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Network Denoising Revisited: A Ricci-Flow-Inspired Graph Diffusion Method

Ye Fang, Chuan-Xian Ren

arXiv:2608.16923v1cs.SIcs.LG

TL;DR

Network denoising must account for heterogeneous transport caused by graphs’ non-Euclidean geometry, which similarity-driven diffusion overlooks. Ricci-Diffusion incorporates edge curvature into graph diffusion, and experiments show improved denoising quality and frequent downstream-task benefits across real-world and synthetic graphs.

  • Problem

    Network denoising seeks refined edge weights that better reflect true relationships and improve downstream performance, but conventional diffusion overlooks geometry-induced differences in information transport.

  • Method

    Ricci-Diffusion updates edge weights with a curvature-aware diffusion kernel that adapts local transport to heterogeneous graph geometry and promotes curvature homogenization.

  • Results

    Experiments across real-world and controlled synthetic graphs improve explicit denoising quality and often benefit downstream tasks, while curvature modulation improves ablation performance.

  • Takeaways & Limitations

    Incorporating curvature into graph diffusion is effective for structure-oriented network denoising and yields more coherent, structurally consistent networks.

Abstract

from arXiv · show

Networks provide a fundamental representation of relationships among entities. However, real-world networks are often corrupted by noise caused by measurement errors and inherent stochasticity, hindering the discovery of meaningful structure. Most denoising methods rely on similarity-driven diffusion and ignore the non-Euclidean geometry of graphs, where local variations induce heterogeneous information transport. This motivates a geometric revisit of network denoising. In this work, we propose Ricci-Diffusion, a curvature-guided graph diffusion method inspired by Ricci flow. Specifically, Ricci-Diffusion exhibits a Ricci-flow-like evolution, in which relative edge-level curvature modulates local transport in the diffusion kernel and guides edge-weight updates toward a more regular graph geometry. We further provide a theoretical analysis showing that curvature can distinguish graph structures that common similarity-driven diffusion kernels fail to separate, and that curvature induces first-order corrections in one-step diffusion updates. The resulting diffusion process explicitly characterizes transport heterogeneity across local geometries and admits theoretical convergence to a stable denoised network. Results on real-world and synthetic graphs show that curvature-guided updates and curvature homogenization improve structure recovery and downstream performance.

I. INTRODUCTION · II. RELATED WORK

The paper reframes network denoising as geometry-aware information transport and proposes Ricci-Diffusion, which uses edge-level curvature to adapt diffusion to heterogeneous graph structure. It positions this approach against similarity-based, probabilistic, and conventional diffusion methods, while establishing theoretical distinctions and evaluating structure recovery and downstream performance.

  • I. INTRODUCTION: Networks model relationships across fields, including weighted biological interaction networks used to identify functional modules and characterize cellular organization.In weighted protein–protein interaction networks, edge weights encode physicochemical interaction strength.
  • I. INTRODUCTION: Network denoising suppresses spurious connections and adjusts edge weights to recover meaningful structural relations.
  • II. RELATED WORK: Prior denoising methods use local similarity heuristics, probabilistic modeling, causal inference, or diffusion based on random walks, paths, spectral inversion, and doubly stochastic operators.These approaches generally rely on static structural criteria, global inference objectives, or similarity-based propagation.
  • I. INTRODUCTION: Ricci-Diffusion reframes denoising as geometry-dependent information transport by embedding edge-level Ricci curvature into diffusion kernels.The method is inspired by Ricci-flow-like evolution and adapts local transport to heterogeneous graph geometry.
  • I. INTRODUCTION: Curvature-aware diffusion distinguishes graph structures that common similarity-driven kernels cannot separate and introduces first-order geometry corrections to standard diffusion.Theorem 1 establishes the existence of such structures, with Fig. 1 providing a representative special case.
  • I. INTRODUCTION: Relative curvature biases transport toward structurally coherent propagation by down-weighting unreliable bridge-like edges during iterative edge-weight updates.This mechanism yields denoised networks with clearer geometric structure.
  • I. INTRODUCTION: Experiments on real-world biological networks and controlled synthetic graphs assess structure recovery, downstream performance, curvature-guided edge evolution, and empirical Ricci-flow-like behavior.The reported results demonstrate improved structure recovery and competitive downstream performance.
  • II. RELATED WORK: Graph curvature, including Ollivier–Ricci and Forman–Ricci curvature, characterizes local geometry through optimal transport or combinatorial constructions.Prior work has used these curvature notions in community detection, while diffusion-based denoising has lacked an explicit geometric perspective on heterogeneous information transport.

