The robustness of democratic consensus

TL;DR

This paper investigates the concept of democratic consensus in large-scale linear opinion dynamics models, where individual influence diminishes as the number of agents grows, highlighting its relevance in networked systems.

Contribution

It introduces the notion of democracy in consensus models and analyzes its properties and implications in large-scale opinion dynamics.

Findings

Democratic consensus occurs when influence weights vanish as agents increase.

The property is relevant for understanding opinion formation in large networks.

Democracy relaxes the requirement of equal influence in average consensus.

Abstract

In linear models of consensus dynamics, the state of the various agents converges to a value which is a convex combination of the agents' initial states. We call it democratic if in the large scale limit (number of agents going to infinity) the vector of convex weights converges to 0 uniformly. Democracy is a relevant property which naturally shows up when we deal with opinion dynamic models and cooperative algorithms such as consensus over a network: it says that each agent's measure/opinion is going to play a negligeable role in the asymptotic behavior of the global system. It can be seen as a relaxation of average consensus, where all agents have exactly the same weight in the final value, which becomes negligible for a large number of agents.