Towards Effective Federated Multimodal Graph Learning via Navigating Multifaceted Heterogeneity

# Towards Effective Federated Multimodal Graph Learning via Navigating Multifaceted Heterogeneity
Source: [https://arxiv.org/html/2608.00623](https://arxiv.org/html/2608.00623)
Yinlin Zhu1, Di Wu1, Yi Zhang2, Xunkai Li3, Wang Luo1, Wei\-Jin Huang1, Miao Hu1, Guocong Quan1

###### Abstract

Multimodal\-attributed graphs \(MAGs\), where nodes carry heterogeneous semantic content across multiple modalities while edges encode relational dependencies, have been widely adopted across diverse domains\. Federated multimodal graph learning \(FMGL\) extends federated graph learning \(FGL\) to MAGs, enabling collaborative optimization across decentralized MAGs without exposing raw data\. However, naively applying existing FGL methods to FMGL is insufficient, as they fail to navigate themultifaceted heterogeneityinherent in decentralized MAGs, including task heterogeneity across diverse client objectives, modality heterogeneity from discrepant modality quality and semantic domains, and topology heterogeneity arising from divergent topological patterns with low cross\-modality correlation\. To address these challenges, we proposeFederated multimodal graph learning withTopology\-awareCross\-modalRouting \(FedTCR\), the first systematic algorithm designed for FMGL\. To handle task heterogeneity, FedTCR employs a two\-stage paradigm that comprises federated task\-agnostic pre\-training followed by isolated task\-oriented fine\-tuning\. To jointly address modality and topology heterogeneity, FedTCR introduces a topology\-aware cross\-modal routing mechanism\. Concretely, each client distills modality\-specific knowledge into compact prototypes via topology\-aware importance\-weighted aggregation informed by graph structure; the server then evaluates cross\-client cross\-modal relationships among these structure\-informed prototypes and routes informative ones as contrastive references, driving a tri\-level cross\-modal contrastive learning scheme that jointly aligns cross\-client modalities while preserving discrimination\. Experiments across 7 domains demonstrate that FedTCR outperforms state\-of\-the\-art baselines on both graph\-centric and modality\-centric tasks\.

## 1Introduction

Multimodal\-attributed graphs \(MAGs\) represent entities as nodes enriched with heterogeneous semantic content across multiple modalities, while edges encode relational dependencies among these entities\. This expressive data structure has been widely adopted across diverse domains\(Yangletet al\.[2025](https://arxiv.org/html/2608.00623#bib.bib50); Li and Nabavi[2024](https://arxiv.org/html/2608.00623#bib.bib51)\)\. Motivated by such scenarios, multimodal graph learning \(MGL\) has garnered increasing attention, both enhancing graph\-centric tasks through rich multimodal attributes\(Zhenget al\.[2025](https://arxiv.org/html/2608.00623#bib.bib67)\)and improving modality\-specific tasks via topological structures\(Maiet al\.[2023](https://arxiv.org/html/2608.00623#bib.bib66)\)\.

Despite these advances, existing MGL approaches predominantly assume centralized data access, where all MAG data reside within a single learning system\. In practice, however, multiple organizations \(e\.g\., Instagram, X\) independently maintain private MAGs that encode complementary knowledge, while privacy regulations and commercial constraints prohibit direct data aggregation, preventing isolated MGL models from leveraging this cross\-platform collective intelligence\.

To this end, recent work\(Liet al\.[2026](https://arxiv.org/html/2608.00623#bib.bib68)\)proposed federated multimodal graph learning \(FMGL\), extending federated graph learning \(FGL\)\(Liet al\.[2024b](https://arxiv.org/html/2608.00623#bib.bib82)\)to MAGs\. FGL enables collaborative optimization through iterative client\-server communication, transmitting model parameters or gradients without exposing raw graph data\. While conceptually promising, this paradigm lacks systematic analysis of key challenges and targeted solutions\. We argue that naively extending traditional FGL methods to FMGL is insufficient, as they fail to address themultifaceted heterogeneityinherent in decentralized MAGs:

#### Task Heterogeneity\.

The rich multimodal semantics of MAGs naturally lead to diverse application requirements across clients\. Beyond conventional graph\-level tasks \(e\.g\., node classification\), clients may pursue modality\-centric objectives such as cross\-modal retrieval or multimodal generation\. However, mainstream FGL algorithms\(Zhanget al\.[2021](https://arxiv.org/html/2608.00623#bib.bib35); Baeket al\.[2023](https://arxiv.org/html/2608.00623#bib.bib34); Huanget al\.[2024](https://arxiv.org/html/2608.00623#bib.bib40)\)are designed for a uniform task type and exclusively focus on graph\-centric objectives\(Liet al\.[2024b](https://arxiv.org/html/2608.00623#bib.bib82); Zhuet al\.[2025](https://arxiv.org/html/2608.00623#bib.bib110)\)\. Their architectures and optimization procedures are tightly coupled to a single task type, preventing collaboration when clients pursue different objectives\.

#### Modality Heterogeneity\.

Distinct MAG sources and collection preferences across clients yield significant discrepancies in both modality quality and semantic domains\. Clients with noisy or domain\-biased modalities may propagate misleading knowledge during cross\-client aggregation, degrading others’ high\-quality representations\. Existing FGL methods address only unimodal attribute heterogeneity, lacking per\-modality reliability assessment and cross\-client cross\-modal alignment capabilities\.

#### Topology Heterogeneity\.

Graph topology provides crucial structural priors for modeling entity relationships\. In FMGL, however, client MAGs often exhibit diverse topological patterns due to varying user behaviors or data collection processes\. Existing FGL methods evaluate topological quality via homophily\-based metrics on local subgraphs\(Zhuet al\.[2024](https://arxiv.org/html/2608.00623#bib.bib33); Liet al\.[2023](https://arxiv.org/html/2608.00623#bib.bib36); Huanget al\.[2024](https://arxiv.org/html/2608.00623#bib.bib40)\), implicitly assuming consistent homophily patterns across clients\. This assumption is fragile in MAGs, where node pairs strongly homophilic under one modality may display weak homophily or even heterophily under another\(Zhenget al\.[2025](https://arxiv.org/html/2608.00623#bib.bib67)\), making traditional FGL methods unable to reliably distill and share topology priors across clients\.

Building upon these insights, we proposeFederated multimodal graph learning withTopology\-awareCross\-modalRouting \(FedTCR\), the first systematic algorithm addressing multifaceted heterogeneity in FMGL\. To tackletask heterogeneity, FedTCR adopts a two\-stage paradigm: federated task\-agnostic pre\-training followed by isolated task\-oriented fine\-tuning, where clients collaboratively learn a shared multimodal graph encoder without assuming unified downstream objectives and then independently adapt it to their specific applications\. To jointly addressmodality and topology heterogeneity, FedTCR introduces a topology\-aware cross\-modal routing mechanism that synergistically integrates two key designs: \(1\)topology\-aware prototype construction, where each client estimates node importance from graph topology and performs importance\-weighted aggregation to distill modality\-specific knowledge into compact prototypes; \(2\)cross\-modal prototype routing, where the server evaluates cross\-client cross\-modal relationships among these structure\-informed prototypes and routes informative ones as positive or negative references for each client\. The routed references drive a tri\-level cross\-modal contrastive learning scheme \(operating at node, neighbor, and client levels\) that jointly aligns cross\-client modalities while preserving discrimination\.

#### Contributions\.

\(1\)Valuable Insights\.We identify themultifaceted heterogeneityin FMGL, revealing fundamental challenges overlooked by existing FGL methods\. \(2\)Novel Method\.We propose FedTCR, the first optimization algorithm explicitly designed for FMGL\. FedTCR adopts a two\-stage paradigm with task\-agnostic federated pre\-training to handle task heterogeneity, and introduces a topology\-aware cross\-modal routing mechanism to jointly address modality and topology heterogeneity\. \(3\)State\-of\-the\-art Performance\.Experiments across 7 domains demonstrate that FedTCR achieves consistent and substantial improvements over state\-of\-the\-art baselines on both graph\-centric and modality\-centric tasks\.

## 2Related Work

#### Multimodal Graph Learning\.

MGL enriches graph learning with heterogeneous semantic content from multiple modalities\. A line of work constructs modality\-specific relational structures and fuses them for recommendation, e\.g\., MMGCN\(Weiet al\.[2019](https://arxiv.org/html/2608.00623#bib.bib63)\), MGAT\(Taoet al\.[2020](https://arxiv.org/html/2608.00623#bib.bib62)\), and LGMRec\(Guoet al\.[2024](https://arxiv.org/html/2608.00623#bib.bib58)\)\. Another line exploits the interplay between multimodal semantics and topology for graph\-oriented objectives\(Guoet al\.[2025](https://arxiv.org/html/2608.00623#bib.bib97); Zhenget al\.[2025](https://arxiv.org/html/2608.00623#bib.bib67); Maiet al\.[2023](https://arxiv.org/html/2608.00623#bib.bib66)\)\. More recent studies inject graph context into generative models, conditioning LLMs on multimodal neighborhoods for graph\-to\-text generation\(Yoonet al\.[2023](https://arxiv.org/html/2608.00623#bib.bib59)\)and steering diffusion models with graph\-guided signals for graph\-to\-image synthesis\(Jinet al\.[2024](https://arxiv.org/html/2608.00623#bib.bib60)\)\. Complementary benchmarks such as OpenMAG\(Wanet al\.[2026](https://arxiv.org/html/2608.00623#bib.bib129)\)systematize MAG datasets, encoders, and task protocols\. Despite this progress, existing MGL studies predominantly assume centralized data access and cannot accommodate privately held MAGs distributed across organizations\.

#### Federated Graph Learning\.

FGL extends federated learning to graph\-structured data, enabling collaborative training without exposing raw graphs\(Fuet al\.[2022](https://arxiv.org/html/2608.00623#bib.bib126)\)\. Existing studies mainly fall into three categories: graph\-level FGL, which mitigates cross\-dataset heterogeneity via gradient\-aware client clustering \(GCFL\+\(Xieet al\.[2021](https://arxiv.org/html/2608.00623#bib.bib46)\)\); subgraph\-level FGL, which addresses subgraph heterogeneity and missing cross\-client links \(Fed\-PUB\(Baeket al\.[2023](https://arxiv.org/html/2608.00623#bib.bib34)\), FedGTA\(Liet al\.[2023](https://arxiv.org/html/2608.00623#bib.bib36)\), FedSage\+\(Zhanget al\.[2021](https://arxiv.org/html/2608.00623#bib.bib35)\), AdaFGL\(Liet al\.[2024a](https://arxiv.org/html/2608.00623#bib.bib121)\), FGSSL\(Huanget al\.[2024](https://arxiv.org/html/2608.00623#bib.bib40)\), FGGP\(Wanet al\.[2024](https://arxiv.org/html/2608.00623#bib.bib41)\), FedTAD\(Zhuet al\.[2024](https://arxiv.org/html/2608.00623#bib.bib33)\), FedDEP\(Zhanget al\.[2024b](https://arxiv.org/html/2608.00623#bib.bib123)\), FedGCN\(Yaoet al\.[2023](https://arxiv.org/html/2608.00623#bib.bib120)\), FedGL\(Chenet al\.[2021](https://arxiv.org/html/2608.00623#bib.bib119)\)\); and node\-level FGL over ego\-networks \(FedEgo\(Zhanget al\.[2023](https://arxiv.org/html/2608.00623#bib.bib122)\)\), standardized by benchmarks such as FedGraphNN\(Heet al\.[2021](https://arxiv.org/html/2608.00623#bib.bib124)\), FederatedScope\-GNN\(Wanget al\.[2022](https://arxiv.org/html/2608.00623#bib.bib125)\), and OpenFGL\(Liet al\.[2024b](https://arxiv.org/html/2608.00623#bib.bib82)\)\. However, these methods are designed for unimodal graphs and a uniform task type, and thus cannot navigate the multifaceted heterogeneity inherent in federated multimodal graph learning, which is the focus of this work\.

