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

Abstract

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.

arXiv comment6 pages, 1 figure

References in corpus (14)

• Emergence of scaling in random networks
• Statistical mechanics of complex networks
• The structure and function of complex networks
• Assortative mixing in networks
• Hierarchical organization of modularity in metabolic networks
• Evolution of networks
• Dynamical and correlation properties of the Internet
• Universal Behavior of Load Distribution in Scale-free Networks
• Class of correlated random networks with hidden variables
• The topological relationship between the large-scale attributes and local interaction patterns of complex networks
• Correlated random networks
• Number of loops of size h in growing scale-free networks
• Loops structure of the Internet at the Autonomous System Level
• An algorithm for counting circuits: application to real-world and random graphs

Cited by in corpus (9)

• Characterization of complex networks: A survey of measurements
• Percolation transition in networks with degree-degree correlation
• Clustering of random scale-free networks
• Response of degree-correlated scale-free networks to stimuli
• Fluctuations analysis in complex networks modeled by hidden variable models. Necessity of a large cut-off in hidden-variable models
• Cycles and Clustering in Multiplex Networks
• Network science Ising states of matter
• Characterizing cycle structure in complex networks
• Fluctuations of motifs and non self-averaging in complex networks. A self- vs non-self-averaging phase transition scenario