Emergence of Long-Range Correlations in Random Networks

TL;DR

This paper analytically investigates how long-range degree correlations emerge in uncorrelated random networks, revealing a divergence at criticality and characterizing the decay of correlations with distance.

Contribution

It introduces a characteristic length scale for degree correlations and analyzes their behavior at and near the critical point in random networks.

Findings

Negative degree correlations occur within a characteristic length.

Correlation length diverges at the critical point where the giant component is fractal.

Correlation function decays exponentially with distance off-critical, with power-law behavior near criticality.

Abstract

We perform an analytical analysis of the long-range degree correlation of the giant component in an uncorrelated random network by employing generating functions. By introducing a characteristic length, we find that a pair of nodes in the giant component is negatively degree-correlated within the characteristic length and uncorrelated otherwise. At the critical point, where the giant component becomes fractal, the characteristic length diverges and the negative long-range degree correlation emerges. We further propose a correlation function for degrees of the $l$-distant node pairs, which behaves as an exponentially decreasing function of distance in the off-critical region. The correlation function obeys a power-law with an exponential cutoff near the critical point. The Erd\H{o}s-R\'{e}nyi random graph is employed to confirm this critical behavior.