A general formulation of long-range degree correlations in complex networks

Abstract

We provide a general framework for analyzing degree correlations between nodes separated by more than one step (i.e., beyond nearest neighbors) in complex networks. One probability and four conditional probabilities are introduced to fully describe long-range degree correlations with respect to $k$ and $k^{\prime }$ of two nodes and shortest path length $l$ between them. We present general relations among these probabilities and clarify the relevance to nearest-neighbor degree correlations. Unlike nearest-neighbor correlations, some of these probabilities are meaningful only in finite-size networks. Furthermore, as a baseline to determine the existence or nonexistence of long-range degree correlations in a network, the functional forms of these probabilities for networks without any long-range degree correlations are analytically evaluated within a mean-field approximation. The validity of our argument is demonstrated by applying it to real-world networks.

9 pages, 4 figures

References in corpus (11)

• Mitigation of Malicious Attacks on Networks
• Fractal and Transfractal Recursive Scale-Free Nets
• Percolation on correlated networks
• Correlations in weighted networks
• Synchronization in weighed scale-free networks with degree-degree correlation
• Percolation transition in networks with degree-degree correlation
• Local clustering in scale-free networks with hidden variables
• Epidemic Extinction Paths in Complex Networks
• Fractality and degree correlations in scale-free networks
• Synchronization in Scale Free networks with degree correlation
• Resilience of antagonistic networks with regard to the effects of initial failures and degree-degree correlations

Cited by in corpus (9)

• Information theoretic network approach to socioeconomic correlations
• Structure of percolating clusters in random clustered networks
• Degree correlations in graphs with clique clustering
• Emergence of Long-Range Correlations in Random Networks
• Analytic solution of the two-star model with correlated degrees
• Identification of intrinsic long-range degree correlations in complex networks
• Percolation on a maximally disassortative network
• Bifractality of fractal scale-free networks
• Revisiting finite size effect of percolation in degree correlated networks