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Consensus Over Activity-Driven Networks

The problem of self-coordination of a network of dynamical systems toward a common state is often referred to as the consensus problem. In view of its wide range of applications, the consensus problem has been extensively studied in the past few decades. However, most of the available results focus on static networks, challenging our mathematical understanding of coordination in temporal networks. In this article, we study discrete-time stochastic consensus over temporal networks, modeled as activity-driven networks. In this paradigm, each node has a specific tendency to create links in the network, measured through an activity potential. Differences in the activity potential of nodes favor the evolution of heterogeneous networks, in which some nodes are more involved in the process of information sharing than others. Through stochastic stability theory, we characterize the expected consensus state, which is found to be dominated by low-activity nodes. By further leveraging eigenvalue perturbation techniques, we derive a closed-form expression for the convergence rate in a mean-square sense, which points at a detrimental effect of moderate levels of heterogeneity for large networks. Simulations are conducted to support and illustrate our analytical findings.