Super Ricci flows for weighted graphs

TL;DR

This paper introduces a new concept of super Ricci flows for finite weighted graphs that can handle singularities and structural changes, extending classical Ricci flow ideas to discrete settings.

Contribution

It defines a robust super Ricci flow framework for time-dependent weighted graphs, including singularities, and connects it with heat flow, Bochner inequalities, and optimal transport.

Findings

Flow can continue past singularities in weighted graphs

Equivalent characterizations via Bochner inequality and transport estimates

Consistency with classical Ricci flows in the continuum limit

Abstract

We present a notion of super Ricci flow for time-dependent finite weighted graphs. A challenging feature is that these flows typically encounter singularities where the underlying graph structure changes. Our notion is robust enough to allow the flow to continue past these singularities. As a crucial tool for this purpose we study the heat flow on such singular time-dependent weighted graphs with changing graph structure. We then give several equivalent characterizations of super Ricci flows in terms of a discrete dynamic Bochner inequality, gradient and transport estimates for the heat flow, and dynamic convexity of the entropy along discrete optimal transport paths. The latter property can be used to show that our notion of super Ricci flow is consistent with classical super Ricci flows for manifolds (or metric measure spaces) in a discrete to continuum limit.