Emergence of long-range correlations in random networks

Abstract

We perform an analysis of the long-range degree correlation of the giant component (GC) in an uncorrelated random network by employing generating functions. By introducing a characteristic length, we find that a pair of nodes in the GC is negatively degree-correlated within the characteristic length and uncorrelated otherwise. At the critical point, where the GC becomes fractal, the characteristic length diverges and the negative long-range degree correlation emerges. We further propose a correlation function for degrees of two nodes separated by the shortest path length l, 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 Erdős-Rényi random graph is employed to confirm this critical behavior.