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arxiv: 2604.08144 · v1 · submitted 2026-04-09 · 🧮 math.CA · math.ST· stat.TH

An Efficient Entropy Flow on Weighted Graphs: Theory and Applications

Juan Zhao , Jicheng Ma , Yunyan Yang , Liang Zhao This is my paper

Pith reviewed 2026-05-10 17:26 UTC · model grok-4.3

classification 🧮 math.CA math.STstat.TH
keywords entropy flowweighted graphscommunity detectiondiscrete Ricci flowgraph geometryprobability evolutionconvergence
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The pith

Entropy flow evolves probabilities on weighted graphs to match Ricci flow geometry for community detection but runs in 1.61 to 3.20 percent of the time.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines a new entropy flow that tracks how probability distributions change across the edges and vertices of a weighted graph. This flow is constructed to share the geometric intuition of discrete Ricci flow while using only local information, avoiding any need to compute optimal transport distances or shortest paths. The authors give a rigorous mathematical formulation, prove that solutions exist for all time and converge to equilibrium, and apply the flow to the task of identifying communities in real networks. Experiments show detection accuracy comparable to Ricci flow methods, yet with dramatically lower runtime because the expensive global calculations are eliminated. This matters for any analysis that must scale to large graphs where previous curvature-based flows become impractical.

Core claim

The central claim is that entropy flow can be formulated directly on weighted graphs as an evolution of probability measures, that its solutions exist globally in time and converge, and that the resulting process detects communities with accuracy equivalent to discrete Ricci flow while requiring only 1.61 percent to 3.20 percent of the computation time by bypassing optimal transport distances and shortest-path searches.

What carries the argument

The entropy flow itself: a local differential evolution rule for probability distributions on the weighted graph that encodes geometric change without global distance computations.

If this is right

  • Community detection becomes feasible on graphs too large for optimal-transport methods.
  • The convergence proof guarantees that the flow reaches a stable state usable for downstream tasks.
  • Other graph-analysis problems that rely on geometric intuition can adopt the same local formulation to gain speed.
  • The framework supplies a new, provably convergent tool for studying probability evolution on discrete structures.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same local flow could be discretized in different ways to handle directed or signed edges.
  • Entropy flow may serve as a computationally cheap surrogate for curvature in theoretical studies of graph limits.
  • Testing the flow on synthetic graphs with planted communities would give a direct check on whether accuracy holds when ground truth is known.

Load-bearing premise

A purely local definition of entropy flow on weighted graphs can still capture the geometric properties required for community detection.

What would settle it

On standard benchmark networks, entropy flow community partitions differ substantially in accuracy from those produced by discrete Ricci flow, or the numerical solutions fail to converge within the time interval predicted by the existence proof.

Figures

Figures reproduced from arXiv: 2604.08144 by Jicheng Ma, Juan Zhao, Liang Zhao, Yunyan Yang.

Figure 1
Figure 1. Figure 1: community detection 4. Proof of the main theorem In this section, we will prove long time existence of the entropy flow (2.4) and the conver￾gence of the weights along the entropy flow. This is based on the ODE theory. We will also provide the proof of Corollary 2.2. Proof of Theorem 2.1. We divide the proof into several steps. Step 1. There holds DKL(R α x , R α y ) ≥ 0 for all x, y ∈ V and all α ∈ (0, 1)… view at source ↗
Figure 2
Figure 2. Figure 2: Effect of iterations on three real-world datasets 5.4. Analysis of edge entropy and weight distributions We investigate how the entropy flow algorithm affects the distributions of edge entropy and edge weights [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Distribution changes of entropy values and edge weights [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Effect of edge weight cutoff on community structure 5.6. Comparison to other methods We compare our community detection method with three classical approaches: the Girvan￾Newman algorithm based on edge betweenness [9], the greedy modularity maximization algo￾rithm [4, 24], and the label propagation algorithm [5]. In addition, five Ricci curvature-based methods are considered, including unnormalized discret… view at source ↗
read the original abstract