## 3Preliminaries

#### Multimodal\-attributed Graphs\.

We consider a FMGL system consisting ofKKclients coordinated by a central server\. Each clientkkprivately maintains a MAG𝒢k=\(𝒱k,ℰk,\{𝐗k,\(m\)\}m∈ℳ\)\\mathcal\{G\}^\{k\}=\(\\mathcal\{V\}^\{k\},\\mathcal\{E\}^\{k\},\\\{\\mathbf\{X\}^\{k,\(m\)\}\\\}\_\{m\\in\\mathcal\{M\}\}\), where𝒱k\\mathcal\{V\}^\{k\}andℰk\\mathcal\{E\}^\{k\}denote the node and edge sets, respectively\.ℳ\\mathcal\{M\}denotes the set of available modalities\. For each nodevi∈𝒱kv\_\{i\}\\in\\mathcal\{V\}^\{k\}under modalitymm, a modality\-specific feature vector𝐱ik,\(m\)∈ℝdm\\mathbf\{x\}\_\{i\}^\{k,\(m\)\}\\in\\mathbb\{R\}^\{d\_\{m\}\}is obtained using a pre\-trained modality encoder \(e\.g\., Sentence\-BERT\(Reimers and Gurevych[2019](https://arxiv.org/html/2608.00623#bib.bib52)\)or T5\(Raffelet al\.[2020](https://arxiv.org/html/2608.00623#bib.bib53)\)for text, ViT\(Wuet al\.[2020a](https://arxiv.org/html/2608.00623#bib.bib54)\)or DINOv2\(Oquabet al\.[2023](https://arxiv.org/html/2608.00623#bib.bib55)\)for images\)\. The node features of modalitymmare organized as a matrix𝐗k,\(m\)∈ℝ\|𝒱k\|×dm\\mathbf\{X\}^\{k,\(m\)\}\\in\\mathbb\{R\}^\{\|\\mathcal\{V\}^\{k\}\|\\times d\_\{m\}\}\. All modalities share a common relational structure represented by the adjacency matrix𝐀k∈ℝ\|𝒱k\|×\|𝒱k\|\\mathbf\{A\}^\{k\}\\in\\mathbb\{R\}^\{\|\\mathcal\{V\}^\{k\}\|\\times\|\\mathcal\{V\}^\{k\}\|\}, with degree matrix𝐃k\\mathbf\{D\}^\{k\}defined as𝐃i​ik=∑j𝐀i​jk\\mathbf\{D\}^\{k\}\_\{ii\}=\\sum\_\{j\}\\mathbf\{A\}^\{k\}\_\{ij\}and𝐃i​jk=0\\mathbf\{D\}^\{k\}\_\{ij\}=0fori≠ji\\neq j\. Following standard graph signal processing, the symmetrically normalized adjacency matrix is defined as𝐀~k=\(𝐃k\)−1/2​𝐀k​\(𝐃k\)−1/2\\tilde\{\\mathbf\{A\}\}^\{k\}=\(\\mathbf\{D\}^\{k\}\)^\{\-1/2\}\\mathbf\{A\}^\{k\}\(\\mathbf\{D\}^\{k\}\)^\{\-1/2\}, and the corresponding graph Laplacian is𝐋k=𝐈−𝐀~k\\mathbf\{L\}^\{k\}=\\mathbf\{I\}\-\\tilde\{\\mathbf\{A\}\}^\{k\}\.

#### Federated Training Protocol\.

Due to privacy constraints, clients cannot share raw graph data; instead, they collaborate through iterative server\-client communication\. FedTCR adopts a two\-stage paradigm:federated task\-agnostic pre\-trainingfollowed byisolated task\-oriented fine\-tuning\. During pre\-training, at each communication round the server samples a subset𝒮\\mathcal\{S\}of clients; each selected clientk∈𝒮k\\in\\mathcal\{S\}downloads the global parameters𝐖^\\hat\{\\mathbf\{W\}\}and any server\-side messages \(e\.g\., routed prototypes\) and performs local optimization on its private MAG𝒢k\\mathcal\{G\}^\{k\}; the server then aggregates the updated parameters via node\-count\-weighted averaging,𝐖^=∑k∈𝒮\|𝒱k\|N​𝐖k\\hat\{\\mathbf\{W\}\}=\\sum\_\{k\\in\\mathcal\{S\}\}\\frac\{\|\\mathcal\{V\}^\{k\}\|\}\{N\}\\mathbf\{W\}^\{k\}withN=∑k∈𝒮\|𝒱k\|N=\\sum\_\{k\\in\\mathcal\{S\}\}\|\\mathcal\{V\}^\{k\}\|, and performs additional procedures \(e\.g\., cross\-modal prototype routing\)\. After pre\-training converges, each client independently fine\-tunes the encoder with a task\-specific head without further communication\.

#### Downstream Tasks\.

We consider two graph\-centric tasks \(node classification and link prediction\) and two modality\-centric tasks \(modality retrieval and modality generation covering G2Text and G2Image\), with formal definitions deferred to Appendix[B](https://arxiv.org/html/2608.00623#A2)\. In this work, we focus on learning a task\-agnostic multimodal graph encoder through federated pre\-training, which can be subsequently adapted to these diverse downstream applications on individual clients\.

## 4Methodology

We present FedTCR \(Fig\.[1](https://arxiv.org/html/2608.00623#S4.F1)\), an optimization algorithm explicitly designed for FMGL to address the multifaceted heterogeneity in Sec\.[1](https://arxiv.org/html/2608.00623#S1)\. FedTCR adopts a two\-stage paradigm to handle task heterogeneity: federated task\-agnostic pre\-training \(Sec\.[4\.1](https://arxiv.org/html/2608.00623#S4.SS1)\) followed by isolated task\-oriented fine\-tuning \(Sec\.[4\.2](https://arxiv.org/html/2608.00623#S4.SS2)\), and jointly addresses modality and topology heterogeneity during pre\-training through a topology\-aware cross\-modal routing mechanism\.

![Refer to caption](https://arxiv.org/html/2608.00623v1/x1.png)Figure 1:Overview of FedTCR, which addresses multifaceted heterogeneity in FMGL through federated task\-agnostic pre\-training with topology\-aware cross\-modal routing, followed by isolated task\-oriented fine\-tuning\.### 4\.1Federated Task\-agnostic Pre\-training

To address task heterogeneity, FedTCR decouples task\-agnostic representation learning from task\-specific adaptation\. During pre\-training, clients collaboratively learn a shared multimodal graph encoder without assuming unified downstream objectives, thereby capturing transferable multimodal and structural knowledge across decentralized MAGs\. At each communication round, pre\-training proceeds in two stages: \(1\) Each client performs local multimodal graph encoding, constructs topology\-aware modality prototypes, and optimizes multimodal representations; \(2\) The server aggregates encoder parameters via weighted averaging and performs cross\-modal prototype routing to deliver personalized contrastive knowledge to each client for the next round of optimization\.

#### Multimodal Graph Encoding\.

For each clientkkwith MAG𝒢k\\mathcal\{G\}^\{k\}, we first project each modality’s raw features into a shareddd\-dimensional latent space via modality\-specific linear transformations𝚽\(m\)∈ℝdm×d\\mathbf\{\\Phi\}^\{\(m\)\}\\in\\mathbb\{R\}^\{d\_\{m\}\\times d\}:

𝐳ik,\(m\)=\(𝐗k,\(m\)​𝚽\(m\)\)i‖\(𝐗k,\(m\)​𝚽\(m\)\)i‖2,∀i∈𝒱k,m∈ℳ,\\mathbf\{z\}\_\{i\}^\{k,\(m\)\}=\\frac\{\(\\mathbf\{X\}^\{k,\(m\)\}\\mathbf\{\\Phi\}^\{\(m\)\}\)\_\{i\}\}\{\\\|\(\\mathbf\{X\}^\{k,\(m\)\}\\mathbf\{\\Phi\}^\{\(m\)\}\)\_\{i\}\\\|\_\{2\}\},\\quad\\forall i\\in\\mathcal\{V\}^\{k\},\\,m\\in\\mathcal\{M\},\(1\)where\(⋅\)i\(\\cdot\)\_\{i\}denotes theii\-th row, and𝐙k,\(m\)∈ℝ\|𝒱k\|×d\\mathbf\{Z\}^\{k,\(m\)\}\\in\\mathbb\{R\}^\{\|\\mathcal\{V\}^\{k\}\|\\times d\}stacks these rows to stabilize training\. The fused representation is obtained by averaging across modalities:𝐙¯k=1\|ℳ\|​∑m∈ℳ𝐙k,\(m\)\\bar\{\\mathbf\{Z\}\}^\{k\}=\\frac\{1\}\{\|\\mathcal\{M\}\|\}\\sum\_\{m\\in\\mathcal\{M\}\}\\mathbf\{Z\}^\{k,\(m\)\}\. We then apply low\-pass graph filtering over the local topology to produce the multimodal graph representation:

𝐇k=1α\+1​∑l=0L\(αα\+1​𝐀~k\)l​𝐙¯k,\\mathbf\{H\}^\{k\}=\\frac\{1\}\{\\alpha\+1\}\\sum\_\{l=0\}^\{L\}\\left\(\\frac\{\\alpha\}\{\\alpha\+1\}\\tilde\{\\mathbf\{A\}\}^\{k\}\\right\)^\{l\}\\bar\{\\mathbf\{Z\}\}^\{k\},\(2\)whereα\>0\\alpha\>0controls the smoothing strength,𝐀~k\\tilde\{\\mathbf\{A\}\}^\{k\}is the symmetrically normalized adjacency matrix, andLLis the propagation depth\. As we show in Theorem[1](https://arxiv.org/html/2608.00623#Thmtheorem1), many widely used GNNs \(e\.g\., GCN\(Kipf and Welling[2017](https://arxiv.org/html/2608.00623#bib.bib88)\), SGC\(Wuet al\.[2019](https://arxiv.org/html/2608.00623#bib.bib93)\)\) can be viewed as special cases of this graph\-filtering formulation\.

#### Topology\-aware Prototype Construction\.

In FMGL, we distill each client’s modality\-specific knowledge into compact prototypes for cross\-client alignment\. Naive mean aggregation treats all nodes equally and ignores the local MAG topology; inspired by\(Blondelet al\.[2008](https://arxiv.org/html/2608.00623#bib.bib1); Lovász[1993](https://arxiv.org/html/2608.00623#bib.bib85)\), we instead assign each node a topology\-aware weight measuring its representativeness for the graph’s community\. Without loss of generality, we use PageRank algorithm\(Pageet al\.[1999](https://arxiv.org/html/2608.00623#bib.bib111)\)to compute these weights, so that importance propagates over multiple hops rather than depending only on 1\-hop degree\. Letα∈\(0,1\)\\alpha\\in\(0,1\)be the damping factor; the PageRank vector𝝅k∈ℝ\|𝒱k\|\\boldsymbol\{\\pi\}^\{k\}\\in\\mathbb\{R\}^\{\|\\mathcal\{V\}^\{k\}\|\}is the stationary distribution

𝝅k=1−α\|𝒱k\|​1\+α​\(\(𝐃k\)−1​𝐀k\)⊤​𝝅k,\\boldsymbol\{\\pi\}^\{k\}=\\frac\{1\-\\alpha\}\{\|\\mathcal\{V\}^\{k\}\|\}\\,\\mathbf\{1\}\+\\alpha\\left\(\(\\mathbf\{D\}^\{k\}\)^\{\-1\}\\mathbf\{A\}^\{k\}\\right\)^\{\\top\}\\boldsymbol\{\\pi\}^\{k\},\(3\)where𝐀k\\mathbf\{A\}^\{k\}is the adjacency matrix and𝐃k\\mathbf\{D\}^\{k\}is the diagonal degree matrix; the transpose gives the transition probabilities from neighbors into each node\. We compute𝝅k\\boldsymbol\{\\pi\}^\{k\}by power iteration, then setωik=πik\\omega\_\{i\}^\{k\}=\\pi\_\{i\}^\{k\}after convergence, formulated as:

πik,\(t\+1\)=1−α\|𝒱k\|\+α​∑j∈𝒱k𝐀j​ikdjk​πjk,\(t\),\\pi\_\{i\}^\{k,\(t\+1\)\}=\\frac\{1\-\\alpha\}\{\|\\mathcal\{V\}^\{k\}\|\}\+\\alpha\\sum\_\{j\\in\\mathcal\{V\}^\{k\}\}\\frac\{\\mathbf\{A\}^\{k\}\_\{ji\}\}\{d\_\{j\}^\{k\}\}\\pi\_\{j\}^\{k,\(t\)\},\(4\)wheredjk=∑l𝐀j​lkd\_\{j\}^\{k\}=\\sum\_\{l\}\\mathbf\{A\}^\{k\}\_\{jl\}is the degree of nodevjv\_\{j\}\.𝝅k\\boldsymbol\{\\pi\}^\{k\}is normalized so that∑i∈𝒱kωik=1\\sum\_\{i\\in\\mathcal\{V\}^\{k\}\}\\omega\_\{i\}^\{k\}=1withωik=πik\\omega\_\{i\}^\{k\}=\\pi\_\{i\}^\{k\}\.

