Vector-Valued Gossip over w-Holonomic Networks

We study the weighted average consensus problem for a gossip network of agents with vector-valued states. For a given matrix-weighted graph, the gossip process is described by a sequence of pairs of adjacent agents communicating and updating their states based on the edge matrix weight. Our key contribution is providing conditions for the convergence of this non-homogeneous Markov process as well as the characterization of its limit set. To this end, we introduce the notion of “w-holonomy” of a set of stochastic matrices, which enables the characterization of sequences of gossiping pairs resulting in reaching a desired consensus in a decentralized manner. Stated otherwise, our result characterizes the limiting behavior of infinite products of (non-commuting, possibly with absorbing states) stochastic matrices.

Recommended citation: Vector-Valued Gossip over $w$-Holonomic Networks E. Bayram, M.-A. Belabbas, T. Başar - arXiv preprint arXiv:2311.04455, 2023 https://arxiv.org/abs/2311.04455