Random walks on simple graphs in connection with electrical resistor networks lead to the definition of Markov chains with transition probability matrix in terms of electrical conductances. We extend this definition to an effective transition matrix $P_{ij}$ to account for the probability of going from vertex $i$ to any vertex $j$ of the original connected graph $G$. Also, we present an algorithm based on the definition of this effective transition matrix among vertices in the network to extract a topological feature related to the manner by which graph $G$ has been organized. This topological feature corresponds to the communities in the graph.

• Received 28 February 2007

DOI:https://doi.org/10.1103/PhysRevE.76.036101