Using these normalized weights, we define modality\-specific prototypes by importance\-weighted aggregation:

𝐏k,\(m\)=∑i∈𝒱kωik⋅𝐳ik,\(m\),∀m∈ℳ¯,\\mathbf\{P\}^\{k,\(m\)\}=\\sum\_\{i\\in\\mathcal\{V\}^\{k\}\}\\omega\_\{i\}^\{k\}\\cdot\\mathbf\{z\}\_\{i\}^\{k,\(m\)\},\\quad\\forall m\\in\\bar\{\\mathcal\{M\}\},\(5\)whereℳ¯=ℳ∪\{\|ℳ\|\+1\}\\bar\{\\mathcal\{M\}\}=\\mathcal\{M\}\\cup\\\{\|\\mathcal\{M\}\|\+1\\\}adds the fused modality;𝐏k,\(m\)\\mathbf\{P\}^\{k,\(m\)\}form∈ℳ¯m\\in\\bar\{\\mathcal\{M\}\}is computed from𝐙k,\(m\)\\mathbf\{Z\}^\{k,\(m\)\}, and𝐙k,\(\|ℳ\|\+1\)=𝐇k\\mathbf\{Z\}^\{k,\(\|\\mathcal\{M\}\|\+1\)\}=\\mathbf\{H\}^\{k\}\(i\.e\., Eq\. \([2](https://arxiv.org/html/2608.00623#S4.E2)\)\)\. Each prototype thus distills the local MAG’s modality\-specific semantics and structure into a single vector\. Together with the updated encoder parameters, they are uploaded to the server for aggregation and used in the subsequent routing\. To protect privacy, each client further applies\(ε,δ\)\(\\varepsilon,\\delta\)\-differential privacy \(DP\)\(Dwork[2008](https://arxiv.org/html/2608.00623#bib.bib30); Abadiet al\.[2016](https://arxiv.org/html/2608.00623#bib.bib31)\)before uploading: each prototype𝐏k,\(m\)\\mathbf\{P\}^\{k,\(m\)\}is L2\-clipped with a boundC\>0C\>0and then perturbed with Gaussian noise calibrated to the target\(ε,δ\)\(\\varepsilon,\\delta\):

𝐏~k,\(m\)\\displaystyle\\tilde\{\\mathbf\{P\}\}^\{k,\(m\)\}=𝐏k,\(m\)⋅min⁡\(1,C‖𝐏k,\(m\)‖2\),\\displaystyle=\\mathbf\{P\}^\{k,\(m\)\}\\cdot\\min\\Bigl\(1,\\frac\{C\}\{\\\|\\mathbf\{P\}^\{k,\(m\)\}\\\|\_\{2\}\}\\Bigr\),\(6\)𝐏^k,\(m\)\\displaystyle\\hat\{\\mathbf\{P\}\}^\{k,\(m\)\}=𝐏~k,\(m\)\+𝒩​\(0,σ2​C2​𝐈d\),\\displaystyle=\\tilde\{\\mathbf\{P\}\}^\{k,\(m\)\}\+\\mathcal\{N\}\(0,\\sigma^\{2\}C^\{2\}\\mathbf\{I\}\_\{d\}\),whereσ\\sigmais the noise scale\. The server performs cross\-modal prototype routing \(Eq\. \([7](https://arxiv.org/html/2608.00623#S4.E7)\)\) on the noisy prototypes, andε\\varepsilonandCCjointly control the privacy\-utility tradeoff\.

#### Cross\-modal Prototype Routing\.

After receiving all clients’ topology\-aware prototypes and encoder parameters, the server \(1\) aggregates encoder parameters\{𝚽\(m\)\}m∈ℳ\\\{\\mathbf\{\\Phi\}^\{\(m\)\}\\\}\_\{m\\in\\mathcal\{M\}\}via a mainstream FL algorithm \(e\.g\., FedAvg\(McMahanet al\.[2017](https://arxiv.org/html/2608.00623#bib.bib29)\)\), and \(2\) performs cross\-modal prototype routing to produce and deliver personalized contrastive knowledge to each client, which guides the next round of federated optimization\.

For each clientaa, modalityi∈ℳ¯i\\in\\bar\{\\mathcal\{M\}\}, and each other modalityj≠ij\\neq i, the server selects from\{𝐏b,\(j\)\}b≠a\\\{\\mathbf\{P\}^\{b,\(j\)\}\\\}\_\{b\\neq a\}the most similar prototype as the positive and the most dissimilar as the negative, which can be formulated as follows:

𝐏posa,\(i,j\)\\displaystyle\\mathbf\{P\}\_\{\\text\{pos\}\}^\{a,\(i,j\)\}=𝐏^b∗,\(j\),b∗=arg​maxb≠a⁡cos⁡\(𝐏^a,\(i\),𝐏^b,\(j\)\),\\displaystyle=\\hat\{\\mathbf\{P\}\}^\{b^\{\*\},\(j\)\},\\quad b^\{\*\}=\\operatorname\*\{arg\\,max\}\_\{b\\neq a\}\\cos\(\\hat\{\\mathbf\{P\}\}^\{a,\(i\)\},\\hat\{\\mathbf\{P\}\}^\{b,\(j\)\}\),\(7\)𝐏nega,\(i,j\)\\displaystyle\\mathbf\{P\}\_\{\\text\{neg\}\}^\{a,\(i,j\)\}=𝐏^b′,\(j\),b′=arg​minb≠a⁡cos⁡\(𝐏^a,\(i\),𝐏^b,\(j\)\)\.\\displaystyle=\\hat\{\\mathbf\{P\}\}^\{b^\{\\prime\},\(j\)\},\\quad b^\{\\prime\}=\\operatorname\*\{arg\\,min\}\_\{b\\neq a\}\\cos\(\\hat\{\\mathbf\{P\}\}^\{a,\(i\)\},\\hat\{\\mathbf\{P\}\}^\{b,\(j\)\}\)\.Herecos⁡\(⋅,⋅\)\\cos\(\\cdot,\\cdot\)denotes cosine similarity\. The server sends the routed pairs\{\(𝐏posa,\(i,j\),𝐏nega,\(i,j\)\)\}i,j\\\{\(\\mathbf\{P\}\_\{\\text\{pos\}\}^\{a,\(i,j\)\},\\mathbf\{P\}\_\{\\text\{neg\}\}^\{a,\(i,j\)\}\)\\\}\_\{i,j\}to each clientaafor client\-level contrastive learning \(Eq\. \([10](https://arxiv.org/html/2608.00623#S4.E10)\)\)\. Enforcing this contrast yields two effects: \(1\) per\-modality cross\-client alignment encourages complementary knowledge transfer and mitigates propagation of noisy or domain\-biased modalities, addressing modality heterogeneity; \(2\) aligning topology\-aware prototypes across modality boundaries harmonizes diverse topological patterns, addressing the topology heterogeneity inherent in decentralized MAGs\.

#### Local Optimization Objectives\.

Each client optimizes a composite objective comprising two local multimodal graph learning losses and one cross\-client alignment loss, following the cross\-contrastive paradigm\. Lettingψ​\(𝐮,𝐯\)=exp⁡\(cos⁡\(𝐮,𝐯\)/τ\)\\psi\(\\mathbf\{u\},\\mathbf\{v\}\)=\\exp\(\\cos\(\\mathbf\{u\},\\mathbf\{v\}\)/\\tau\)with temperatureτ\\tau, and denoting𝐳ik,\(\|ℳ\|\+1\)=𝐡ik\\mathbf\{z\}\_\{i\}^\{k,\(\|\\mathcal\{M\}\|\+1\)\}=\\mathbf\{h\}\_\{i\}^\{k\}as the fused graph\-filtered representation treated as an additional modality, we writeψi​j\(m1,m2\)\\psi\_\{ij\}^\{\(m\_\{1\},m\_\{2\}\)\}as shorthand forψ​\(𝐳ik,\(m1\),𝐳jk,\(m2\)\)\\psi\\big\(\\mathbf\{z\}\_\{i\}^\{k,\(m\_\{1\}\)\},\\mathbf\{z\}\_\{j\}^\{k,\(m\_\{2\}\)\}\\big\):

\(1\)Node\-level Contrastive Lossmaximizes the inter\-modality agreement of the same node while contrasting it against imposter nodes from the other modality:

ℒm=∑m1∈ℳ¯m2∈ℳ¯∖\{m1\}∑i∈𝒱k−log⁡ψi​i\(m1,m2\)∑j∈𝒱kψi​j\(m1,m2\)\.\\mathcal\{L\}\_\{m\}=\\sum\_\{\\begin\{subarray\}\{c\}m\_\{1\}\\in\\bar\{\\mathcal\{M\}\}\\\\ m\_\{2\}\\in\\bar\{\\mathcal\{M\}\}\\setminus\\\{m\_\{1\}\\\}\\end\{subarray\}\}\\sum\_\{i\\in\\mathcal\{V\}^\{k\}\}\-\\log\\frac\{\\psi\_\{ii\}^\{\(m\_\{1\},m\_\{2\}\)\}\}\{\\sum\_\{j\\in\\mathcal\{V\}^\{k\}\}\\psi\_\{ij\}^\{\(m\_\{1\},m\_\{2\}\)\}\}\.\(8\)
\(2\)Neighbor\-level Contrastive Losscaptures the semantic relations underlying the graph topology\. For each nodevi∈𝒱kv\_\{i\}\\in\\mathcal\{V\}^\{k\}, we simulate random walks on clientkk’s graph to collect a set of topological neighbors𝒫ik\\mathcal\{P\}\_\{i\}^\{k\}as positives, and randomly sample𝒩ik⊂𝒱k∖𝒫ik\\mathcal\{N\}\_\{i\}^\{k\}\\subset\\mathcal\{V\}^\{k\}\\setminus\\mathcal\{P\}\_\{i\}^\{k\}as negatives:

ℒg=∑m1∈ℳ¯m2∈ℳ¯∖\{m1\}∑i∈𝒱k−log⁡∑j∈𝒫ikψi​j\(m1,m2\)∑j∈𝒫ik∪𝒩ikψi​j\(m1,m2\)\.\\mathcal\{L\}\_\{g\}=\\sum\_\{\\begin\{subarray\}\{c\}m\_\{1\}\\in\\bar\{\\mathcal\{M\}\}\\\\ m\_\{2\}\\in\\bar\{\\mathcal\{M\}\}\\setminus\\\{m\_\{1\}\\\}\\end\{subarray\}\}\\sum\_\{i\\in\\mathcal\{V\}^\{k\}\}\-\\log\\frac\{\\sum\_\{j\\in\\mathcal\{P\}\_\{i\}^\{k\}\}\\psi\_\{ij\}^\{\(m\_\{1\},m\_\{2\}\)\}\}\{\\sum\_\{j\\in\\mathcal\{P\}\_\{i\}^\{k\}\\cup\\mathcal\{N\}\_\{i\}^\{k\}\}\\psi\_\{ij\}^\{\(m\_\{1\},m\_\{2\}\)\}\}\.\(9\)
\(3\)Client\-level Contrastive Lossleverages the personalized contrastive knowledge from the server \(Eq\. \([7](https://arxiv.org/html/2608.00623#S4.E7)\)\) to perform prototype\-level contrastive learning across clients and modalities\. Writingψpos\(i,j\)=ψ​\(𝐏k,\(i\),𝐏posk,\(i,j\)\)\\psi\_\{\\mathrm\{pos\}\}^\{\(i,j\)\}=\\psi\\big\(\\mathbf\{P\}^\{k,\(i\)\},\\mathbf\{P\}\_\{\\text\{pos\}\}^\{k,\(i,j\)\}\\big\)andψneg\(i,j\)=ψ​\(𝐏k,\(i\),𝐏negk,\(i,j\)\)\\psi\_\{\\mathrm\{neg\}\}^\{\(i,j\)\}=\\psi\\big\(\\mathbf\{P\}^\{k,\(i\)\},\\mathbf\{P\}\_\{\\text\{neg\}\}^\{k,\(i,j\)\}\\big\):

ℒc=∑i∈ℳ¯j∈ℳ¯∖\{i\}−log⁡ψpos\(i,j\)ψpos\(i,j\)\+ψneg\(i,j\)\.\\mathcal\{L\}\_\{c\}=\\sum\_\{\\begin\{subarray\}\{c\}i\\in\\bar\{\\mathcal\{M\}\}\\\\ j\\in\\bar\{\\mathcal\{M\}\}\\setminus\\\{i\\\}\\end\{subarray\}\}\-\\log\\frac\{\\psi\_\{\\mathrm\{pos\}\}^\{\(i,j\)\}\}\{\\psi\_\{\\mathrm\{pos\}\}^\{\(i,j\)\}\+\\psi\_\{\\mathrm\{neg\}\}^\{\(i,j\)\}\}\.\(10\)Notably, Eq\. \([10](https://arxiv.org/html/2608.00623#S4.E10)\) is activated from the second round after the server collects all client prototypes\.

