Network analysis based on statistical-thermodynamics formalism

Abstract

Random walk on a real social networking service consisting of 2271 nodes is analyzed on the basis of the statistical-thermodynamics formalism to find phase transitions in network structure. Each phase can be related to a characteristic local structure of the network such as a cluster or a hub. For this purpose, the generalized transition matrix is introduced, whose largest eigenvalue yields statistical structure functions. The weighted visiting frequency related to the Gibbs probability measure, which is useful for extracting characteristic local structures, is obtained from the products of the right and left eigenvectors corresponding to the largest eigenvalue. An algorithm to extract the characteristic local structure of each phase is also suggested on the basis of this weighted visiting frequency.