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Structural Entropy-Driven Graph Diffusion Generation for One-Shot Federated Graph Learning

Shutong Zheng, Lele Fu, Sheng Huang, Wei Yang Bryan Lim, Chuan Chen

arXiv:2609.06499v1cs.LG

TL;DR

One-shot FGL must estimate client contributions from compressed information, but volume-based weighting overlooks graph connectivity. SPIRE uses structural entropy to guide client weighting and server-side pseudograph generation. Across seven datasets, it achieves superior classification performance and robust scalability, particularly on large graphs.

  • Problem

    One-shot FGL methods generally weight clients uniformly or by data scale without explicitly evaluating local graph structure, while pseudograph generation may fail to preserve meaningful topological semantics.

    Source: §2.3 · p. 2

  • Method

    SPIRE uses first-order structural entropy to derive topology-aware client influence and guides server-side pseudograph synthesis with that bias.

    Source: §1 · p. 2, §4.1 · p. 4, §4.2 · p. 6

  • Results

    Across seven datasets SPIRE achieves superior classification performance and robust scalability, particularly on large graphs; in pathological-topology stress tests it outperforms volume-based weighting by +2.47%–+3.47%.

    Source: §6 · p. 9, §5.6 · p. 9

  • Takeaways & Limitations

    The paper positions structural entropy as an efficient O(1) inductive prior for differentiating client influence under degree heterogeneity, not as a universal graph-quality oracle.

    Source: §5.6 · p. 9

    Degree-level statistics cannot distinguish non-isomorphic graphs with identical degree sequences or determine whether highly centralized connectivity benefits downstream learning.

    Source: §5.6 · p. 9

Abstract

from arXiv · show

One-shot federated graph learning (FGL) requires the server to estimate client contributions from highly compressed information, yet conventional volume-based weighting captures the amount of client data while overlooking how its connectivity is organized. In this paper, we propose SPIRE, a Structural Entropy-Driven Graph Diffusion Generation method that introduces topology-aware client differentiation into one-shot FGL. Specifically, we employ first-order degree-distribution structural entropy as a compact descriptor of degree-mass dispersion and use it to derive structural client weights, providing an inductive bias that accounts for differences in graph topology beyond data volume. On the generation side, a graph diffusion model on the server synthesizes pseudographs conditioned on the weighted client prototypes, capturing both semantic and structural information without requiring additional client-side training. The generated pseudographs are then assembled via disjoint union fusion to train a global graph neural network. Extensive experiments on seven real-world graph datasets demonstrate that SPIRE consistently outperforms conventional and one-shot FGL methods, with particularly strong gains under highly heterogeneous (non-IID) and graph-perturbed settings.

1 Introduction

Graph clients differ in both data distribution and topology, while one-shot FGL asks the server to estimate their influence from compressed information. SPIRE uses structural entropy to guide client weighting and server-side pseudograph generation, addressing the volume–structure mismatch identified in the paper.

1 Introduction

  • One-shot FGL requires influence estimation from highly compressed statistics, whereas iterative FGL methods rely on repeated communication.The paper notes that repeated synchronization can incur prohibitive bandwidth costs and exacerbate privacy risks.Source: §1 · p. 1
  • Volume-based one-shot weighting can favor larger clients without accounting for how their connectivity is organized.Figure 1 contrasts FedAvg’s volume preference for C1 with SPIRE’s structural-entropy differentiation.Source: §1 · pp. 1–2, Figure 1 · p. 1
  • SPIRE uses first-order structural entropy as a compact descriptor of connectivity distribution beyond graph volume.The method conditions server-side graph diffusion on weighted client prototypes and avoids dense adjacency optimization.Source: §1 · p. 2
  • SPIRE propagates an entropy-regularized structural influence bias from client weighting into downstream global reconstruction.Its contributions include feature-first pseudograph synthesis via conditional diffusion and kNN assembly, plus sensitivity bounds for entropy perturbations.Source: §1 · p. 2
  • Evaluations across seven datasets confirm SPIRE’s state-of-the-art performance, robustness to severe heterogeneity, and practical large-scale scalability.The paper identifies these as its theoretical and empirical validation outcomes.Source: §1 · p. 2

2 Related Work

Related work spans conventional federated learning, federated graph learning, and one-shot learning. The paper identifies a gap in one-shot FGL: existing approaches generally weight clients by uniformity or data scale and may not preserve meaningful topological semantics in generated pseudographs.