Finally, the overall client\-side optimization objective during pre\-training is formulated as follows:

ℒ=λm​ℒm\+λg​ℒg\+λc​ℒc,\\mathcal\{L\}=\\lambda\_\{m\}\\mathcal\{L\}\_\{m\}\+\\lambda\_\{g\}\\mathcal\{L\}\_\{g\}\+\\lambda\_\{c\}\\mathcal\{L\}\_\{c\},\(11\)whereλm\\lambda\_\{m\},λg\\lambda\_\{g\}, andλc\\lambda\_\{c\}are balancing coefficients\.

### 4\.2Isolated Task\-oriented Fine\-tuning

Once federated pre\-training converges, each client retains the aggregated encoder and independently fine\-tunes on its local MAG without further communication\. Clientkkattaches a task\-specific headgkg^\{k\}and optimizes it jointly with the encoder via a task\-specific lossℒtask\\mathcal\{L\}\_\{\\text\{task\}\}: forgraph\-centric tasks\(e\.g\., node classification, link prediction\), the head takes the graph\-filtered representation𝐇k\\mathbf\{H\}^\{k\}\(Eq\. \([2](https://arxiv.org/html/2608.00623#S4.E2)\)\); formodality\-centric tasks\(e\.g\., modality retrieval\), it takes the modality\-specific embeddings𝐙k,\(m\)\\mathbf\{Z\}^\{k,\(m\)\}\(Eq\. \([1](https://arxiv.org/html/2608.00623#S4.E1)\)\) to retain modality\-level characteristics\. This isolation is deliberate: it avoids conflicting gradients from heterogeneous downstream objectives while preserving the transferable knowledge from pre\-training\. The complete procedure is given in Appendix[J](https://arxiv.org/html/2608.00623#A10)\.

## 5Experiments

We evaluate FedTCR along six research questions, with full reproducibility details deferred to Appendix[D](https://arxiv.org/html/2608.00623#A4)and Appendix[B](https://arxiv.org/html/2608.00623#A2):Q1: Can FedTCR outperform existing methods on graph\-centric and modality\-centric tasks under homogeneous\-task clients \(Sec\.[5\.2](https://arxiv.org/html/2608.00623#S5.SS2)\)?Q2: Can it enable effective collaboration across task\-heterogeneous clients \(Sec\.[5\.3](https://arxiv.org/html/2608.00623#S5.SS3)\)?Q3: What does each module of FedTCR contribute \(Sec\.[5\.4](https://arxiv.org/html/2608.00623#S5.SS4)\)?Q4: How robust is it under data sparsity \(Sec\.[5\.5](https://arxiv.org/html/2608.00623#S5.SS5)\)?Q5: How sensitive is it to key hyperparameters \(Sec\.[5\.6](https://arxiv.org/html/2608.00623#S5.SS6)\)?Q6: How does it perform under differentially private prototype perturbation \(Sec\.[5\.7](https://arxiv.org/html/2608.00623#S5.SS7)\)?

### 5\.1Experimental Setup

#### Datasets and Simulation Strategy\.

We evaluate FedTCR on 8 publicly available MAG datasets spanning 7 domains: Movies\(Niet al\.[2019](https://arxiv.org/html/2608.00623#bib.bib79)\), Grocery and Toys\(Niet al\.[2019](https://arxiv.org/html/2608.00623#bib.bib79); Houet al\.[2024](https://arxiv.org/html/2608.00623#bib.bib74)\), RedditS\(Desaiet al\.[2021](https://arxiv.org/html/2608.00623#bib.bib77)\), BiliDance\(Zhanget al\.[2024a](https://arxiv.org/html/2608.00623#bib.bib78)\), ele\-fashion\(Niet al\.[2019](https://arxiv.org/html/2608.00623#bib.bib79); Houet al\.[2024](https://arxiv.org/html/2608.00623#bib.bib74)\), Flickr30k\(Plummeret al\.[2015](https://arxiv.org/html/2608.00623#bib.bib73)\), and SemArt\(Garcia and Vogiatzis[2018](https://arxiv.org/html/2608.00623#bib.bib72)\)\. Following subgraph FL benchmarks\(Liet al\.[2024b](https://arxiv.org/html/2608.00623#bib.bib82)\), each MAG is partitioned into client subgraphs with the Louvain algorithm\(Blondelet al\.[2008](https://arxiv.org/html/2608.00623#bib.bib1)\): 5 clients per MAG in thehomogeneous\-taskconfiguration, and 4 clients per MAG in theheterogeneous\-taskone \(one MAG per task, 20 clients across the 5 tasks in total\)\. Dataset statistics and descriptions are in Appendix[D](https://arxiv.org/html/2608.00623#A4)\.

#### Baselines\.

We compare against four method families: \(1\)Isolated Training\(MMGCN\(Weiet al\.[2019](https://arxiv.org/html/2608.00623#bib.bib63)\), MGAT\(Taoet al\.[2020](https://arxiv.org/html/2608.00623#bib.bib62)\), LGMRec\(Guoet al\.[2024](https://arxiv.org/html/2608.00623#bib.bib58)\)\); \(2\)Classical FL\(FedAvg\(McMahanet al\.[2017](https://arxiv.org/html/2608.00623#bib.bib29)\), SCAFFOLD\(Karimireddyet al\.[2020](https://arxiv.org/html/2608.00623#bib.bib24)\), FedDC\(Gaoet al\.[2022](https://arxiv.org/html/2608.00623#bib.bib48)\)\) with an MMGCN backbone; \(3\)Multimodal FL\(PEPSY\(Nguyenet al\.[2025](https://arxiv.org/html/2608.00623#bib.bib114)\), FedMVP\(Cheet al\.[2024](https://arxiv.org/html/2608.00623#bib.bib112)\), FedMAC\(Nguyenet al\.[2024](https://arxiv.org/html/2608.00623#bib.bib113)\)\); and \(4\)Unimodal FGL\(FedSage\+\(Zhanget al\.[2021](https://arxiv.org/html/2608.00623#bib.bib35)\), FedGTA\(Liet al\.[2023](https://arxiv.org/html/2608.00623#bib.bib36)\), FedSPA\(Tanet al\.[2025b](https://arxiv.org/html/2608.00623#bib.bib44)\), FedIIH\(Yuet al\.[2025](https://arxiv.org/html/2608.00623#bib.bib43)\), FedSSP\(Tanet al\.[2024](https://arxiv.org/html/2608.00623#bib.bib115)\), S2FGL\(Tanet al\.[2025a](https://arxiv.org/html/2608.00623#bib.bib45)\), FedLap\(Aliakbariet al\.[2025](https://arxiv.org/html/2608.00623#bib.bib116)\)\)\. Unimodal FGL methods are graph\-centric only and thus inapplicable to modality\-centric tasks; label\-dependent methods \(e\.g\., FedGTA, FedIIH\) cannot perform link prediction\. These cases are marked as “N/A”\. Baseline details are in Appendix[E](https://arxiv.org/html/2608.00623#A5)\.

#### Evaluation Metrics\.

We adopt task\-specific metrics: ACC for node classification, AUC for link prediction, R@1 for modality retrieval, and R\-L \(ROUGE\-L\) for G2Text with C\-S \(CLIP\-Score\) for G2Image\.

Node Classification \(ACC\)Link Prediction \(AUC\)Modality Retrieval \(R@1\)Modality Generation \(R\-L/C\-S\)MethodsMoviesGroceryRedditSBiliDanceToysele\-fashionFlickr30kSemArtMMGCN53\.47±\\pm0\.3280\.39±\\pm0\.5189\.52±\\pm0\.2376\.06±\\pm0\.4249\.29±\\pm0\.4681\.74±\\pm0\.2824\.56±\\pm0\.2366\.12±\\pm0\.28MGAT52\.12±\\pm0\.3480\.41±\\pm0\.4990\.23±\\pm0\.2576\.21±\\pm0\.4448\.89±\\pm0\.4382\.12±\\pm0\.3124\.34±\\pm0\.2165\.78±\\pm0\.30LGMRec53\.34±\\pm0\.2980\.56±\\pm0\.5390\.89±\\pm0\.2276\.68±\\pm0\.3849\.78±\\pm0\.4582\.45±\\pm0\.2625\.12±\\pm0\.2466\.89±\\pm0\.26Isolated53\.07±\\pm0\.3680\.93±\\pm0\.5788\.34±\\pm0\.2662\.89±\\pm0\.4649\.44±\\pm0\.5281\.37±\\pm0\.3123\.95±\\pm0\.2665\.34±\\pm0\.31FedAvg49\.07±\\pm0\.4378\.18±\\pm0\.6892\.32±\\pm0\.3172\.30±\\pm0\.5645\.31±\\pm0\.6268\.96±\\pm0\.3724\.02±\\pm0\.3365\.56±\\pm0\.37SCAFFOLD49\.56±\\pm0\.4578\.66±\\pm0\.6392\.78±\\pm0\.2872\.78±\\pm0\.5245\.89±\\pm0\.5869\.67±\\pm0\.3923\.89±\\pm0\.2965\.89±\\pm0\.34FedDC49\.87±\\pm0\.4078\.43±\\pm0\.6693\.01±\\pm0\.3073\.01±\\pm0\.5446\.23±\\pm0\.6469\.89±\\pm0\.3524\.45±\\pm0\.3166\.12±\\pm0\.38PEPSY48\.67±\\pm0\.4778\.13±\\pm0\.7393\.45±\\pm0\.3373\.56±\\pm0\.6047\.21±\\pm0\.6670\.12±\\pm0\.4024\.78±\\pm0\.3566\.45±\\pm0\.40FedMVP48\.36±\\pm0\.5077\.98±\\pm0\.7193\.21±\\pm0\.3674\.18±\\pm0\.6344\.73±\\pm0\.6969\.52±\\pm0\.4425\.01±\\pm0\.3266\.78±\\pm0\.43FedMAC49\.01±\\pm0\.4478\.35±\\pm0\.7693\.67±\\pm0\.3174\.02±\\pm0\.5746\.89±\\pm0\.6370\.44±\\pm0\.4125\.23±\\pm0\.3467\.01±\\pm0\.38FedSage\+50\.23±\\pm0\.3679\.21±\\pm0\.5692\.89±\\pm0\.2672\.67±\\pm0\.46N/AN/AN/AN/AFedGTA50\.89±\\pm0\.3279\.56±\\pm0\.51N/AN/AN/AN/AN/AN/AFedSPA48\.89±\\pm0\.6479\.34±\\pm1\.0193\.12±\\pm0\.4673\.12±\\pm0\.82N/AN/AN/AN/AFedIIH51\.03±\\pm0\.3479\.62±\\pm0\.53N/AN/AN/AN/AN/AN/AFedSSP48\.35±\\pm0\.4378\.26±\\pm0\.6793\.52±\\pm0\.3173\.45±\\pm0\.5543\.11±\\pm0\.6170\.31±\\pm0\.36N/AN/AS2FGL48\.56±\\pm0\.8979\.12±\\pm1\.3892\.97±\\pm0\.6372\.89±\\pm1\.13N/AN/AN/AN/AFedLap50\.77±\\pm0\.3079\.43±\\pm0\.4793\.23±\\pm0\.2273\.23±\\pm0\.39N/AN/AN/AN/AFedTCR \(Ours\)54\.27±\\pm0\.2581\.01±\\pm0\.4097\.84±\\pm0\.1878\.45±\\pm0\.3253\.64±\\pm0\.3683\.87±\\pm0\.2226\.89±\\pm0\.1869\.45±\\pm0\.22

Table 1:Performance comparison under the homogeneous\-task setting \(5 clients\)\. ‘N/A’ indicates the algorithm is not applicable to the task\. Best results are in bold and the second\-best are underlined\.

### 5\.2Homogeneous Tasks \(Answer forQ1\)

To answerQ1, we compare FedTCR against all four baseline families on the 8 MAGs under the homogeneous\-task setting \(5 clients per MAG, all sharing the MAG’s designated task\); Table[1](https://arxiv.org/html/2608.00623#S5.T1)reports the full per\-dataset comparison\.

#### Graph\-centric Tasks\.

FedTCR consistently achieves the best performance across all graph\-centric benchmarks \(Table[1](https://arxiv.org/html/2608.00623#S5.T1)\), improving over the second\-best method by \+1\.50% ACC on Movies and \+0\.10% on Grocery for node classification, and surpassing the runner\-up by \+4\.45% AUC on RedditS and \+2\.31% on BiliDance for link prediction\. Classical FL methods with the MMGCN backbone outperform isolated training on the link prediction benchmarks, confirming the benefit of federated collaboration on structure\-dependent tasks, but they cannot exploit modality\- and topology\-specific knowledge across clients as FedTCR does\.