III. RICCI-DIFFUSION · A. Network Denoising Problem · B. Curvature-Aware Diffusion Kernel

Ricci-Diffusion formulates network denoising as geometry-aware graph diffusion, using curvature-guided transport to update edge weights and suppress unreliable connections. Its curvature-aware kernel combines local similarity structure with Ollivier–Ricci curvature to produce geometry-adaptive, mass-preserving diffusion.

  • III. RICCI-DIFFUSION: Ricci-Diffusion denoises weighted graphs by updating edge weights through a curvature-aware diffusion kernel that adapts aggregation and diffusion strength within three-hop neighborhoods.Relative edge-level curvature reduces unreliable bridge-like propagation while promoting structurally coherent transport.
  • A. Network Denoising Problem: The method models graph denoising as generalized diffusion, in which the diffusion kernel determines information transportation across heterogeneous graph geometry.The target is a refined graph whose edge weights better reflect true relationships and improve downstream performance.
  • A. Network Denoising Problem: The denoising problem starts from an undirected weighted graph G = (V, E, W) with symmetric nonnegative weights, producing a refined graph eG with improved relationship estimates.Each weight encodes the similarity or confidence of a relationship between two nodes.
  • B. Curvature-Aware Diffusion Kernel: Ricci flow motivates graph evolution in which curvature guides metric updates and promotes curvature homogenization, yielding clearer network organization.The graph analogue contracts or expands regions according to curvature-related geometric behavior.
  • B. Curvature-Aware Diffusion Kernel: The kernel combines a base similarity field with curvature-guided transport so relative edge-level curvature biases diffusion away from unreliable connections and toward geometrically consistent ones.This explicitly models heterogeneous information transport rather than relying only on similarity aggregation.
  • B. Curvature-Aware Diffusion Kernel: The similarity field is built from a k-nearest-neighbors sparsification that retains each node’s top-k strongest connections, assuming local neighborhoods remain robust under noise.The resulting sparse graph supports multi-hop transition structure and higher-order consistency.
  • B. Curvature-Aware Diffusion Kernel: Ollivier–Ricci curvature is computed after mapping edge weights to distances dij = −log(wij), using Wasserstein geometry to characterize local transport contraction or expansion.Positive curvature indicates densely connected regions, whereas negative curvature appears on sparse or bridge-like edges.
  • B. Curvature-Aware Diffusion Kernel: Exponential curvature modulation is normalized and projected onto doubly stochastic matrices, producing a stable kernel with geometry-adaptive information transport and mass preservation.This kernel forms the foundation of Ricci-Diffusion as a curvature-guided graph denoising method.

C. Ricci-Diffusion Process

Ricci-Diffusion iteratively updates edge weights with a curvature-aware kernel, combining three-hop transport with regularization. Its dynamic-to-static strategy adapts curvature during an initial stage, then fixes the kernel for a convergent diffusion process.