2.1 Conventional Federated Learning

  • Conventional FL methods address heterogeneity through distillation, generative modeling, and prototype aggregation, but are primarily designed for Euclidean data.The cited work describes conventional FL as facing statistical heterogeneity and communication inefficiency under non-IID settings [5] [18] [22].Source: §2.1 · p. 2

2.2 Federated Graph Learning

  • FGL work covers graph-level representation sharing, node-level cross-client structure preservation, and topology-aware distillation or causal decoupling.SEFGL uses structural entropy for local optimization and iterative personalized prototype clustering in multi-round settings.Source: §2.2 · p. 2

2.3 One-Shot Federated Learning

  • One-shot FGL methods include proxy-model and topological-codebook approaches, but generally use uniform or data-scale weighting and may lose meaningful topological semantics.One-shot methods also include public-data distillation, data-free generators, and synthetic-data sharing strategies.Source: §2.3 · p. 2

3 Preliminaries and Motivation

The preliminaries define the graph, GNN, and federated setting used in the paper. The motivation shows that client volume alone may misrepresent structural contribution, motivating first-order structural entropy as a compact influence signal.

3.1 Background

  • A client graph G=(V,E,X,Y) has nodes, edges, features, and labels, with its topology represented by an adjacency matrix.Each node has a feature vector and corresponding label.Source: §3.1 · pp. 2–3
  • A GNN updates node embeddings by aggregating information from neighboring nodes and applying an update rule.The initial node embedding is its feature vector.Source: §3.1 · p. 3
  • In standard FGL, the server aggregates client model parameters using each client’s node count and the total number of nodes.The paper uses this volume-based rule to motivate separating graph size from structural utility.Source: §3.1 · p. 3

3.2 Motivation

  • On Cora, a large structurally dispersed client frequently underperforms a small concentrated client, and heterogeneous one-shot aggregation can cause weight divergence and global performance drops.The study constructs four client archetypes by varying volume and degree-distribution entropy independently.Source: §3.2 · p. 3
  • First-order structural entropy summarizes concentration of the local degree-mass distribution; lower entropy indicates a more concentrated topology.SPIRE uses it to differentiate client influence beyond volume without transmitting raw adjacency matrices.Source: §3.2 · p. 3

4 Methodology

SPIRE uses first-order degree-distribution entropy to weight clients, then propagates those weights through prototype aggregation, barycentric alignment, and pseudograph generation. The server trains a global GNN on the assembled pseudograph.

4.1 Topology-Aware Influence Allocation

  • Clients summarize each graph with normalized first-order degree entropy, class prototypes, and node count; no raw features, adjacency matrices, or model parameters are transmitted.The reported communication cost is O(C×F + 1) per client.Source: §4.1 · p. 3
  • Entropy-derived weights favor clients with concentrated degree distributions; τ trades structural selectivity against uniform regularization and dampens transmitted-statistic error.SPIRE carries the same weights into subsequent global reconstruction rather than using them only for initial aggregation.Source: §4.1 · pp. 4–5

4.2 Structure-Guided Pseudograph Synthesis

  • The server uses the weights to aggregate semantic prototypes and initialize a geometric target with a Wasserstein barycenter.The same weights are used for barycentric alignment to maintain structural consistency.Source: §4.2 · p. 5
  • Under bounded barycenters and transport costs, Lemma 1 bounds objective deviation by M∥α̂ − α∥1, while Corollary 1 bounds entropy-error propagation by 2M tanh(δ/τ).The corollary combines the weight-perturbation result with barycentric objective stability.Source: §4.2 · p. 5
  • Conditional diffusion generates features from noise under weighted prototypes, while centroid alignment and structural-entropy regularization guide the generator.The entropy term encourages generated topology to follow the degree-level preference used for client weighting.Source: §4.2 · pp. 5–6
  • Entropy weights set generated sample counts and edge weights; synthesized features receive KNN topology, and client subgraphs are merged by disjoint union for global GNN training.Controlled Gaussian noise is added to synthesized features before topology reconstruction.Source: §4.2 · pp. 5–6

