A Proposed Ollivier–Ricci–Bourguignon Type Flow

In the current work, we propose a discrete version of the Ricci–Bourguignon flow, based on the notion of Ollivier–Ricci curvature, to investigate the evolution of weighted graphs. This flow generalizes the Ollivier–Ricci flow in community detection problems. By deriving essential Lipschitz estimates for curvature operators and using standard ODE theory, we prove local results, including existence, uniqueness, continuous dependence, and a blow-up criterion for solutions to the proposed equation. In addition, we derive global-in-time results for some specific cases with idleness parameter larger than <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:math> . Precisely, considering a path graph of length 3 and the Wasserstein distance defined with the resistance mobility function, we obtain not only the global well-posedness of the solution but also its asymptotic behaviors. For a star graph with one center and the Wasserstein distance defined with the linear mobility function, we show that the proposed flow coincides with a normalized Ollivier–Ricci flow which preserves the total volume of the graph. Furthermore, we prove that under the action of the flow, the graph converges to a uniform star graph.