Randomized Consensus with Attractive and Repulsive Links

TL;DR

This paper analyzes a randomized consensus algorithm on graphs with both attractive and repulsive links, establishing conditions for convergence or divergence based on the strength of repulsive interactions.

Contribution

It introduces a model incorporating both attractive and repulsive links, providing probabilistic convergence/divergence conditions and threshold values for stability.

Findings

A threshold for repulsive link strength determines convergence or divergence.

A single repulsive link can significantly alter consensus behavior.

Robustness depends on graph size and structure.

Abstract

We study convergence properties of a randomized consensus algorithm over a graph with both attractive and repulsive links. At each time instant, a node is randomly selected to interact with a random neighbor. Depending on if the link between the two nodes belongs to a given subgraph of attractive or repulsive links, the node update follows a standard attractive weighted average or a repulsive weighted average, respectively. The repulsive update has the opposite sign of the standard consensus update. In this way, it counteracts the consensus formation and can be seen as a model of link faults or malicious attacks in a communication network, or the impact of trust and antagonism in a social network. Various probabilistic convergence and divergence conditions are established. A threshold condition for the strength of the repulsive action is given for convergence in expectation: when the repulsive weight crosses this threshold value, the algorithm transits from convergence to divergence. An explicit value of the threshold is derived for classes of attractive and repulsive graphs. The results show that a single repulsive link can sometimes drastically change the behavior of the consensus algorithm. They also explicitly show how the robustness of the consensus algorithm depends on the size and other properties of the graphs.