ4\\ell\_\{4\},ℓ7\\ell\_\{7\}\) carry the majority of attention mass\. Center: the nine FDU motifs in the bank, each represented as a3×33\\times 3binary mask pair \(M↑\(ℓ\),M↓\(ℓ\)M\_\{\\uparrow\}^\{\(\\ell\)\},M\_\{\\downarrow\}^\{\(\\ell\)\}\); blue cells indicate activating edges and red cells indicate inhibiting edges, each at uniform full intensity\. Right: the learned interaction magnitude matricesα\\alpha\(activating, blue\) andβ\\beta\(inhibiting, red\); cell intensity reflects magnitude, encoding interaction strength independently of the FDU\-derived structural topology\.\(d\)Hill\-type signed channel dynamics\. Left column: activating Hill functionH↑​\(x;κ,K\)H\_\{\\uparrow\}\(x;\\kappa,K\)and its derivative∂H↑/∂x\>0\\partial H\_\{\\uparrow\}/\\partial x\>0for cooperativity coefficientsκ=1,2,4,8\\kappa=1,2,4,8\(K=1K=1\)\. Right column: inhibiting Hill functionH↓​\(x;κ,K\)H\_\{\\downarrow\}\(x;\\kappa,K\)and its derivative∂H↓/∂x<0\\partial H\_\{\\downarrow\}/\\partial x<0\. The sign\-definite derivatives establish that edges withαi​jeff\>0\\alpha\_\{ij\}^\{\\mathrm\{eff\}\}\>0induce small\-signal activation \(∂Fi/∂xj\>0\\partial F\_\{i\}/\\partial x\_\{j\}\>0\) and edges withβi​jeff\>0\\beta\_\{ij\}^\{\\mathrm\{eff\}\}\>0induce small\-signal inhibition \(∂Fi/∂xj<0\\partial F\_\{i\}/\\partial x\_\{j\}<0\), preserving a direct correspondence between learned dynamics and the signed motif representation\.![[Uncaptioned image]](https://arxiv.org/html/2609.11934v1/x5.png)

Fig\. 5:FDU\-regularized physics\-informed inference recovers signed interaction structure, motif identity, and dynamics across six representative motifs spanning all five structural classes\.Each panel\(a–f\)presents results for one representative motif in three rows\. The same six motifs appear in\(Fig\.[3](https://arxiv.org/html/2609.11934#Sx11.F3)\)\. Top row: ground\-truth connectivity diagram \(left; arrows denote activating edges, filled circles denote inhibiting edges\) and inferred normalised effective interaction strengths as dotplots \(right\), with activating \(α\\alpha\) and inhibiting \(β\\beta\) channels shown separately; rows \(y\-axis\) index target nodes, columns \(x\-axis\) index source nodes, and dot size encodes normalised effective strength\. Interactions at or above50%of the globally normalised maximum are displayed\. Second row: range\-normalised root mean square error heatmap \(rows: perturbation conditions; columns: state components A, B, C\), evaluated on held\-out time points \(left\), and FDU attention distribution over the motif bank \(right\), with attention weight on the x\-axis and motif barcodes on the y\-axis; the pink arrow marks the generating motif\. Third row: element\-wise activation \(αx​y\\alpha\_\{xy\}, left\) and inhibiting \(βx​y\\beta\_\{xy\}, right\) coefficient trajectories, with training epoch on the x\-axis and normalised coefficient value on the y\-axis; legend entries denote source→\\totarget pairs\.\(a\)002100010\(classS−S\_\{\-\}\)\.\(b\)001100010\(classS\+S\_\{\+\}\)\.\(c\)000100110\(classPcP\_\{c\}\)\.\(d\)000200210\(classPcP\_\{c\}\)\.\(e\)000100120\(classPin​\-​cP\_\{\\mathrm\{in\\text\{\-\}c\}\}\)\.\(f\)000201220\(classPxP\_\{x\}\): attention distributes across two dominant FDUs \(000201200and000200220\), consistent with the unique minimal two\-FDU decomposition for this motif\.![[Uncaptioned image]](https://arxiv.org/html/2609.11934v1/x6.png)

