Ollivier Ricci-flow on weighted graphs

abstract: We study the existence of solutions of Ricci flow equations of Ollivier-Lin-Lu-Yau curvature defined on weighted graphs. Our work is motivated by the work of Ni-Lin-Luo-Gao in which the discrete time Ricci flow algorithm has been applied successfully as a discrete geometric approach in detecting communities. Our main result is the existence and uniqueness theorem for solutions to a continuous time normalized Ricci flow. We also display possible solutions to the Ricci flow on path graph and prove the Ricci flow on finite star graph with at least three leaves converges to constant-weighted star.