We propose a novel entropy flow on weighted graphs, which provides a principled framework that characterizes the evolution of probability distributions over graph structures while sharing geometric intuition with discrete Ricci flow. We provide its rigorous formulation, establish its fundamental theoretical properties, and prove the long-time existence and convergence of its solutions. To demonstrate its applicability, we employ entropy flow for community detection in real-world networks. Empirically, it achieves detection accuracy fully comparable to that of discrete Ricci flow. Crucially, by avoiding computations of optimal transport distances and shortest paths, our approach overcomes the fundamental computational bottleneck of Ollivier and Lin-Lu-Yau Ricci flows. As a result, entropy flow requires only $1.61\%$-$3.20\%$ of the computation time of Ricci flow. These results indicate that entropy flow provides a theoretically rigorous and computationally efficient framework for large-scale graph analysis.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper proposes a novel entropy flow on weighted graphs that characterizes the evolution of probability distributions over graph structures while sharing geometric intuition with discrete Ricci flow. It claims to provide a rigorous formulation, establish fundamental theoretical properties, prove long-time existence and convergence of solutions, and apply the flow to community detection on real-world networks, achieving accuracy comparable to discrete Ricci flow while requiring only 1.61%-3.20% of the computation time by avoiding optimal transport distances and shortest-path computations.

Significance. If the theoretical claims and empirical results hold, the work offers a computationally tractable local alternative to Ollivier and Lin-Lu-Yau Ricci flows for large-scale graph analysis. The avoidance of expensive global computations while retaining utility for community detection could enable broader applications in network science, provided the flow preserves the necessary geometric properties.

major comments (2)
  1. [Abstract] The abstract asserts a 'rigorous formulation' and 'proofs of long-time existence and convergence' for the entropy flow, but the explicit discrete scheme, curvature interpretation, and derivation of the evolution equation are not supplied. Without these (presumably in §2–3), it is impossible to verify whether the local definition indeed preserves the geometric properties required for the community-detection claims to hold without implicit reliance on transport or path computations.
  2. [Abstract (empirical claims)] The empirical claim that entropy flow achieves 'fully comparable' detection accuracy to discrete Ricci flow while using 1.61%–3.20% of the time rests on unspecified datasets, baseline implementations, error analysis, and timing protocols. These details are load-bearing for the central efficiency-and-parity assertion and must be provided with reproducible code or tables to allow verification.
minor comments (2)
  1. Notation for the entropy functional and its discrete gradient should be introduced with a clear comparison to the corresponding Ricci curvature terms to aid readers familiar with the literature.
  2. [Abstract] The abstract would benefit from a one-sentence statement of the precise update rule or evolution equation to convey the local character of the flow immediately.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We are grateful to the referee for their thorough review and valuable feedback on our paper 'An Efficient Entropy Flow on Weighted Graphs: Theory and Applications'. Their comments highlight important aspects for improving the clarity and reproducibility of our work. We address each major comment in detail below and outline the revisions we plan to make.

read point-by-point responses

  1. Referee: [Abstract] The abstract asserts a 'rigorous formulation' and 'proofs of long-time existence and convergence' for the entropy flow, but the explicit discrete scheme, curvature interpretation, and derivation of the evolution equation are not supplied. Without these (presumably in §2–3), it is impossible to verify whether the local definition indeed preserves the geometric properties required for the community-detection claims to hold without implicit reliance on transport or path computations.

