Evolving edge weights via local entropy flow and cohesion flow on graphs

Abstract: In this paper, we first propose two different quantities on graphs, namely local entropy and cohesion, then design two corresponding flows for edge weights: the local entropy flow and the cohesion flow. We establish the global existence and uniqueness of solutions for both flows and investigate their asymptotic behaviors, including the case that the limit goes to positive infinity. Moreover, they can be applied to fundamental network analysis tasks, including community detection and node classification. Empirical evaluations demonstrate that our method achieves performance competitive with Ollivier Ricci flow and Lin-Lu-Yau Ricci flow on benchmark network analysis tasks. In experimental scenarios, we first apply the cohesion flow to evolve the edge weights of the graph, and then apply the local entropy flow to further update the resulting weighted graph. Both flows are computationally efficient, leading to a significant reduction in overall computational cost and improved scalability.