Statistical mechanics of community detection
Abstract
Starting from a general ansatz, we show how community detection can be interpreted as finding the ground state of an infinite range spin glass. Our approach applies to weighted and directed networks alike. It contains the ad hoc introduced quality function from [J. Reichardt and S. Bornholdt, Phys. Rev. Lett. 93, 218701 (2004)] and the modularity Q as defined by Newman and Girvan [Phys. Rev. E 69, 026113 (2004)] as special cases. The community structure of the network is interpreted as the spin configuration that minimizes the energy of the spin glass with the spin states being the community indices. We elucidate the properties of the ground state configuration to give a concise definition of communities as cohesive subgroups in networks that is adaptive to the specific class of network under study. Further, we show how hierarchies and overlap in the community structure can be detected. Computationally efficient local update rules for optimization procedures to find the ground state are given. We show how the ansatz may be used to discover the community around a given node without detecting all communities in the full network and we give benchmarks for the performance of this extension. Finally, we give expectation values for the modularity of random graphs, which can be used in the assessment of statistical significance of community structure.
 Publication:

Physical Review E
 Pub Date:
 July 2006
 DOI:
 10.1103/PhysRevE.74.016110
 arXiv:
 arXiv:condmat/0603718
 Bibcode:
 2006PhRvE..74a6110R
 Keywords:

 89.75.Hc;
 89.65.s;
 05.50.+q;
 64.60.Cn;
 Networks and genealogical trees;
 Social and economic systems;
 Lattice theory and statistics;
 Orderdisorder transformations;
 statistical mechanics of model systems;
 Condensed Matter  Disordered Systems and Neural Networks;
 Condensed Matter  Statistical Mechanics;
 Physics  Physics and Society
 EPrint:
 Phys. Rev. E 74 (2006) 016110