The bridges to consensus: Network effects in a bounded confidence opinion dynamics model

Abstract: In this work we present novel results to the problem of the Hegselmann-Krause dynamics in networks obtained by an extensive study of the behavior of the standard order parameter sensitive to the onset of consensus: the normalized size of the giant cluster. This order parameter reveals the non trivial effect of the network topology on the steady states of the dynamics, overlooked by previous works, which concentrated on the onset of unanimity, and allows to detect regions of polarization between the fragmented and the consensus phases. While the previous results on unanimity are confirmed, the consensus threshold shifts in the opposite direction compared to the threshold for unanimity. A detailed finite size scaling analysis shows that, in general, consensus is easier to obtain in networks than in mixed populations. At a difference with previous studies, we show that the network topology is relevant beyond the finitness of the average degree with increasing system size. In particular, in pure random networks (either uniform random graphs or scale free networks), the consensus threshold seems to vanish in the thermodynamic limit. A detailed analysis of the time evolution of the dynamics reveals the role of bridges in the network, which allow for the interaction between agents belonging to clusters of very different opinions, after several repeated interaction steps. These bridges are at the origin of the shift of the confidence threshold to lower values in networks as compared to lattices or the mixed population.