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

TL;DR

This paper investigates how degree correlations influence the loop structures in scale-free networks, revealing their effects on clustering and Hamiltonian paths through theoretical analysis and real-world data validation.

Contribution

It introduces a novel analysis of degree correlations' impact on loop subgraphs, including clustering and Hamiltonian paths, in scale-free networks.

Findings

Degree correlations affect the scaling of the clustering coefficient.

Correlations influence the number of Hamiltonian paths in real networks.

Results are validated through exact loop counting in real graphs.

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.