Consensus in random WSNs with correlated symmetric links

In this paper we study the convergence of consensus algorithms in wireless sensor networks with random topologies and instantaneous symmetric links which exist with different probabilities independently in time but correlated in space. Aiming at minimizing the convergence time of the algorithm, we adopt an optimization criterion based on the spectral radius of a positive semidefinite matrix for which we derive a closed-form expression, and propose a general formulation for the spatial correlation among the links that allows us to compute the optimum parameters ensuring almost sure convergence. We show that our general formulation can be particularized for existing protocols found in literature, and derive additionally closed-form expressions for the optimal parameters in two particular cases of equal probability of connection for all the links.