# Opinion Dynamics with Decaying Confidence: Application to Community Detection in Graphs

## TL;DR

This paper introduces a decaying confidence opinion dynamics model that captures community formation in networks and proposes a novel eigenvalue-based approach for community detection, demonstrating effectiveness through examples.

## Contribution

It presents a new opinion dynamics framework with decaying confidence and applies it to develop an eigenvalue-based community detection method in graphs.

## Key findings

- Communities correspond to asymptotically connected components.
- Eigenvalues of the dynamics matrix characterize community structure.
- The proposed method effectively detects communities in example networks.

## Abstract

We study a class of discrete-time multi-agent systems modelling opinion dynamics with decaying confidence. We consider a network of agents where each agent has an opinion. At each time step, the agents exchange their opinion with their neighbors and update it by taking into account only the opinions that differ from their own less than some confidence bound. This confidence bound is decaying: an agent gives repetitively confidence only to its neighbors that approach sufficiently fast its opinion. Essentially, the agents try to reach an agreement with the constraint that it has to be approached no slower than a prescribed convergence rate. Under that constraint, global consensus may not be achieved and only local agreements may be reached. The agents reaching a local agreement form communities inside the network. In this paper, we analyze this opinion dynamics model: we show that communities correspond to asymptotically connected component of the network and give an algebraic characterization of communities in terms of eigenvalues of the matrix defining the collective dynamics. Finally, we apply our opinion dynamics model to address the problem of community detection in graphs. We propose a new formulation of the community detection problem based on eigenvalues of normalized Laplacian matrix of graphs and show that this problem can be solved using our opinion dynamics model. We consider three examples of networks, and compare the communities we detect with those obtained by existing algorithms based on modularity optimization. We show that our opinion dynamics model not only provides an appealing approach to community detection but that it is also effective.