  • One Step Iteration: Ricci-Diffusion updates weights by W_t+1 = τT_κW_tT_κ + (1 − τ)T_κ, combining curvature-weighted three-hop transport with regularization against numerical drift.Curvature enters through direct kernel corrections and source- and target-side biases in propagated edge weights.
  • Generalized Diffusion Representation: The resulting weight matrix admits a generalized graph diffusion representation with coefficients θ_2k = 0 and θ_2k+1 = (1 − τ)τ^k for k ≥ 0.This places Ricci-Diffusion within the generalized graph diffusion framework.
  • Convergence: If T_κ is a DSM, the iteration converges to a unique nontrivial fixed point f W, which is also a DSM.The convergence follows from the kernel’s unit spectral radius and the associated matrix geometric series.
  • Dynamic-to-Static Strategy: During the dynamic stage, curvature and the diffusion kernel are periodically recomputed as edge weights evolve; afterward, the kernel is fixed and diffusion continues until convergence.The two-stage strategy adapts transport geometry while avoiding excessive recomputation, reducing computational cost and preserving theoretical tractability.

D. Theoretical Interpretation of Ricci-Diffusion

Theoretical results explain why curvature provides geometric information that similarity-driven diffusion misses and how it enters Ricci-Diffusion updates. Curvature induces a first-order, neighborhood-relative correction that guides local edge-weight evolution toward curvature homogenization.

  • Similarity-kernel limitation and curvature separation: Theorem 1 shows that similarity-driven kernels can assign identical transport strength to edges with different geometric roles, whereas curvature separates them.This separation supplies the local geometric signal used by Ricci-Diffusion.
  • Curvature-induced first-order correction: Proposition 3 characterizes curvature’s effect on one-step diffusion through a first-order correction to the parameterized kernel update.The detailed proofs are provided in Appendix I.
  • Similarity-kernel limitation and curvature separation: κua −κuv = 2k k + 3 > 0.The constructed graph family therefore exhibits an explicit positive curvature separation between the two edges.
  • Curvature-induced first-order correction: The correction depends on each edge’s curvature relative to its local neighborhood, rather than on absolute curvature alone.Specifically, the curvature signal is represented by Ts(i, j)(κij −¯κi) for neighboring edges and 0 otherwise.
  • Curvature-induced first-order correction: The first-order update combines direct kernel adjustment with source-side and target-side curvature biases in propagated edge-weight updates.These terms are represented by (1 −τ) ˙T0, τ ˙T0WT0, and τT0W ˙T0, respectively.

IV. EXPERIMENTS

The experiments evaluate Ricci-Diffusion as a general-purpose network denoising module from four complementary perspectives, including three biologically motivated downstream scenarios. Real-world comparisons use representative diffusion-based baselines, while synthetic experiments add BORF and GSR.

  • Experimental design: Ricci-Diffusion is evaluated across four complementary perspectives as a general-purpose network denoising module.The reported downstream scenarios include tissue-specific gene function prediction, Hi-C network denoising for TAD detection, and fine-grained species identification using species similarity networks.
  • Experimental design: The downstream scenarios cover tissue-specific gene function prediction, Hi-C network denoising for TAD detection, and fine-grained species identification from species similarity networks.These biologically motivated tasks depend on network fidelity because edges encode meaningful information.
  • Baselines: Real-world denoising experiments compare Ricci-Diffusion with Network Deconvolution (ND), Network Enhancement (NE), and Network Refinement (NR).These are representative diffusion-based methods.
  • Baselines: Synthetic experiments additionally include BORF and GSR as baselines.These baselines are used in addition to the real-world comparison methods.
  • Implementation: The method reports dynamic RD-Dyn and static RD-Sta iteration strategies when computationally feasible.RD-Dyn uses a dynamic-to-static strategy, whereas RD-Sta operates without the dynamic stage.
  • Experimental details: Dataset construction, parameter settings, and evaluation protocols are provided in Appendix II.The passage identifies Appendix II as the location of these experimental details.

A. Gene Function Prediction on Tissue Networks

Ricci-Diffusion denoising is evaluated on 16 tissue-specific gene interaction networks using AUROC for gene function prediction. RD-Sta improves AUROC over raw networks across all tissues, with pronounced gains in blood platelet and b lymphocyte networks.

