  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.
