Wireless network capacity versus Ollivier-Ricci curvature under Heat-Diffusion (HD) protocol

Negative curvature in the relatively new Ollivier-Ricci sense of a wireless network graph is shown to be the culprit behind large queue occupancy, large routing energy, and restricted capacity region under any throughput optimal protocol. This is the wireless counterpart of the congestion phenomenon occurring in a Gromov negatively curved wired network under least cost path routing. Significantly different protocols call for significantly different curvature concepts to explain the “congestion” phenomenon. The rationale for the Ollivier-Ricci curvature is that it is defined in terms of a transportation cost—formalized by the Wasserstein distance—under a diffusion process. The Heat-Diffusion protocol used in this paper is driven by the queue differential, so that interpreting packets as calories, it lends itself to a genuine heat diffusion, yet retaining the 3-stage process of weighting-scheduling-forwarding of well-known Back-Pressure protocol. The main result is that the transportation definition of the Ollivier-Ricci curvature allows for the direct connection—without resorting to the Laplacian operator of heat calculus—between curvature and queue occupancy, routing energy, and capacity region.