Communicability Graph and Community Structures in Complex Networks

We use the concept of network communicability (Phys. Rev. E 77, 036111 (2008)) to define communities in a complex network. The communities are defined as the cliques of a “communicability graph”. The communicability graph is defined here as the graph having the same set of vertices as the complex network and the connections between them are determined by the communicability function. Two nodes are connected in this graph if their internal communicability is larger than the external communicability with the rest of nodes. We define that the connected nodes in the communicability graph are members of the same network community. Then, the problem of finding network communities is expressed in terms of the all-cliques problem. In addition, we extend here the concept of communicability to account for the strength of the interactions between the nodes. This “strength parameter” is identified as the inverse temperature of the network and can account for the spring constant of a spring network, the conductivity of a resister network or the bandwith of a telephone network. Furthermore, it can represent the level of stress which the node interactions receive, like the level of social agitation in a social network.