Fig\. 6:FDU\-regularized structural inference recovers signed interaction structure and motif identity from a network\-embedded triad under open\-system confounding\.\(a\)Ground\-truth 20\-node directed regulatory network comprising 43 directed signed edges across 10 overlapping signed triad subnetworks\. Target nodes 1, 16, and 18 \(orange\) form a classPin​\-​cP\_\{\\mathrm\{in\\text\{\-\}c\}\}triad\. Blue arrows: activating edges; red arrows: inhibiting edges\.\(b\)Per\-node perturbation effect profile across all 20 nodes, with perturbation effect magnitude on the x\-axis \(log scale; measured as median standard\-deviation\-normalised root mean square error relative to the unperturbed reference across all non\-reference conditions\) and nodes sorted by effect magnitude on the y\-axis\. Points indicate median perturbation effect magnitude across all non\-reference conditions; bars extend from median to Q75 \(solid\) and Q75 to maximum \(dashed\)\. Target nodes \(orange\) are sorted to the top\.\(c\)Inferred normalised effective interaction strengths as dotplots for the three target source columns \(nodes 1, 16, 18\), with activating \(α\\alpha\) and inhibiting \(β\\beta\) channels shown separately; rows \(y\-axis\) index all 20 nodes, and dot size encodes normalised effective strength\. Interactions at or above25%of the globally normalised maximum are displayed\.\(d\)Element\-wise activation \(αx​y\\alpha\_\{xy\}, left\) and inhibiting \(βx​y\\beta\_\{xy\}, right\) coefficient trajectories for the three target source columns, with training epoch on the x\-axis and normalised coefficient value on the y\-axis\. Labeled curves identify the seven directed edges from target nodes 1, 16, and 18\.\(e\)Range\-normalised root mean square error heatmap \(rows: 13 perturbation conditions; columns: 3 target nodes\), evaluated on held\-out time points\.\(f\)Top 10 FDU attention weights, with attention weight on the x\-axis and motif barcodes on the y\-axis\. Pink arrows mark the FDU components of both minimal decompositions of the targetPin​\-​cP\_\{\\mathrm\{in\\text\{\-\}c\}\}triad\. Orange arrows mark FDUs from the surrounding regulatory environment \(ranks 1, 2, and 5\), all involving nodes 4 and 5\. The barcode010000220appears at ranks 2 and 5, reflecting two distinct node\-set embeddings of the same FDU type, spanning nodes\{4,5,7\}\\\{4,5,7\\\}and\{4,5,15\}\\\{4,5,15\\\}respectively; both carry confirmed edges from the full network\.Table 1:Minimal perturbation\-panel assignments across permutation\-invariant structural classes of signed triads\.Counts over all 512 pairwise\-connected signed triads\. Each triad is assigned to the minimal perturbation\-panel category sufficient to cancel relayed contributions on all parallel ordered pairs and to support attribution across all coherence configurations present\. Column\(u,0\)\(u,0\): single\-node perturbations suffice \(classSS; no parallel ordered pairs\) \(Rule 1\)\. Columns\(u,−u\)\(u,\-u\)and\(u,u\)\(u,u\): relay cancellation requires one co\-perturbation polarity, determined by the first relay leg sign \(Rule 2\)\. Column\(u,−u&u\)\(u,\-u\\,\\&\\,u\): both co\-perturbation polarities,\(u,−u\)\(u,\-u\)and\(u,u\)\(u,u\), are required in the panel, arising from conflicting relay\-cancellation requirements across parallel ordered pairs \(Rule 2\), or from mixed coherence structure in classPxP\_\{x\}\(Rule 3\)\.
### Description of Additional Supplementary Files

Supplementary Material\.Complete catalog of all 512 pairwise\-connected signed three\-node triads with structural classifications, perturbation panel assignments, and FDU decompositions\. Each row corresponds to one admissible signed triad\. Columns:Triad ID, the 9\-character ternary motif code obtained by flattening theT∈\{0,1,2\}3×3T\\in\\\{0,1,2\\\}^\{3\\times 3\}encoding row\-wise \(0: no edge; 1: activating; 2: inhibiting\);Category, the permutation\-invariant structural class \(PcoherentP\_\{\\mathrm\{coherent\}\},PincoherentP\_\{\\mathrm\{incoherent\}\},PmixedP\_\{\\mathrm\{mixed\}\},SpositiveS\_\{\\mathrm\{positive\}\},SnegativeS\_\{\\mathrm\{negative\}\}; corresponding toPcP\_\{c\},Pin​\-​cP\_\{\\mathrm\{in\\text\{\-\}c\}\},PxP\_\{x\},S\+S\_\{\+\},S−S\_\{\-\}in the main text\);Panel, the minimal perturbation panel assignment prescribed by the structural class \(\(u,0\)\(u,0\): single\-component;\(u,−u\)\(u,\-u\): opposite\-sign;\(u,u\)\(u,u\): same\-sign;\(u,−u&u\)\(u,\-u\\,\\&\\,u\): both polarities\);Number of mutual pairs, the number of unordered node pairs carrying edges in both directions;Number of edges, the total number of directed signed edges;Number of compatible FDUs, the number of tournament FDUs compatible with the triad under sign\-consistent superposition;FDU Decompositions, the minimal FDU decomposition\(s\) as pipe\-separated barcode pairs, with multiple decompositions separated by commas\.

Supplementary Information\.ContainsSupplementary Fig\. 1,Supplementary Fig\. 2,Supplementary Fig\. 3,Supplementary Table 1, andSupplementary Note\.