Consensus Based Distributed Spectral Radius Estimation

A consensus based distributed algorithm to compute the spectral radius of a network is proposed. The spectral radius of the graph is the largest eigenvalue of the adjacency matrix, and is a useful characterization of the network graph. Conventionally, centralized methods are used to compute the spectral radius, which involves eigenvalue decomposition of the adjacency matrix of the underlying graph. Our distributed algorithm uses a simple update rule to reach consensus on the spectral radius, using only local communications. We consider time-varying graphs to model packet loss and imperfect transmissions, and provide the convergence characteristics of our algorithm, for both static and time-varying graphs. We prove that the convergence error is a function of principal eigenvector of adjacency matrix of the graph and reduces as $\mathcal {O}(1/t)$, where $t$ is the number of iterations. The algorithm works for any connected graph structure. Simulation results supporting the theory are also presented.