Agreement in presence of noise: pseudogradients on random geometric networks

We consider the agreement problem over realizations of a (Poisson) random geometric network with noisy interconnections. The vertices of random geometric networks are assumed to be uniformly distributed on the unit square; an edge exists between a pair of vertices if the distance between them is less than or equal to a given threshold. Our treatment of the agreement problem in such a setting relies upon notions from stochastic stability. In this venue, we show that the noisy agreement protocol has a guaranteed convergence with probability one, provided that an embedded step size parameter meets certain constraints. These constraints turn out to closely related to the spectra of the underlying graph Laplacian. Moreover, we point out the ramifications of having noisy networks by establishing connections between rate of convergence of the protocol and the range threshold in random geometric graphs.