  In this paper, we introduce a new notion of curvature on the edges of a graph
that is defined in terms of effective resistances. We call this the
Ricci--Foster curvature. We study the Ricci flow resulting from this curvature.
We prove the existence of solutions to Ricci flow on short time intervals, and
prove that Ricci flow preserves graphs with nonnegative (resp. positive)
curvature.