  • Evaluation: The evaluation uses the standard gene-function prediction protocol on tissue-specific gene interaction networks, with denoised networks assessed by AUROC.The benchmark covers 16 tissue-specific networks.
  • Results: RD-Sta consistently improves AUROC over raw networks across all 16 tissues.This indicates enhanced recovery of biologically meaningful signals across diverse tissue contexts.
  • Results: 0.759 vs. 0.570: RD-Sta achieves this AUROC gain on the blood platelet network.The passage identifies blood platelet as showing a particularly pronounced improvement.
  • Results: 0.776 vs. 0.606: RD-Sta achieves this AUROC gain on the b lymphocyte network.The passage identifies b lymphocyte as showing a particularly pronounced improvement.

B. Hi-C Network Denoising for TAD Detection

On GM12878 Hi-C contact networks at 1 k and 5 k resolutions, Ricci-Diffusion improves TAD detection measured by NMI. RD-Sta and RD-Dyn outperform NR and ND, while RD-Dyn is especially effective at the sparser, noisier 5 k resolution and surpasses RD-Sta across both resolutions.

  • Evaluation: The evaluation uses GM12878 Hi-C contact networks across all autosomes at 1 k and 5 k resolutions, with TAD detection assessed by NMI.Louvain community detection is applied to raw and denoised networks, and higher NMI indicates better agreement with reference domain structures.
  • Results: RD-Sta and RD-Dyn consistently outperform NR and ND at both resolutions, while both RD variants show larger gains over NE at 5 k.RD-Dyn is slightly higher than NE at 1 k resolution, with stronger relative improvement at 5 k.
  • Results: RD-Dyn outperforms RD-Sta across both resolutions, indicating that adapting the diffusion kernel to evolving graph geometry better captures the network backbone and improves TAD detection.The resolution-dependent gains suggest Ricci-Diffusion is more effective when Hi-C networks are sparser and more affected by noise.

C. Fine-Grained Species Identification

On the Leeds Butterfly dataset, curvature-guided Ricci-Diffusion improves fine-grained species retrieval accuracy over the evaluated baselines. RD-Dyn also produces cleaner species-specific communities by suppressing spurious inter-species connections.

  • Evaluation setup: The Leeds Butterfly evaluation uses 832 images from 10 species and measures accuracy as the proportion of same-species images among top-ranked neighbors.The task evaluates image retrieval for fine-grained species identification.
  • Retrieval accuracy: 0.817 mean retrieval accuracy is achieved by RD-Dyn, followed by RD-Sta at 0.783, exceeding NE (0.759), NR (0.546), Raw (0.517), and ND (0.476).These gains remain stable across different numbers of retrieved neighbors.
  • Structural visualization: RD-Dyn yields compact species-specific clusters and strongly suppressed inter-community connections compared with Raw, NR, and NE.The visualization indicates cleaner community separation and structural sharpening consistent with Ricci-flow-like behavior.

D. Geometric Validation of Ricci-Flow-Like Behavior

On the Leeds Butterfly dataset, Ricci-Diffusion exhibits Ricci-flow-like behavior: curvature is coupled to edge-weight evolution, while diffusion progressively concentrates curvature distributions. More negatively curved edges are suppressed more strongly, and RD-Dyn produces the most concentrated final distribution among compared denoising methods.

  • Curvature–weight coupling: Relative edge-weight updates remain negatively correlated with Ollivier–Ricci curvature throughout diffusion.This indicates curvature-guided coupling between edge-weight evolution and local geometry.
  • Curvature–weight coupling: More negatively curved edges undergo stronger weight suppression, whereas positively curved edges are more likely to be preserved.The observed update pattern links curvature sign to heterogeneous transport and edge-weight change.
  • Curvature homogenization: Curvature distributions progressively concentrate as diffusion proceeds from initialization to convergence.The evolution is reported on the Leeds Butterfly dataset as geometric evidence of Ricci-flow-like behavior.
  • Curvature homogenization: RD-Dyn yields the most concentrated final curvature distribution among the compared denoising methods.More concentrated final distributions correspond to clearer geometric structure in the reported comparison.

