Communicability Betweenness in Complex Networks

We implement the communicability betweenness in complex network as presented in the following paper

http://arxiv.org/abs/0905.4102

We have networkx in python for the implementations.

communicability in complex networkz

1. communicability in complex networks

Many topological and dynamical properties of complex networks are
defined by assuming that most of the transport on the network flows along
the shortest paths. However, there are different scenarios in which non-
shortest paths are used to reach the network destination. Thus the
consideration of the shortest paths only does not account for the global
communicability of a complex network. Here we propose a new measure of
the communicability of a complex network, which is a broad generalization
of the concept of the shortest path. According to the new measure, most of
real-world networks display the largest communicability between the most
connected (popular) nodes of the network (assortative communicability).
There are also several networks with the disassortative communicability,
where the most “popular” nodes communicate very poorly to each other.
Using this information we classify a diverse set of real-world complex
systems into a small number of universality classes based on their structure-
dynamic correlation. In addition, the new communicability measure is able
to distinguish finer structures of networks, such as communities into which a
network is divided. A community is unambiguously defined here as a set of
nodes displaying larger communicability among them than to the rest of
nodes in the network.

The communicability is compute using the formula in this paper

http://arxiv.org/abs/0707.0756v1

Method of identifying networks communities based on the Green's function of the network

Community identification has been an active area of research in complex 
networks. Many topological and dynamical properties of complex networks  are
defined by assuming that most of the transport on the network flows  along the
shortest  paths http://en.wikipedia.org/wiki/Centrality.  However there are
scenarios in which non shortest path are used to reach  network destination.
Thus the consideration of the shortest paths only  does not account for the
global communicability of complex network. New measures of the communicability
of a complex networks has proposed in 
http://www.ncbi.nlm.nih.gov/pubmed/18517465. Communicability is then view as the
Green function of the networks. The correlation  between the node degree and the
communicability (Green's function) is used in order to investigate the
structure-dynamic relationship in complex network. It has been shown in 
http://www.ncbi.nlm.nih.gov/pubmed/18517465 how communicability, or  Green
function can be used to identify network communities. The method  use the
spectral decomposition of the Green function.