Resistance-Based Performance Analysis of the Consensus Algorithm over Geometric Graphs

The performance of the linear consensus algorithm is studied by using a Linear Quadratic (LQ) cost. The objective is to understand how the communication topology influences this algorithm. This is achieved by exploiting an analogy between Markov Chains and electrical resistive networks. Indeed, this permits to uncover the relation between the LQ performance cost and the average effective resistance of a suitable electrical network and, moreover, to show that, if the communication graph fulfils some local properties, then its behavior can be approximated by that of a grid, which is a graph whose associated LQ cost is well-known.