#### Modality\-centric Tasks\.

FedTCR also demonstrates clear superiority on modality\-centric tasks: \+7\.75% R@1 on Toys and \+1\.72% on ele\-fashion for modality retrieval, and \+6\.58% R\-L on Flickr30k \(G2Text\) and \+3\.64% C\-S on SemArt \(G2Image\) for modality generation, over the best competing method\. This consistent superiority shows that topology\-aware cross\-modal routing effectively transfers complementary multimodal knowledge across clients, while task\-agnostic pre\-training learns representations that generalize across downstream applications\.

![Refer to caption](https://arxiv.org/html/2608.00623v1/x2.png)Figure 2:Performance under the task\-heterogeneous setting \(4 clients for each dataset, i\.e\., one MAG per task\)\. FedTCR \(homo\.\) federates only the 4 same\-task clients, while FedTCR \(hete\.\) pre\-trains all 20 clients across the 5 tasks\.

### 5\.3Heterogeneous Tasks \(Answer forQ2\)

To answerQ2, we evaluate FedTCR under a task\-heterogeneous setting comprising 5 downstream tasks, each instantiated on a different MAG partitioned into 4 clients: node classification on Movies, link prediction on RedditS, modality retrieval on Toys, G2Text on Flickr30k, and G2Image on SemArt\. Existing FL and FGL methods cannot join such a federation at all, as their architectures couple the federated parameters and the training objective to a single task type; their only viable collaboration is same\-task federation, which is already covered by Table[1](https://arxiv.org/html/2608.00623#S5.T1)where all of them trail FedTCR\. We therefore conduct a controlled self\-comparison that varies only the collaboration scope over identical per\-task partitions:Isolated\(no federation\), FedTCR \(homo\.\) \(4 same\-task clients\), and FedTCR \(hete\.\) \(all 20 clients\)\. As shown in Fig\.[2](https://arxiv.org/html/2608.00623#S5.F2), FedTCR \(homo\.\) already improves over Isolated training by \+7\.06% on average, and FedTCR \(hete\.\) further improves over FedTCR \(homo\.\) by \+2\.44% on average\. The gains are the largest on the modality\-generation tasks \(\+3\.87% on average\), which depend most heavily on the complementary cross\-modal knowledge transferred by the topology\-aware routing, and remain positive on graph\-centric tasks \(\+1\.00%\)\. Since the arms differ only in the collaboration scope, these results establish that cross\-task collaboration carries genuine net benefit rather than interference, enabling cross\-task collective intelligence that is architecturally out of reach for existing methods\.

### 5\.4Ablation Study \(Answer forQ3\)

To answerQ3, we ablate five components of FedTCR: \(1\)w/o TW: replaces PageRank importance weighting with uniform averaging in prototype construction \(Eq\. \([5](https://arxiv.org/html/2608.00623#S4.E5)\)\); \(2\)w/o CR: removes cross\-modal prototype routing and the associated client\-level lossℒc\\mathcal\{L\}\_\{c\}\(Eq\. \([10](https://arxiv.org/html/2608.00623#S4.E10)\)\); \(3\)w/oℒm\\mathcal\{L\}\_\{m\}and \(4\)w/oℒg\\mathcal\{L\}\_\{g\}: remove the node\-level \(Eq\. \([8](https://arxiv.org/html/2608.00623#S4.E8)\)\) and neighbor\-level \(Eq\. \([9](https://arxiv.org/html/2608.00623#S4.E9)\)\) contrastive losses, respectively; and \(5\)w/o TS: removes the pre\-training stage and directly performs end\-to\-end federated task\-specific training\. All variants are trained under the identical protocol and hyperparameters as the full model, so any performance gap is attributable solely to the removed component\.

Table 2:Ablation study \(5 clients\)\. Each row removes one component from FedTCR\.
![Refer to caption](https://arxiv.org/html/2608.00623v1/x3.png)Figure 3:Retained ACC on Movies under three sparsity types at varying ratios, relative to the unperturbed setting\. Shaded bands denote the standard deviation over five seeds\.As shown in Table[2](https://arxiv.org/html/2608.00623#S5.T2), removing any component consistently degrades performance, demonstrating their complementary contributions\.w/o CRincurs the largest drop \(e\.g\.,−\-2\.28% ACC on Movies and−\-4\.70% R@1 on Toys\), highlighting the importance of cross\-modal prototype routing for exchanging task\-relevant knowledge beyond parameter aggregation\. The degradation ofw/o TSfurther verifies that task\-agnostic pre\-training is crucial for learning transferable representations before task\-specific adaptation\. In addition,w/o TWconsistently underperforms the full model, validating the benefit of topology\-aware prototype construction, while the larger impact ofw/oℒg\\mathcal\{L\}\_\{g\}compared withw/oℒm\\mathcal\{L\}\_\{m\}suggests that neighborhood\-level supervision provides stronger guidance than node\-level alignment\.

### 5\.5Robustness Analysis \(Answer forQ4\)

To answerQ4, we evaluate FedTCR under three data sparsity scenarios controlled by ratioα\\alpha:feature sparsity\(feature masking\),topology sparsity\(edge removal\), andlabel sparsity\(label removal\), in both task settings\. As shown in Fig\.[3](https://arxiv.org/html/2608.00623#S5.F3), FedTCR remains robust across all sparsity types and task settings: atα=50%\\alpha=50\\%in the homogeneous setting it preserves 93\.2%, 94\.1%, and 97\.1% of its original ACC under feature, topology, and label sparsity, respectively\. The degradation pattern is setting\-dependent: feature sparsity hurts most under homogeneous clients, where corrupted modality representations propagate directly into prototype construction, whereas topology sparsity dominates in the heterogeneous\-task federation, where divergent objectives make topology\-aware routing the main channel of cross\-client support; label sparsity stays the mildest in both settings, since task\-agnostic pre\-training requires no downstream supervision\.

### 5\.6Hyperparameter Sensitivity \(Answer forQ5\)

Table 3:Hyperparameter sensitivity on Movies\. Each row sweeps one hyperparameter over the grid shown, with the others fixed at their default values\.
To answerQ5, we sweep the loss weightsλm\\lambda\_\{m\},λg\\lambda\_\{g\},λc\\lambda\_\{c\}\(summing to11by default\), the temperatureτ\\tau, and the propagation depthLL\. As shown in Table[3](https://arxiv.org/html/2608.00623#S5.T3), FedTCR performs best at the default configuration and remains stable around it; the model is primarily sensitive toλc\\lambda\_\{c\}andτ\\tau: an overly smallλc\\lambda\_\{c\}weakens the cross\-client alignment signal, while an overly largeτ\\taumakes the contrastive distribution overly uniform\. In contrast, varyingλm\\lambda\_\{m\},λg\\lambda\_\{g\}, orLLchanges the accuracy only marginally, indicating that the defaults already lie in a flat optimal region and that FedTCR requires little hyperparameter tuning in practice\.

### 5\.7Privacy\-Utility Tradeoff \(Answer forQ6\)

Table 4:Performance under\(ε,δ\)\(\\varepsilon,\\delta\)\-DP prototype perturbation withδ=10−5\\delta=10^\{\-5\}and clipping boundC=1\.0C=1\.0, mean and standard deviation over five random seeds \(percentages\)\.
To answerQ6, we evaluate FedTCR with\(ε,δ\)\(\\varepsilon,\\delta\)\-differentially private prototype uploading \(Eq\. \([6](https://arxiv.org/html/2608.00623#S4.E6)\)\), fixingδ=10−5\\delta=10^\{\-5\}andC=1\.0C=1\.0while varyingε\\varepsilon\. As shown in Table[4](https://arxiv.org/html/2608.00623#S5.T4), performance is nearly lossless atε≥8\\varepsilon\\geq 8\(over 99\.4% retained\), since prototypes are importance\-weighted aggregates over many nodes, which limits per\-client sensitivity, and the server\-side aggregation further averages the perturbations out across clients\. Even atε=1\\varepsilon=1, FedTCR retains over 97\.4% of its non\-private performance, staying clearly ahead of the strongest non\-private baseline on RedditS and Toys and within 1\.1% of MMGCN on Movies\.

## 6Conclusion

We identify themultifaceted heterogeneityinherent in federated multimodal graph learning \(FMGL\), spanning task, modality, and topology heterogeneity across decentralized MAGs\. We propose FedTCR, the first systematic algorithm for FMGL, which employs a two\-stage paradigm \(federated task\-agnostic pre\-training followed by isolated task\-oriented fine\-tuning\) and a topology\-aware cross\-modal routing mechanism that distills modality\-specific knowledge into structure\-informed prototypes for cross\-client contrastive alignment\. Experiments across 8 datasets show that FedTCR consistently achieves state\-of\-the\-art performance on graph\-centric and modality\-centric tasks, laying a foundation for privacy\-preserving multimodal graph learning\.

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- K\. Zhang, Y\. Zhu, X\. Xie, C\. Yang, S\. M\. Yiu, and J\. Han \(2024b\)Deep efficient private neighbor generation for subgraph federated learning\.InProceedings of the 2024 SIAM International Conference on Data Mining,Cited by:[§2](https://arxiv.org/html/2608.00623#S2.SS0.SSS0.Px2.p1.1)\.
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## Appendix AMore Related Works

#### Graph Neural Networks \(GNNs\)\.

Earlier research on deep graph learning extends convolution to handle graphs\(Brunaet al\.[2013](https://arxiv.org/html/2608.00623#bib.bib86)\)but comes with notable parameter counts\. To this end, GCN\(Kipf and Welling[2017](https://arxiv.org/html/2608.00623#bib.bib88)\)simplifies graph convolution by utilizing a 1\-order Chebyshev filter to capture local neighborhood information\. Moreover, GAT\(Veličkovićet al\.[2018](https://arxiv.org/html/2608.00623#bib.bib90)\)adopts graph attention, allowing weighted aggregation\. GraphSAGE\(Hamiltonet al\.[2017](https://arxiv.org/html/2608.00623#bib.bib89)\)introduces a variety of learnable aggregation functions for performing message aggregation\. Moreover, GIN\(Xuet al\.[2019](https://arxiv.org/html/2608.00623#bib.bib117)\)aims to preserve structural information maximally and theoretically proves its discriminative power matches the Weisfeiler\-Lehman graph isomorphism test\. Further details on GNN research can be found in surveys\(Wuet al\.[2020b](https://arxiv.org/html/2608.00623#bib.bib80); Zhouet al\.[2020](https://arxiv.org/html/2608.00623#bib.bib81)\)\.

#### Multimodal Federated Learning \(MFL\)\.

MFL studies federated optimization over multimodal data\. PEPSY\(Nguyenet al\.[2025](https://arxiv.org/html/2608.00623#bib.bib114)\)handles both missing modalities and missing input features by learning client\-side data\-missing profiles for robust aggregation\. FedMVP\(Cheet al\.[2024](https://arxiv.org/html/2608.00623#bib.bib112)\)leverages frozen pre\-trained foundation models for cross\-modal completion and representation knowledge transfer, with CKA\-based importance\-aware aggregation\. FedMAC\(Nguyenet al\.[2024](https://arxiv.org/html/2608.00623#bib.bib113)\)aligns heterogeneous modality representations via cross\-modal contrastive learning with modality\-specific adaptive aggregation\. These methods address modality missingness at the data level, but they are not designed for graph\-structured multimodal data and assume a uniform downstream task across clients\.

## Appendix BAdditional Experimental Details

### B\.1Training Protocol

Unless otherwise specified, all reported results are presented as mean±\\pmstandard deviation over five random seeds\. For each seed, all methods share the same data split and, for robustness experiments, the same realization of sparsity patterns\. Baseline implementations use the official released code with the recommended hyperparameter settings, while ablated variants inherit the same configuration with only the corresponding component removed\. Federated pre\-training runs forT=200T=200communication rounds with full client participation andE=2E=2local epochs per round; task\-oriented fine\-tuning runs for 100 epochs per client\.

#### Perturbation model\.

For the robustness study \(Sec\.[5\.5](https://arxiv.org/html/2608.00623#S5.SS5)\), three types of sparsity are injected before training with ratioα\\alpha, and for a given seed every method receives the identical realization\.Feature sparsityrandomly masksα\\alpha% of the feature dimensions of every node in the target modality, setting them to zero\.Topology sparsityremovesα\\alpha% of the existing edges uniformly at random\.Label sparsityremovesα\\alpha% of the available node labels for fine\-tuning, sampled uniformly per client\.

