Topology for Global Average Consensus

We consider the problem of global average-consensus in a network of sensors. The communication among sensors is restricted by the underlying network, where each sensor can exchange information only with its neighbors. We analyze the influence of the network topology on the convergence speed of the iterative consensus algorithm and focus on the design of network topology with respect to this optimality criterion. Topology optimization is a difficult combinatorial optimization problem. We reduce it to a spectral graph design problem, namely, maximizing the ratio gamma of two eigenvalues of the Laplacian matrix L of the graph. We consider a class of expander graphs, called Ramanujan graphs, for which we find a lower bound on gamma. Through numerical studies, we show that the consensus algorithm converges much faster on the Ramanujan graphs than on structured graphs, on Erdos-Renyi random graphs, and on graphs exhibiting small-world properties.