Coevolution of Opinions and Directed Adaptive Networks in a Social Group

In the interactions of a social group, people usually update and express their opinions through the observational learning behaviors. The formed directed networks are adaptive which are influenced by the evolution of opinions; while in turn modify the dynamic process of opinions. We extend the Hegselmann-Krause (HK) model to investigate the coevolution of opinions and observational networks (directed Erdös-Rényi network). Directed links can be broken with a probability if the difference of two opinions exceeds a certain confidence level ε, but new links can form randomly. Simulation results reveal that both the static networks and adaptive networks have three types: more than one cluster (fragmented) with small ε, consensus with a certain probability with moderate ε, always consensus with large ε. Also, on both networks, the tendencies of average of opinion clusters, consensus probability and average of convergence rounds are similar, and the fewest of average of opinion clusters satisfies the rough 1/(2 ε)-rule. On static networks, final opinions are influenced by percolation properties of networks; but on directed adaptive networks, it is basically determined by the rewiring probability, which increases the average degree of networks. When rewired probability is larger than zero, the results of adaptive networks are getting better than static networks. However, after the final average in- and out-degree of both networks exceeds a threshold, there is little improvement on the results.