Effect of degree correlations on the loop structure of scale-free networks

Ginestra Bianconi, Matteo Marsili

Published 2005-11-11, updated 2006-07-14Version 2

In this paper we study the impact of degree correlations in the subgraphs statistics of scale-free networks. In particular we consider loops: a simple case of network subgraphs which encode the redundancy of the paths passing through every two nodes of the network. We provide an understanding of the scaling of the clustering coefficient in modular networks in terms of the eigenvector of the correlation matrix associated with the maximal eigenvalue and we show that correlations affect in a relevant way the average number of Hamiltonian paths in a 3-core of real world networks. We prove our results in the two point correlated hidden variable ensemble and we check the results with exact counting of small loops in real graphs.