  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.
