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Evolution of weights on a connected finite graph

Jicheng Ma, Yunyan Yang

TL;DR

This work introduces a generalized edge-weight evolution on connected finite graphs, where each edge weight changes according to the difference between a Wasserstein distance between chosen probability measures and the graph distance. By proving local Lipschitz properties and applying standard ODE theory, the authors establish global existence and uniqueness for both the continuous flow and a quasi-normalized variant, without requiring surgery. They translate the continuous model into discrete algorithms using $\alpha$-lazy one-step and two-step random walks for community detection, and demonstrate strong performance on real networks with stability across parameter choices. The approach yields high modularity and robust clustering on large-scale graphs, is simple to implement, and is supported by publicly available code.

Abstract

On a connected finite graph, we propose an evolution of weights including Ollivier's Ricci flow as a special case. During the evolution process, on each edge, the speed of change of weight is exactly the difference between the Wasserstein distance related to two probability measures and certain graph distance. Here the probability measure may be chosen as an $α$-lazy one-step random walk, an $α$-lazy two-step random walk, or a general probability measure. Based on the ODE theory, we show that the initial value problem has a unique global solution. A discrete version of the above evolution is applied to the problem of community detection. Our algorithm is based on such a discrete evolution, where probability measures are chosen as $α$-lazy one-step random walk and $α$-lazy two-step random walk respectively. Note that the later measure has not been used in previous works [2, 16, 21, 24]. Here, as in [21], only one surgery needs to be performed after the last iteration. Moreover, our algorithm is much easier than those of [2, 16, 21], which were all based on Lin-Lu-Yau's Ricci curvature. The code is available at https://github.com/mjc191812/Evolution-of-weights-on-a-connected-finite-graph.

Evolution of weights on a connected finite graph

TL;DR

This work introduces a generalized edge-weight evolution on connected finite graphs, where each edge weight changes according to the difference between a Wasserstein distance between chosen probability measures and the graph distance. By proving local Lipschitz properties and applying standard ODE theory, the authors establish global existence and uniqueness for both the continuous flow and a quasi-normalized variant, without requiring surgery. They translate the continuous model into discrete algorithms using -lazy one-step and two-step random walks for community detection, and demonstrate strong performance on real networks with stability across parameter choices. The approach yields high modularity and robust clustering on large-scale graphs, is simple to implement, and is supported by publicly available code.

Abstract

On a connected finite graph, we propose an evolution of weights including Ollivier's Ricci flow as a special case. During the evolution process, on each edge, the speed of change of weight is exactly the difference between the Wasserstein distance related to two probability measures and certain graph distance. Here the probability measure may be chosen as an -lazy one-step random walk, an -lazy two-step random walk, or a general probability measure. Based on the ODE theory, we show that the initial value problem has a unique global solution. A discrete version of the above evolution is applied to the problem of community detection. Our algorithm is based on such a discrete evolution, where probability measures are chosen as -lazy one-step random walk and -lazy two-step random walk respectively. Note that the later measure has not been used in previous works [2, 16, 21, 24]. Here, as in [21], only one surgery needs to be performed after the last iteration. Moreover, our algorithm is much easier than those of [2, 16, 21], which were all based on Lin-Lu-Yau's Ricci curvature. The code is available at https://github.com/mjc191812/Evolution-of-weights-on-a-connected-finite-graph.

Paper Structure

This paper contains 15 sections, 3 theorems, 79 equations, 22 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Notations and main results
  3. A key lemma
  4. Proof of Theorems \ref{['theorem1']} and \ref{['theorem2']}
  5. Examples
  6. Experimental results
  7. Discretization and algorithm design
  8. Real-world datasets and criteria
  9. Analysis of key parameters
  10. Impact of on experimental results
  11. Effect of iterations on experimental results
  12. -lazy one-step random walk
  13. -lazy two-step random walk
  14. Comparison with other methods
  15. Concluding remarks

Key Result

Theorem 2.1

Let $G=(V,E,\mathbf{w}_0)$ be a connected weighted finite graph, where $V$ is the vertex set, $E=\{e_1,e_2,\cdots,e_m\}$ is the edge set, and $\mathbf{w}_0= (w_{0,1},w_{0,2},\cdots,w_{0,m})\in\mathbb{R}^m_+$ is an arbitrary weight on $E$. If $\mu\in\mathscr{M}$ satisfies that for each vertex $x\in V

Figures (22)

  • Figure 1: A line segment
  • Figure 2: A path graph G with length 2
  • Figure 3: A triangle
  • Figure 4: A square
  • Figure 5: A complete graph with 4 vertices $K_4$
  • ...and 17 more figures

Theorems & Definitions (3)

  • Theorem 2.1
  • Theorem 2.2
  • Lemma 3.1