  We examine two-layer networks and centrality measures defined on them. The
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 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.
