Centrality measures and opinion dynamics in two-layer networks with replica nodes

We examine two-layer networks and centrality measures defined on them. We propose two fast and accurate algorithms to approximate the game-theoretic centrality measures and examine connection between centrality measures and characteristics of opinion dynamic processes on such networks. As an example, we consider a Zachary's karate club social network and extend it by adding the second (internal) layer of communication. Internal layer represents the idea that individuals can share their real opinions with their close friends. The structures of the external and internal layers may be different. As characteristics of opinion dynamic processes we mean consensus time and winning rate of a particular opinion. We find significantly strong positive correlation between internal graph density and consensus time, and significantly strong negative correlation between centrality of authoritative nodes and consensus time.