Consensus in Correlated Random Topologies: Weights for Finite Time Horizon

We consider the weight design problem for the consensus algorithm under a finite time horizon. We assume that the underlying network is random where the links fail at each iteration with certain probability and the link failures can be spatially correlated. We formulate a family of weight design criteria (objective functions) that minimize n, n = 1,...,N (out of N possible) largest (slowest) eigenvalues of the matrix that describes the mean squared consensus error dynamics. We show that the objective functions are convex; hence, globally optimal weights (with respect to the design criteria) can be efficiently obtained. Numerical examples on large scale, sparse random networks with spatially correlated link failures show that: 1) weights obtained according to our criteria lead to significantly faster convergence than the choices available in the literature; 2) different design criteria that corresponds to different n, exhibits very interesting tradeoffs: faster transient performance leads to slower long time run performance and vice versa. Thus, n is a valuable degree of freedom and can be appropriately selected for the given time horizon.