#### Task definitions\.

We consider two graph\-centric tasks: \(1\)Node Classification: assigning category labels to unlabeled nodes; \(2\)Link Prediction: estimating whether an edge\(u,v\)\(u,v\)should exist inℰk\\mathcal\{E\}^\{k\}; and two modality\-centric tasks: \(1\)Modality Retrieval: given a query from one modality \(e\.g\., text\), retrieving its corresponding representation in another modality; \(2\)Modality Generation: producing content in a target modality conditioned on a target node along with task instructions and its graph neighborhood, covering Graph\-to\-Text \(G2Text\) generation of textual descriptions and Graph\-to\-Image \(G2Image\) synthesis of visual content\.

### B\.2Model Configuration

Each node is associated with text and image features extracted from frozen pre\-trained encoders \(Sentence\-BERT\(Reimers and Gurevych[2019](https://arxiv.org/html/2608.00623#bib.bib52)\)or T5\(Raffelet al\.[2020](https://arxiv.org/html/2608.00623#bib.bib53)\)for text, ViT\(Wuet al\.[2020a](https://arxiv.org/html/2608.00623#bib.bib54)\)or DINOv2\(Oquabet al\.[2023](https://arxiv.org/html/2608.00623#bib.bib55)\)for images; Qwen2\-VL\-7B\-Instruct\(Baiet al\.[2023](https://arxiv.org/html/2608.00623#bib.bib128)\)is used for the G2Text/G2Image feature extraction\), producing 768\-dimensional representations for all datasets\. Each modality is encoded by a linear projection \(Eq\. \([1](https://arxiv.org/html/2608.00623#S4.E1)\)\) followed by low\-pass graph filtering \(Eq\. \([2](https://arxiv.org/html/2608.00623#S4.E2)\)\)\.

Unless otherwise stated, the default hyperparameters are shared across all datasets: the latent dimension isd=128d=128; the propagation depth isL=3L=3; the smoothing strength isα=2\.0\\alpha=2\.0in Eq\. \([2](https://arxiv.org/html/2608.00623#S4.E2)\); the PageRank damping factor is0\.850\.85with 20 power iterations; the contrastive temperature isτ=0\.5\\tau=0\.5; the loss weights areλm=0\.3\\lambda\_\{m\}=0\.3,λg=0\.2\\lambda\_\{g\}=0\.2, andλc=0\.5\\lambda\_\{c\}=0\.5, which are normalized to sum to11; and the differential\-privacy configuration isδ=10−5\\delta=10^\{\-5\}with clip boundC=1\.0C=1\.0\(ε=∞\\varepsilon=\\inftyunless the privacy budget is explicitly swept in Sec\.[5\.7](https://arxiv.org/html/2608.00623#S5.SS7)\)\. For the neighbor\-level contrastive loss, we simulate 5 random walks of length 10 per node and sample an equal number of negatives\. A sensitivity analysis of the main hyperparameters is reported in Sec\.[5\.6](https://arxiv.org/html/2608.00623#S5.SS6)\.

#### Optimization\.

All models are optimized using Adam with learning rate10−310^\{\-3\}and weight decay10−510^\{\-5\}under full\-batch training\. Encoder parameters are aggregated at the server by node\-count\-weighted averaging, and no server\-side model is maintained for the routing outputs\.

## Appendix CEfficiency Analysis

We analyze the computational and communication complexity of FedTCR per federated round\. On the client side, multimodal graph encoding costs𝒪​\(\|ℳ\|⋅\|𝒱k\|​dm​d\)\\mathcal\{O\}\(\|\\mathcal\{M\}\|\\cdot\|\\mathcal\{V\}^\{k\}\|d\_\{m\}d\)for the modality projections and𝒪​\(L⋅\|ℰk\|​d\)\\mathcal\{O\}\(L\\cdot\|\\mathcal\{E\}^\{k\}\|d\)for the low\-pass graph filtering with sparse adjacency multiplications\. The topology\-aware weights are computed via PageRank power iteration at𝒪​\(\|ℰk\|\)\\mathcal\{O\}\(\|\\mathcal\{E\}^\{k\}\|\)per iteration, and only once as a preprocessing step before training\. Prototype construction costs𝒪​\(\|ℳ¯\|⋅\|𝒱k\|​d\)\\mathcal\{O\}\(\|\\bar\{\\mathcal\{M\}\}\|\\cdot\|\\mathcal\{V\}^\{k\}\|d\), and the three contrastive losses are computed over per\-client sampled node pairs, sharing the same order as standard cross\-contrastive multimodal training\. On the server side, parameter aggregation costs𝒪​\(\|𝒮\|⋅\|ℳ¯\|​dm​d\)\\mathcal\{O\}\(\|\\mathcal\{S\}\|\\cdot\|\\bar\{\\mathcal\{M\}\}\|d\_\{m\}d\), while cross\-modal routing requires pairwise cosine similarities among\|𝒮\|×\|ℳ¯\|\|\\mathcal\{S\}\|\\times\|\\bar\{\\mathcal\{M\}\}\|prototypes, i\.e\.,𝒪​\(\|𝒮\|2​\|ℳ¯\|2​d\)\\mathcal\{O\}\(\|\\mathcal\{S\}\|^\{2\}\|\\bar\{\\mathcal\{M\}\}\|^\{2\}d\), which is negligible compared with client\-side training since each prototype is a singledd\-dimensional vector\. In terms of communication, each client uploads the shared encoder parameters\{𝚽\(m\)\}m∈ℳ\\\{\\mathbf\{\\Phi\}^\{\(m\)\}\\\}\_\{m\\in\\mathcal\{M\}\}, identical to standard FL methods such as FedAvg, plus\|ℳ¯\|\|\\bar\{\\mathcal\{M\}\}\|prototype vectors, a marginal overhead of\|ℳ¯\|⋅d\|\\bar\{\\mathcal\{M\}\}\|\\cdot dvalues per round\. The isolated fine\-tuning stage involves no further communication\. Overall, FedTCR introduces only lightweight overhead over classical FL methods while enabling cross\-task and cross\-modal collaboration\.

## Appendix DDataset Details

Detailed statistical information on the datasets is presented in Table[5](https://arxiv.org/html/2608.00623#A4.T5), where the graph statistics follow the OpenMAG construction\(Wanet al\.[2026](https://arxiv.org/html/2608.00623#bib.bib129)\), with textual descriptions as follows\.

Table 5:Statistics of the experimental datasets\.#### Movies

\(Niet al\.[2019](https://arxiv.org/html/2608.00623#bib.bib79)\)is sourced from Amazon’s Movies and TV category\. Nodes correspond to DVD/Blu\-ray products, and edges reflect consumer co\-purchasing behavior\. Node attributes include textual plot synopses and customer reviews, alongside visual features derived from official cover art\.

#### Grocery

\(Niet al\.[2019](https://arxiv.org/html/2608.00623#bib.bib79); Houet al\.[2024](https://arxiv.org/html/2608.00623#bib.bib74)\)is sourced from Amazon’s Grocery and Gourmet Food category\. Nodes correspond to food and household products, and edges reflect co\-purchasing behavior\. Node attributes include textual product descriptions and visual features from product images\. This dataset is used for node classification, where products are categorized into fine\-grained grocery categories\.

#### RedditS

\(Desaiet al\.[2021](https://arxiv.org/html/2608.00623#bib.bib77)\)is a social network derived from Reddit, where nodes correspond to posts linked by user interaction and community relations\. Each node carries textual post content and associated visual context\. This dataset is utilized for link prediction, forecasting potential interaction links between posts\.

#### BiliDance

\(Zhanget al\.[2024a](https://arxiv.org/html/2608.00623#bib.bib78)\)is a video entertainment network collected from the Bilibili platform, where nodes correspond to dance\-video content items linked by user co\-viewing behavior\. Nodes carry textual descriptions and visual cover features\. This dataset is utilized for link prediction over content\-association structures\.

#### Toys

\(Niet al\.[2019](https://arxiv.org/html/2608.00623#bib.bib79); Houet al\.[2024](https://arxiv.org/html/2608.00623#bib.bib74)\)is sourced from Amazon’s Toys and Games category\. Nodes correspond to toy products, and edges reflect co\-purchasing behavior\. Node attributes include textual product descriptions and visual features from product images\. This dataset is utilized for modality retrieval between textual and visual product content\.

#### ele\-fashion

\(Niet al\.[2019](https://arxiv.org/html/2608.00623#bib.bib79); Houet al\.[2024](https://arxiv.org/html/2608.00623#bib.bib74)\)is a heterogeneous graph merging Amazon’s Electronics and Fashion categories\. Nodes are connected via cross\-category co\-purchasing links, revealing latent consumer preferences across disparate domains\. Features combine technical specs with style descriptions and product imagery\. This dataset is utilized for modality retrieval between textual and visual product content\.

#### Flickr30k

\(Plummeret al\.[2015](https://arxiv.org/html/2608.00623#bib.bib73)\)is a canonical image\-text reasoning dataset\. In the OpenMAG setting\(Wanet al\.[2026](https://arxiv.org/html/2608.00623#bib.bib129)\), we construct a graph where nodes represent image regions and caption phrases, linked by semantic grounding annotations\. This dataset is utilized for Graph\-to\-Text \(G2Text\) generation, evaluating the model’s ability to generate descriptive captions by traversing grounded visual\-textual relationships\.

#### SemArt

\(Garcia and Vogiatzis[2018](https://arxiv.org/html/2608.00623#bib.bib72)\)is an artwork understanding dataset of fine\-art paintings, where nodes correspond to artworks linked by artistic, historical, and semantic relations\. Nodes carry visual painting features and textual descriptions of their content and context\. This dataset is utilized for Graph\-to\-Image \(G2Image\) synthesis, producing visual content conditioned on multimodal graph context\.

## Appendix EBaseline Details

### E\.1Isolated Training

#### MMGCN

\(Weiet al\.[2019](https://arxiv.org/html/2608.00623#bib.bib63)\)introduces a multimodal framework for micro\-video recommendation by modeling user preferences across visual, acoustic, and textual channels\. It builds separate modality\-specific bipartite graphs, captures high\-order interactions within each, and fuses them via a structured integration layer, effectively reflecting user\-item dynamics in each sensory modality\.

#### MGAT

\(Taoet al\.[2020](https://arxiv.org/html/2608.00623#bib.bib62)\)applies gated attention over parallel multimodal interaction graphs for personalized recommendation\. By adaptively weighting different modalities, it disentangles fine\-grained user interests and filters out noisy or conflicting signals, enhancing preference modeling robustness\.

#### LGMRec

\(Guoet al\.[2024](https://arxiv.org/html/2608.00623#bib.bib58)\)is a multimodal recommender that models both local and global user interests through graph learning\. It separates collaborative and multimodal signals in user embeddings and addresses sparsity in local interest modeling\. A local graph embedding module learns collaborative and modality\-specific embeddings independently, while a global hypergraph module captures overall dependencies among users and items\. Combining these decoupled local and global embeddings improves recommendation accuracy and robustness\.

### E\.2Classical FL Methods

#### FedAvg\.

\(McMahanet al\.[2017](https://arxiv.org/html/2608.00623#bib.bib29)\)serves as a foundational method in FL, enabling decentralized model training across diverse devices while preserving data privacy\. Initiated by a central server that distributes a global model, clients independently execute local updates through stochastic gradient descent\. Subsequently, these updates are aggregated by the server via averaging to refine the global model, with the cycle repeating until convergence\.

#### SCAFFOLD\.

\(Karimireddyet al\.[2020](https://arxiv.org/html/2608.00623#bib.bib24)\)employs control variates to mitigate client\-drift in FL\. Demonstrating significant reductions in communication rounds, Scaffold is resilient to data heterogeneity and client sampling\.

#### FedDC\.

\(Gaoet al\.[2022](https://arxiv.org/html/2608.00623#bib.bib48)\)is a novel FL algorithm that corrects local drift through lightweight modifications\. Each client tracks the deviation between local and global model parameters using an auxiliary variable, enhancing parameter\-level consistency\.

### E\.3Multimodal FL Methods

#### PEPSY

\(Nguyenet al\.[2025](https://arxiv.org/html/2608.00623#bib.bib114)\)addresses multimodal federated learning with both missing modalities and missing input features by learning client\-side data\-missing profiles that encode local missing patterns as embedding controls\. These profiles are probabilistically aligned and aggregated on the server to reconfigure shared representations toward each client’s incomplete data view, enabling robust aggregation and stable performance under severe data incompleteness\.