    Authors: The manuscript does provide these elements in the main body. Specifically, Section 2 presents the rigorous formulation of the entropy flow on weighted graphs, including the explicit discrete scheme based on the entropy gradient. The curvature interpretation is discussed by connecting it to the discrete Ricci curvature concepts, and the derivation of the evolution equation is detailed in Section 3, demonstrating its local nature without dependence on optimal transport or shortest-path computations. The proofs of long-time existence and convergence are established in Section 4. We believe this addresses the geometric properties preservation locally. To assist the reader, we will revise the introduction to include a brief overview of the key equations and their derivations, making the connection more explicit. revision: partial

  2. Referee: [Abstract (empirical claims)] The empirical claim that entropy flow achieves 'fully comparable' detection accuracy to discrete Ricci flow while using 1.61%–3.20% of the time rests on unspecified datasets, baseline implementations, error analysis, and timing protocols. These details are load-bearing for the central efficiency-and-parity assertion and must be provided with reproducible code or tables to allow verification.

    Authors: We acknowledge that the abstract is concise and does not detail the experimental setup. The full manuscript includes descriptions of the datasets, which are standard real-world networks, and comparisons with discrete Ricci flow. However, to enhance reproducibility, we will add a dedicated subsection in the experiments with detailed tables reporting the datasets, accuracy results with standard deviations, timing measurements on specified hardware, and implementation details for baselines. We will also release the source code for the entropy flow algorithm and community detection experiments upon publication, allowing full verification of the reported 1.61%-3.20% computation time savings while maintaining comparable accuracy. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper introduces an entropy flow via an independent local definition on weighted graphs, then establishes its formulation, proves existence and convergence as first-principles results, and validates community detection performance through external empirical comparison to Ricci flow. No load-bearing step reduces by construction to fitted inputs, self-citations, or renamed known results; the efficiency gain is shown by explicit avoidance of OT and shortest-path steps rather than by definitional equivalence. The derivation chain remains self-contained against the stated benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Review performed on abstract only; no explicit free parameters, ad-hoc axioms, or invented entities beyond the new flow itself can be extracted. The framework rests on standard domain assumptions of weighted graphs and probability measures.

axioms (1)
  • domain assumption Weighted graphs admit probability distributions that can evolve under a continuous-time flow.
    Implicit in the statement that the flow characterizes evolution of distributions on graph structures.
invented entities (1)
  • Entropy flow no independent evidence
    purpose: To evolve probability distributions on weighted graphs while retaining geometric intuition analogous to discrete Ricci flow.
    Newly proposed object whose definition and properties are the central contribution.

pith-pipeline@v0.9.0 · 5450 in / 1273 out tokens · 65857 ms · 2026-05-10T17:26:11.026963+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

27 extracted references · 27 canonical work pages

  1. [1]

    Anand, G

    K. Anand, G. Bianconi, Entropy measures for networks: Toward an information theory of complex topologies, Phys. Rev. E 80 (4) (2009) 045102

    work page 2009

  2. [2]

    S. Bai, B. Hua, Y . Lin, L. Shuang, On the Ricci flow on trees, arXiv: 2509.22140

    work page arXiv

  3. [3]

    S. Bai, Y . Lin, L. Lu, Z. Wang, S.-T. Yau, Ollivier Ricci-flow on weighted graphs, Am. J. Math. 146 (6) (2024) 1723–1747

    work page 2024

  4. [4]

    Clauset, M

    A. Clauset, M. E. J. Newman, C. Moore, Finding community structure in very large net- works, Phys. Rev. E 70 (2004) 066111

    work page 2004

  5. [5]

    Cordasco, G

    G. Cordasco, G. Luisa, Community detection via semi-synchronous label propagation algo- rithms, IEEE International Workshop on: Business Applications of Social Network Analy- sis (BASNA) (2010) 1–8

    work page 2010

  6. [6]

    T. M. Cover, J. A. Thomas, Elements of Information Theory, Wiley, 2nd edn., 2005

    work page 2005

  7. [7]

    Danon, A

    L. Danon, A. Díaz-Guilera, J. Duch, A. Arenas, Comparing community structure identifi- cation, J. Stat. Mech. Theory Exp. 2005 (09) (2005) P09008

    work page 2005

  8. [8]

    Dehmer, Information processing in complex networks: Graph entropy and information functionals, Appl

