The Role of Persistent Graphs in the Agreement Seeking of Social Networks

TL;DR

This paper explores how persistent arcs in social network graphs influence the convergence to a common belief, establishing conditions under which persistent communication links guarantee agreement.

Contribution

It introduces the concept of persistent graphs with necessary and sufficient conditions for agreement, linking convergence to the structure of persistent arcs.

Findings

Persistent arcs determine convergence to consensus.

Convergence rates depend on the persistent graph's diameter.

Only persistent arcs contribute to global agreement.

Abstract

This paper investigates the role persistent arcs play for a social network to reach a global belief agreement under discrete-time or continuous-time evolution. Each (directed) arc in the underlying communication graph is assumed to be associated with a time-dependent weight function which describes the strength of the information flow from one node to another. An arc is said to be persistent if its weight function has infinite $\mathscr{L}_1$ or $\ell_1$ norm for continuous-time or discrete-time belief evolutions, respectively. The graph that consists of all persistent arcs is called the persistent graph of the underlying network. Three necessary and sufficient conditions on agreement or $\epsilon$-agreement are established, by which we prove that the persistent graph fully determines the convergence to a common opinion in social networks. It is shown how the convergence rates explicitly depend on the diameter of the persistent graph. The results adds to the understanding of the fundamentals behind global agreements, as it is only persistent arcs that contribute to the convergence.