#### FedMVP

\(Cheet al\.[2024](https://arxiv.org/html/2608.00623#bib.bib112)\)addresses modality missing in multimodal federated learning by leveraging frozen pre\-trained foundation models for cross\-modal completion and representation knowledge transfer\. It trains a lightweight joint encoder via multimodal contrastive objectives and performs CKA\-based importance\-aware aggregation on the server, achieving robust performance under severe modality incompleteness\.

#### FedMAC

\(Nguyenet al\.[2024](https://arxiv.org/html/2608.00623#bib.bib113)\)addresses cross\-modal heterogeneity in multimodal federated learning by introducing a modality\-aware collaboration framework that aligns heterogeneous modality representations via cross\-modal contrastive learning\. It further employs modality\-specific aggregation with adaptive weighting to mitigate negative transfer across clients with inconsistent modality availability\.

### E\.4Unimodal FGL Methods

#### FedSage\+

\(Zhanget al\.[2021](https://arxiv.org/html/2608.00623#bib.bib35)\)extends FedSage to the subgraph federated learning setting by explicitly addressing missing cross\-client neighbors\. It jointly trains a GraphSAGE classifier with a local missing\-neighbor generator that synthesizes potential cross\-subgraph neighbors, enabling more complete neighborhood aggregation under federation and improving global generalization without sharing raw graph data\.

#### FedGTA

\(Liet al\.[2023](https://arxiv.org/html/2608.00623#bib.bib36)\)innovatively merges large\-scale graph learning with federated graph learning\. Clients encode topology and node attributes, compute local smoothing confidence and mixed moments of neighbor features, and then upload these to the server\. The server uses this data to perform personalized model aggregation, utilizing local smoothing confidence as weights for effective integration\.

#### FedSPA

\(Tanet al\.[2025b](https://arxiv.org/html/2608.00623#bib.bib44)\)addresses homophily heterogeneity in federated graph learning by explicitly modeling both homophily conflict and homophily bias across clients\. It introduces Subgraph Feature Propagation Decoupling \(SFPD\) to separate homophilic and heterophilic message passing, enabling collaboration under unified homophily levels, and proposes Homophily Bias\-Driven Aggregation \(HBDA\) to adaptively weight client contributions based on spectral and parameter\-sensitivity cues, thereby improving global generalization\.

#### FedIIH

\(Yuet al\.[2025](https://arxiv.org/html/2608.00623#bib.bib43)\)addresses heterogeneity in federated graph learning by jointly modeling inter\-client and intra\-client heterogeneity\. It infers subgraph distribution similarities via a hierarchical variational framework from a global perspective, while disentangling local subgraphs into multiple latent factors to enable factor\-wise personalized federation, leading to robust collaboration across both homophilic and heterophilic graphs\.

#### FedSSP

\(Tanet al\.[2024](https://arxiv.org/html/2608.00623#bib.bib115)\)proposes a personalized federated graph learning framework that addresses cross\-domain structural heterogeneity from a spectral perspective\. It shares generic spectral knowledge across clients to mitigate knowledge conflict induced by domain shifts, while retaining client\-specific components to preserve personalization\.

#### S2FGL

\(Tanet al\.[2025a](https://arxiv.org/html/2608.00623#bib.bib45)\)tackles subgraph federated graph learning by jointly addressing spatial label\-signal disruption and spectral client drift\. It reinforces missing label semantics via prototype\-based semantic sharing and aligns graph\-frequency components across clients to improve robustness and generalization under heterogeneous subgraph distributions\.

#### FedLap

\(Aliakbariet al\.[2025](https://arxiv.org/html/2608.00623#bib.bib116)\)leverages global graph structure in subgraph federated learning via Laplacian smoothing, introducing a structural regularization term that implicitly enforces representation consistency among neighboring nodes without explicit message passing or feature sharing\. By operating in the spectral domain, FedLap captures inter\-subgraph dependencies while achieving strong privacy guarantees and low communication overhead\.

## Appendix FTheoretical Proofs

###### Theorem 1\(Unified view of propagation as graph filtering\)\.

Let𝐀~∈ℝn×n\\tilde\{\\mathbf\{A\}\}\\in\\mathbb\{R\}^\{n\\times n\}be the symmetrically normalized adjacency matrix of an undirected graph and𝐙∈ℝn×d\\mathbf\{Z\}\\in\\mathbb\{R\}^\{n\\times d\}be node representations\. Consider the general depth\-LLpolynomial graph filter, which can be formulated as follows:

𝐇=\(∑l=0Lγl​𝐀~l\)​𝐙,γl≥0,∑l=0Lγl=1\.\\mathbf\{H\}=\\Big\(\\sum\_\{l=0\}^\{L\}\\gamma\_\{l\}\\tilde\{\\mathbf\{A\}\}^\{l\}\\Big\)\\mathbf\{Z\},\\qquad\\gamma\_\{l\}\\geq 0,\\ \\sum\_\{l=0\}^\{L\}\\gamma\_\{l\}=1\.\(12\)Then: \(i\) SGC and the linearizedLL\-layer GCN correspond to the one\-hot choiceγL=1\\gamma\_\{L\}=1; \(ii\) APPNP\-style personalized\-PageRank propagation corresponds to the geometric choiceγl=\(1−β\)​βl\\gamma\_\{l\}=\(1\-\\beta\)\\,\\beta^\{l\}withβ∈\(0,1\)\\beta\\in\(0,1\)andL→∞L\\to\\infty; \(iii\) the FedTCR filter of Eq\. \([2](https://arxiv.org/html/2608.00623#S4.E2)\) is exactly the geometric choice withβ=α/\(α\+1\)\\beta=\\alpha/\(\\alpha\+1\), i\.e\., the finite\-LLtruncation of \(ii\), whose approximation error decays exponentially:

‖𝐇L−𝐇∞‖F≤βL\+1​‖𝐙‖F\.\\big\\\|\\mathbf\{H\}\_\{L\}\-\\mathbf\{H\}\_\{\\infty\}\\big\\\|\_\{F\}\\;\\leq\\;\\beta^\{L\+1\}\\,\\\|\\mathbf\{Z\}\\\|\_\{F\}\.\(13\)

###### Proof\.

Since𝐀~\\tilde\{\\mathbf\{A\}\}is symmetric with spectrum in\[−1,1\]\[\-1,1\], we have‖𝐀~‖2≤1\\\|\\tilde\{\\mathbf\{A\}\}\\\|\_\{2\}\\leq 1, so for anyβ∈\(0,1\)\\beta\\in\(0,1\)the Neumann series∑l=0∞\(β​𝐀~\)l\\sum\_\{l=0\}^\{\\infty\}\(\\beta\\tilde\{\\mathbf\{A\}\}\)^\{l\}converges to\(𝐈−β​𝐀~\)−1\(\\mathbf\{I\}\-\\beta\\tilde\{\\mathbf\{A\}\}\)^\{\-1\}\.

\(i\) WithγL=1\\gamma\_\{L\}=1and all other coefficients zero, Eq\. \([12](https://arxiv.org/html/2608.00623#A6.E12)\) reduces to𝐇=𝐀~L​𝐙\\mathbf\{H\}=\\tilde\{\\mathbf\{A\}\}^\{L\}\\mathbf\{Z\}, which is precisely the propagation operator of SGC; anLL\-layer GCN without intermediate nonlinearities and with shared collapsed weights realizes the same operator as a special case of the general filter\.

\(ii\) Withγl=\(1−β\)​βl\\gamma\_\{l\}=\(1\-\\beta\)\\beta^\{l\}andL→∞L\\to\\infty, the coefficient sequence sums to one, and∑l=0∞\(1−β\)​\(β​𝐀~\)l=\(1−β\)​\(𝐈−β​𝐀~\)−1\\sum\_\{l=0\}^\{\\infty\}\(1\-\\beta\)\(\\beta\\tilde\{\\mathbf\{A\}\}\)^\{l\}=\(1\-\\beta\)\(\\mathbf\{I\}\-\\beta\\tilde\{\\mathbf\{A\}\}\)^\{\-1\}, which is the personalized\-PageRank propagation matrix with teleport probability1−β1\-\\beta, i\.e\., the exact solution that APPNP approximates with a finite number of power\-iteration steps as its propagation matrix\.

\(iii\) Settingβ=α/\(α\+1\)\\beta=\\alpha/\(\\alpha\+1\)gives1−β=1/\(α\+1\)1\-\\beta=1/\(\\alpha\+1\), so the coefficients of Eq\. \([2](https://arxiv.org/html/2608.00623#S4.E2)\) satisfyγl=1α\+1​\(αα\+1\)l=\(1−β\)​βl\\gamma\_\{l\}=\\frac\{1\}\{\\alpha\+1\}\\big\(\\frac\{\\alpha\}\{\\alpha\+1\}\\big\)^\{l\}=\(1\-\\beta\)\\beta^\{l\}, i\.e\., FedTCR’s filter is the finite\-LLtruncation of \(ii\)\. For the error, writing𝐇∞=\(1−β\)​∑l=0∞\(β​𝐀~\)l​𝐙\\mathbf\{H\}\_\{\\infty\}=\(1\-\\beta\)\\sum\_\{l=0\}^\{\\infty\}\(\\beta\\tilde\{\\mathbf\{A\}\}\)^\{l\}\\mathbf\{Z\},

‖𝐇L−𝐇∞‖F\\displaystyle\\\|\\mathbf\{H\}\_\{L\}\-\\mathbf\{H\}\_\{\\infty\}\\\|\_\{F\}≤\(1−β\)​∑l=L\+1∞βl​‖𝐀~‖2l​‖𝐙‖F\\displaystyle\\leq\(1\-\\beta\)\\sum\_\{l=L\+1\}^\{\\infty\}\\beta^\{l\}\\\|\\tilde\{\\mathbf\{A\}\}\\\|\_\{2\}^\{l\}\\,\\\|\\mathbf\{Z\}\\\|\_\{F\}≤\(1−β\)​βL\+11−β​‖𝐙‖F=βL\+1​‖𝐙‖F,\\displaystyle\\leq\(1\-\\beta\)\\,\\frac\{\\beta^\{L\+1\}\}\{1\-\\beta\}\\,\\\|\\mathbf\{Z\}\\\|\_\{F\}=\\beta^\{L\+1\}\\\|\\mathbf\{Z\}\\\|\_\{F\},where the first inequality applies submultiplicativity of the Frobenius norm and the second uses‖𝐀~‖2≤1\\\|\\tilde\{\\mathbf\{A\}\}\\\|\_\{2\}\\leq 1\. ∎

## Appendix GMetric Details

All metrics are computed on each client’s held\-out split after isolated task\-oriented fine\-tuning, and are reported as mean±\\pmstandard deviation over five random seeds \(Appendix[B\.1](https://arxiv.org/html/2608.00623#A2.SS1)\)\. Task definitions follow Appendix[B](https://arxiv.org/html/2608.00623#A2)\.

#### Node Classification\.

We report classification accuracy \(ACC\):

ACC=1N​∑i=1N𝕀​\(y^i=yi\),\\text\{ACC\}=\\frac\{1\}\{N\}\\sum\_\{i=1\}^\{N\}\\mathbb\{I\}\(\\hat\{y\}\_\{i\}=y\_\{i\}\),\(14\)whereNNis the number of test nodes,yiy\_\{i\}the ground\-truth label,y^i\\hat\{y\}\_\{i\}the prediction, and𝕀​\(⋅\)\\mathbb\{I\}\(\\cdot\)the indicator function\.

#### Link Prediction\.

We report the Area Under the ROC Curve \(AUC\), which measures the probability that a randomly chosen positive \(existent\) edge receives a higher score than a randomly chosen negative \(non\-existent\) node pair:

AUC=P​\(s​\(u,v\)\>s​\(u′,v′\)\),\\text\{AUC\}=P\\big\(s\(u,v\)\>s\(u^\{\\prime\},v^\{\\prime\}\)\\big\),\(15\)wheres​\(⋅,⋅\)s\(\\cdot,\\cdot\)is the model’s similarity score,\(u,v\)\(u,v\)is a positive edge, and\(u′,v′\)\(u^\{\\prime\},v^\{\\prime\}\)is a sampled negative pair\.

#### Modality Retrieval\.

We report Recall@1 \(R@1\), the fraction of queries whose ground\-truth cross\-modal target is ranked first:

R@1=1\|Q\|​∑i=1\|Q\|𝕀​\(ranki=1\),\\text\{R@1\}=\\frac\{1\}\{\|Q\|\}\\sum\_\{i=1\}^\{\|Q\|\}\\mathbb\{I\}\(\\text\{rank\}\_\{i\}=1\),\(16\)whereQQis the query set andranki\\text\{rank\}\_\{i\}is the rank of the correct target for queryii\.