E. Ablation Study on Curvature Modulation

The ablation isolates curvature modulation from the diffusion backbone on the Butterfly dataset. Curvature improves denoising performance, with ORC outperforming random, degree-based, and no-curvature alternatives.

  • Ablation design: The ablation compares ORC, FRC, random signals, degree-based proxies, and no curvature (η = 0) while retaining the same diffusion backbone.Extended sensitivity analyses are provided in Appendix IV.
  • Curvature modulation: Removing curvature with η = 0 reduces performance for both RD-Dyn and RD-Sta, while curvature modulation improves over the pure diffusion backbone.The comparison isolates the effect of curvature modulation from the diffusion backbone.
  • Curvature modulation: Random and degree-based signals perform more weakly, while FRC improves in most cases and ORC gives the strongest results.The findings indicate that gains are not explained by arbitrary perturbations or simple structural proxies, and that transport-based curvature better captures local geometric heterogeneity.

F. Controlled Synthetic Denoising and Downstream Validation · V. CONCLUSION

Controlled synthetic experiments show that Ricci-Diffusion, especially RD-Dyn, substantially improves explicit denoising quality and can enhance downstream GCN classification in suitable regimes. The conclusion frames these gains as consequences of curvature-guided edge-weight evolution and progressive curvature homogenization across real-world and synthetic networks.

  • F. Controlled Synthetic Denoising and Downstream Validation: Controlled experiments on GN and LFR graphs with known community structure evaluate explicit denoising using inv-SNR and downstream GCN classification.Lower inv-SNR indicates better separation between intra- and inter-community edges.
  • F. Controlled Synthetic Denoising and Downstream Validation: RD-Dyn achieves the best explicit denoising results across all tested settings, while RD-Sta is usually the second-best method.The synthetic results use inv-SNR, for which lower values are better.
  • F. Controlled Synthetic Denoising and Downstream Validation: 0.170 inv-SNR: RD-Dyn improves GN (0.25/0.02) with ρ = 0.2 from 0.382 on Raw, outperforming BORF (0.492) and GSR (0.366).The reported metric is inv-SNR, where lower values indicate better denoising.
  • F. Controlled Synthetic Denoising and Downstream Validation: 0.017 inv-SNR: RD-Dyn on LFR with µ = 0.2 outperforms BORF (0.419) and GSR (0.304), reducing the metric to 0.017.This result further supports the strong explicit denoising performance of RD-Dyn on controlled synthetic graphs.
  • F. Controlled Synthetic Denoising and Downstream Validation: 0.763 accuracy: RD-Dyn on GN (0.25/0.02) with ρ = 0.2 outperforms Raw (0.620) and GSR (0.757) in downstream GCN classification.The passage notes that GSR achieves the strongest overall classification performance because of its learning-oriented design.
  • F. Controlled Synthetic Denoising and Downstream Validation: 0.934 accuracy: RD-Sta on LFR with µ = 0.1 exceeds Raw (0.850) and GSR (0.896), showing downstream benefits in a suitable regime.The results are reported for GCN node classification on refined graphs.
  • V. CONCLUSION: Ricci-Diffusion revisits network denoising from a geometric perspective through a Ricci-flow-inspired, curvature-guided graph diffusion method.The conclusion presents this as the paper’s central methodological contribution.
  • V. CONCLUSION: Across real-world and controlled synthetic settings, Ricci-Diffusion improves explicit denoising quality and often benefits downstream tasks through curvature-guided edge-weight evolution and progressive curvature homogenization.These behaviors provide geometric insight into how curvature-aware diffusion yields more coherent and structured networks.
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