    M. Dehmer, Information processing in complex networks: Graph entropy and information functionals, Appl. Math. Comput. 201 (2012) 82–94

    work page 2012

  9. [9]

    Girvan, M

    M. Girvan, M. E. Newman, Community structure in social and biological networks, Proc. Natl. Acad. Sci. USA 99 (12) (2002) 7821–7826

    work page 2002

  10. [10]

    R. S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differ. Geom. 17 (2) (1982) 255–306

    work page 1982

  11. [11]

    Hubert, P

    L. Hubert, P. Arabie, Comparing partitions, J. Classification 2 (1985) 193–218

    work page 1985

  12. [12]

    Kullback, R

    S. Kullback, R. A. Leibler, On information and sufficiency, Ann. Math. Statist. 22 (1951) 79–86

    work page 1951

  13. [13]

    X. Lai, S. Bai, Y . Lin, Normalized discrete Ricci flow used in community detection, Phys. A Stat. Mech. Appl. 597 (2022) 127251

    work page 2022

  14. [14]

    Leskovec, SNAP datasets: Stanford large network dataset collection, http://snap.stanford.edu/data

    J. Leskovec, SNAP datasets: Stanford large network dataset collection, http://snap.stanford.edu/data . 19

    work page

  15. [15]

    R. Li, F. Münch, The convergence and uniqueness of a discrete-time nonlinear Markov chain, J. Funct. Anal. 290 (9) (2026) 111367

    work page 2026

  16. [16]

    Y . Lin, L. Lu, S.-T. Yau, Ricci curvature of graphs, Tohoku Math. J. 63 (4) (2011) 605–627

    work page 2011

  17. [17]

    J. Ma, Y . Yang, Evolution of weights on a connected finite graph, arXiv: 2411.06393

    work page arXiv

  18. [18]

    J. Ma, Y . Yang, A modified Ricci flow on arbitrary weighted graph, J. Geom. Anal. 35 (2025) 332

    work page 2025

  19. [19]

    J. Ma, Y . Yang, Piecewise-linear Ricci curvature flows on weighted graphs, arXiv: 2505.15395

    work page arXiv

  20. [20]

    Newman, Networks, Oxford Univ

    M. Newman, Networks, Oxford Univ. Press, 2018

    work page 2018

  21. [21]

    C.-C. Ni, Y . Lin, F. Luo, J. Gao, Community detection on networks with Ricci flow, Sci. Rep. 9 (1) (2019) 9984

    work page 2019

  22. [22]

    Ni, Y .-Y

    C.-C. Ni, Y .-Y . Lin, J. Gao, X. Gu, Network alignment by discrete Ollivier-Ricci flow, in: Graph drawing and network visualization, vol. 11282 ofLecture Notes in Comput. Sci., Springer, Cham, 447–462, 2018

    work page 2018

  23. [23]

    Ollivier, Ricci curvature of Markov chains on metric spaces, J

    Y . Ollivier, Ricci curvature of Markov chains on metric spaces, J. Funct. Anal. 256 (3) (2009) 810–864

    work page 2009

  24. [24]

    Reichardt, S

    J. Reichardt, S. Bornholdt, Statistical mechanics of community detection, Phys. Rev. E 74 (2006) 016110

    work page 2006

  25. [25]

    Sandhu, T

    R. Sandhu, T. Georgiou, E. Reznik, L. Zhu, I. Kolesov, Y . Senbabaoglu, T. Allen, Graph Curvature for Differentiating Cancer Networks, Sci. Rep. 5 (2015) 12323

    work page 2015

  26. [26]

    C. E. Shannon, A mathematical theory of communication, Bell System Tech. J. 27 (1948) 379–423, 623–656

    work page 1948

  27. [27]

    W. W. Zachary, An information flow model for conflict and fission in small groups, Journal of Anthropological Research 33 (4) (1977) 452–473. 20

    work page 1977