#### Graph\-to\-Text \(G2Text\)\.

Following MMGL\(Yoonet al\.[2023](https://arxiv.org/html/2608.00623#bib.bib59)\), we report ROUGE\-L \(R\-L\), which measures sentence\-level recall based on the longest common subsequence between the generated description and the reference:

ROUGE\-L=\(1\+β2\)​Rl​c​s​Pl​c​sRl​c​s\+β2​Pl​c​s\.\\text\{ROUGE\-L\}=\\frac\{\(1\+\\beta^\{2\}\)R\_\{lcs\}P\_\{lcs\}\}\{R\_\{lcs\}\+\\beta^\{2\}P\_\{lcs\}\}\.\(17\)

#### Graph\-to\-Image \(G2Image\)\.

Following InstructG2I\(Jinet al\.[2024](https://arxiv.org/html/2608.00623#bib.bib60)\), we report CLIP\-Score \(C\-S\), which quantifies cross\-modal semantic consistency between the generated image and its conditioning description via pre\-trained CLIP encoders \(ETE\_\{T\}for text andEIE\_\{I\}for images\):

CLIP\-Score​\(I,T\)=max⁡\(100⋅cos⁡\(EI​\(I\),ET​\(T\)\),0\)\.\\text\{CLIP\-Score\}\(I,T\)=\\max\\left\(100\\cdot\\cos\(E\_\{I\}\(I\),E\_\{T\}\(T\)\),0\\right\)\.\(18\)

## Appendix HLimitations

While FedTCR demonstrates strong empirical performance across diverse FMGL scenarios, several limitations warrant discussion\. First, each client distills its modality\-specific knowledge into a single prototype vector per modality, which may not fully capture the internal multi\-community structure of large\-scale MAGs with diverse semantic clusters\. Extending to multiple prototypes per modality \(e\.g\., cluster\-aware prototypes\) could improve expressiveness at the cost of increased communication overhead\. Second, our evaluation focuses on MAGs with text and image modalities; the generalizability to other modality types \(e\.g\., audio, video, tabular data\) remains to be validated\. Third, the two\-stage paradigm assumes that the pre\-training phase has sufficiently converged before fine\-tuning begins, and the potential gap between the task\-agnostic objective and specific downstream tasks may limit the adaptability for highly specialized applications\. Exploring progressive or continual pre\-training strategies could mitigate this issue\.

## Appendix IEnvironment

All experiments are conducted on a workstation equipped with Intel Xeon Scalable processors and NVIDIA RTX 6000 Ada Generation GPUs with 96 GB of VRAM, supported by 256 GB of system RAM\. The computational environment utilizes CUDA 12\.9, while software implementations are developed using Python 3\.10\.18 and PyTorch 2\.8\.

## Appendix JPseudocode

The complete procedure of FedTCR is presented in Algorithm[1](https://arxiv.org/html/2608.00623#algorithm1), which consists of two stages: federated task\-agnostic pre\-training and isolated task\-oriented fine\-tuning\.

Input:

KKclients with MAGs

\{𝒢k\}k=1K\\\{\\mathcal\{G\}^\{k\}\\\}\_\{k=1\}^\{K\}; modality set

ℳ\\mathcal\{M\}; extended modality set

ℳ¯=ℳ∪\{\|ℳ\|\+1\}\\bar\{\\mathcal\{M\}\}=\\mathcal\{M\}\\cup\\\{\|\\mathcal\{M\}\|\+1\\\}; communication rounds

TT; local epochs

EE; smoothing strength

α\\alpha; propagation depth

LL; temperature

τ\\tau; loss weights

λm,λg,λc\\lambda\_\{m\},\\lambda\_\{g\},\\lambda\_\{c\}
Output:Task\-specific predictions for each client

/\*Stage 1: Federated Task\-agnostic Pre\-training \*/

1Initialize modality\-specific encoders

\{𝚽\(m\)\}m∈ℳ\\\{\\mathbf\{\\Phi\}^\{\(m\)\}\\\}\_\{m\\in\\mathcal\{M\}\};

2for*each clientk∈\[K\]k\\in\[K\]in parallel*do

3Compute PageRank importance weights

\{ωik\}i∈𝒱k\\\{\\omega\_\{i\}^\{k\}\\\}\_\{i\\in\\mathcal\{V\}^\{k\}\}via power iteration \(Eq\. \([4](https://arxiv.org/html/2608.00623#S4.E4)\)\);

4Simulate random walks to obtain positive neighbor sets

\{𝒫ik\}i∈𝒱k\\\{\\mathcal\{P\}\_\{i\}^\{k\}\\\}\_\{i\\in\\mathcal\{V\}^\{k\}\}and sample negative sets

\{𝒩ik\}i∈𝒱k\\\{\\mathcal\{N\}\_\{i\}^\{k\}\\\}\_\{i\\in\\mathcal\{V\}^\{k\}\};

5

6for*each roundt=1,2,…,Tt=1,2,\\ldots,T*do

7Server distributes global encoders

\{𝚽\(m\)\}m∈ℳ\\\{\\mathbf\{\\Phi\}^\{\(m\)\}\\\}\_\{m\\in\\mathcal\{M\}\}to all clients;

8if*t≥2t\\geq 2*thenServer distributes routed prototypes

\{\(𝐏posk,\(i,j\),𝐏negk,\(i,j\)\)\}i,j\\\{\(\\mathbf\{P\}\_\{\\mathrm\{pos\}\}^\{k,\(i,j\)\},\\mathbf\{P\}\_\{\\mathrm\{neg\}\}^\{k,\(i,j\)\}\)\\\}\_\{i,j\}to each client

kk;

/\*Client\-side Local Training \*/

9for*each clientk∈\[K\]k\\in\[K\]in parallel*do

10for*each local epoche=1,2,…,Ee=1,2,\\ldots,E*do

/\*Multimodal Graph Encoding \(Eq\. \([1](https://arxiv.org/html/2608.00623#S4.E1)\), \([2](https://arxiv.org/html/2608.00623#S4.E2)\)\) \*/

11

𝐙k,\(m\)←L2Norm​\(𝐗k,\(m\)​𝚽\(m\)\)\\mathbf\{Z\}^\{k,\(m\)\}\\leftarrow\\mathrm\{L2Norm\}\(\\mathbf\{X\}^\{k,\(m\)\}\\mathbf\{\\Phi\}^\{\(m\)\}\),

∀m∈ℳ\\forall m\\in\\mathcal\{M\};

12

𝐙¯k←1\|ℳ\|​∑m∈ℳ𝐙k,\(m\)\\bar\{\\mathbf\{Z\}\}^\{k\}\\leftarrow\\frac\{1\}\{\|\\mathcal\{M\}\|\}\\sum\_\{m\\in\\mathcal\{M\}\}\\mathbf\{Z\}^\{k,\(m\)\};

13

𝐇k←1α\+1​∑l=0L\(αα\+1​𝐀~k\)l​𝐙¯k\\mathbf\{H\}^\{k\}\\leftarrow\\frac\{1\}\{\\alpha\+1\}\\sum\_\{l=0\}^\{L\}\\left\(\\frac\{\\alpha\}\{\\alpha\+1\}\\tilde\{\\mathbf\{A\}\}^\{k\}\\right\)^\{l\}\\bar\{\\mathbf\{Z\}\}^\{k\};

/\*Topology\-aware Prototype Construction \(Eq\. \([5](https://arxiv.org/html/2608.00623#S4.E5)\)\) \*/

14

𝐏k,\(m\)←∑i∈𝒱kωik⋅𝐳ik,\(m\)\\mathbf\{P\}^\{k,\(m\)\}\\leftarrow\\sum\_\{i\\in\\mathcal\{V\}^\{k\}\}\\omega\_\{i\}^\{k\}\\cdot\\mathbf\{z\}\_\{i\}^\{k,\(m\)\},

∀m∈ℳ¯\\forall m\\in\\bar\{\\mathcal\{M\}\};

/\*Local Optimization \(Eq\. \([8](https://arxiv.org/html/2608.00623#S4.E8)\), \([9](https://arxiv.org/html/2608.00623#S4.E9)\), \([10](https://arxiv.org/html/2608.00623#S4.E10)\)\) \*/

15Compute node\-level contrastive loss

ℒm\\mathcal\{L\}\_\{m\};

16Compute neighbor\-level contrastive loss

ℒg\\mathcal\{L\}\_\{g\};

17if*t≥2t\\geq 2*thenCompute client\-level contrastive loss

ℒc\\mathcal\{L\}\_\{c\}using routed prototypes;

18Update

\{𝚽\(m\)\}m∈ℳ\\\{\\mathbf\{\\Phi\}^\{\(m\)\}\\\}\_\{m\\in\\mathcal\{M\}\}via

∇\(λm​ℒm\+λg​ℒg\+λc​ℒc\)\\nabla\(\\lambda\_\{m\}\\mathcal\{L\}\_\{m\}\+\\lambda\_\{g\}\\mathcal\{L\}\_\{g\}\+\\lambda\_\{c\}\\mathcal\{L\}\_\{c\}\)\(Eq\. \([11](https://arxiv.org/html/2608.00623#S4.E11)\)\);

19

20Upload updated encoders

\{𝚽k,\(m\)\}m∈ℳ\\\{\\mathbf\{\\Phi\}^\{k,\(m\)\}\\\}\_\{m\\in\\mathcal\{M\}\}and final prototypes

\{𝐏k,\(m\)\}m∈ℳ¯\\\{\\mathbf\{P\}^\{k,\(m\)\}\\\}\_\{m\\in\\bar\{\\mathcal\{M\}\}\}to server;

21

/\*Server\-side Aggregation and Routing \*/

22Aggregate encoders:

𝚽\(m\)←∑k=1K\|𝒱k\|N​𝚽k,\(m\)\\mathbf\{\\Phi\}^\{\(m\)\}\\leftarrow\\sum\_\{k=1\}^\{K\}\\frac\{\|\\mathcal\{V\}^\{k\}\|\}\{N\}\\mathbf\{\\Phi\}^\{k,\(m\)\},

∀m∈ℳ\\forall m\\in\\mathcal\{M\};

/\*Cross\-modal Prototype Routing \(Eq\. \([7](https://arxiv.org/html/2608.00623#S4.E7)\)\) \*/

23for*each clienta∈\[K\]a\\in\[K\]*do

24for*each modality pair\(i,j\),i∈ℳ¯,j∈ℳ¯∖\{i\}\(i,j\),\\;i\\in\\bar\{\\mathcal\{M\}\},\\;j\\in\\bar\{\\mathcal\{M\}\}\\setminus\\\{i\\\}*do

25

𝐏posa,\(i,j\)←𝐏^b∗,\(j\)\\mathbf\{P\}\_\{\\mathrm\{pos\}\}^\{a,\(i,j\)\}\\leftarrow\\hat\{\\mathbf\{P\}\}^\{b^\{\*\},\(j\)\}, where

b∗=arg​maxb≠a⁡cos⁡\(𝐏^a,\(i\),𝐏^b,\(j\)\)b^\{\*\}=\\operatorname\*\{arg\\,max\}\_\{b\\neq a\}\\cos\(\\hat\{\\mathbf\{P\}\}^\{a,\(i\)\},\\hat\{\\mathbf\{P\}\}^\{b,\(j\)\}\);

26

𝐏nega,\(i,j\)←𝐏^b′,\(j\)\\mathbf\{P\}\_\{\\mathrm\{neg\}\}^\{a,\(i,j\)\}\\leftarrow\\hat\{\\mathbf\{P\}\}^\{b^\{\\prime\},\(j\)\}, where

b′=arg​minb≠a⁡cos⁡\(𝐏^a,\(i\),𝐏^b,\(j\)\)b^\{\\prime\}=\\operatorname\*\{arg\\,min\}\_\{b\\neq a\}\\cos\(\\hat\{\\mathbf\{P\}\}^\{a,\(i\)\},\\hat\{\\mathbf\{P\}\}^\{b,\(j\)\}\);

27

28

29

/\*Stage 2: Isolated Task\-oriented Fine\-tuning \*/

30for*each clientk∈\[K\]k\\in\[K\]in parallel*do

31Attach task\-specific head

gkg^\{k\}to the pre\-trained encoder;

32Fine\-tune

\{𝚽\(m\)\}m∈ℳ\\\{\\mathbf\{\\Phi\}^\{\(m\)\}\\\}\_\{m\\in\\mathcal\{M\}\}and

gkg^\{k\}on local MAG

𝒢k\\mathcal\{G\}^\{k\}with task\-specific loss

ℒtask\\mathcal\{L\}\_\{\\mathrm\{task\}\};

33

Algorithm 1FedTCR: FMGL with Topology\-aware Cross\-modal Routing