Heat Calculus Approach to Queue Stability versus Ricci Curvature of Sensor Networks

In this paper, we develop a continuousgeometry/continuous-time model of a sensor network, which under a variant of the back-pressure protocol—the so-called diffusion protocol—sends data received from the various sensors to a central “sink.” This model is motivated by the necessity to understand how the wireless link graph topology affects network stability. Here network stability is viewed as the issue as to whether, for a given input data rate and a given queue length, none of the queues will overflow. In this continuous-geometry/continuous-time paradigm, the queue occupancy satisfies the heat equation over the manifold coarsely equivalent to the graph of the wireless links. It is shown that the maximum input data rate that can be tolerated without queue overflow is inversely proportional to some appropriately defined norm of the heat kernel of the Laplace-Beltrami operator. From heat calculus, it then follows that the norm of the heat kernel is larger for manifolds of negative Ricci curvature than for those of vanishing curvature. As a corollary and major result, networks of negative Ricci curvature are more difficult to stabilize. Finally, taking scheduling and interference into consideration leads to a hybrid model switching among many diffusion evolutions.