5 Experiments

The experiments compare SPIRE with conventional, multi-round, and one-shot baselines across seven node-classification benchmarks. They examine performance, communication, robustness, sensitivity, component contributions, and the scope of its structural descriptor.

5.1 Experimental Setup

  • Evaluation spans seven node-classification benchmarks, with standard, official, or random splits; comparisons include conventional, multi-round, and one-shot baselines.Source: §5.1 · p. 6

5.2 Performance Comparison

  • SPIRE leads one-shot baselines across all seven datasets; on Cora it beats OASIS by 22.40% and GHOST by 25.87%, and on ogbn-arxiv beats GHOST by 5.51%.Source: §5.2 · p. 7
  • Per-client communication stays below 0.01s on Cora, CiteSeer, and PubMed and is 0.11s on ogbn-arxiv, while server-side synthesis raises Cora training time to 1265s.Source: §5.2 · p. 7, Table 2 · p. 7

5.3 Robustness Analysis

  • Under β=0.05, SPIRE stays stable and widens its lead as heterogeneity intensifies; under ρ/η=0.5, its Cora edge-perturbation drop is 1.96% and feature accuracy is 64.10%.Source: Figure 4 · p. 7, §5.3 · p. 8, Table 3 · p. 8
  • On Cora, SPIRE achieves 60.35% and 58.58% at K=50 and K=100, above OASIS’s 40.59%/37.59% and GHOST’s 37.10%/35.33%.OASIS runs out of memory on ogbn-arxiv across the tested client counts.Source: §5.3 · p. 8, Table 4 · p. 8

5.4 Sensitivity Analysis

  • Across 10^-7 to 10^-3, structural-regularization accuracy varies from 62.80% to 64.06%; alignment weight 0.001 yields 44.50%, while 0.1 yields 64.06%.Source: §5.4 · p. 8

5.5 Ablation Study

  • GFM and SE synergize (64.06% on Cora; 55.32% on CiteSeer), while removing Wasserstein initialization, jitter, or structural regularization lowers Cora accuracy by 4.66%, 1.73%, or 0.27%.Source: §5.5 · p. 9

5.6 Discussion

  • Despite degree-sequence ambiguity, pathological-client stress tests retain a 2.47%–3.47% advantage over volume weighting; SPIRE also lacks formal differential-privacy guarantees.The paper positions the descriptor as an efficient inductive prior, not a universal graph-quality oracle.Source: §5.6 · p. 9

6 Conclusion

SPIRE uses first-order structural entropy to address volume–structure mismatch in client contribution estimation and carries topology-aware weighting into server-side pseudograph synthesis. Across seven datasets, it achieves superior classification performance and robust scalability, while its analysis identifies richer structural representations and formal privacy mechanisms as future work.

6 Conclusion

  • Across seven datasets, SPIRE achieves superior classification performance and robust scalability, particularly on large graphs where conventional methods face communication or memory bottlenecks.Its theoretical analysis also provides explicit bounds on how local entropy perturbations propagate to global barycentric initialization.Source: §6 · pp. 9–10
  • SPIRE uses first-order structural entropy in entropy-regularized client weighting and propagates topology-aware bias into server-side pseudograph synthesis.The paper presents first-order entropy as an effective degree-level proxy that does not require raw topology exchange.Source: §6 · pp. 9–10
  • Future work will explore richer structural representations and formal privacy mechanisms for broader graph-mining applications.Source: §6 · p. 10

Ethical Considerations

The experiments use publicly available standard graph benchmarks in simulation and do not involve human subjects, personally identifiable information, or collected private user data.

Ethical Considerations

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