Uncorrelated Random Networks

TL;DR

This paper introduces a statistical ensemble of uncorrelated, non-degenerate graphs with arbitrary degree distributions, providing an efficient algorithm and analyzing finite-size effects in scale-free networks, highlighting the necessity of correlations in real-world networks.

Contribution

It defines a new ensemble of uncorrelated, non-degenerate graphs with arbitrary degree distributions and studies finite-size effects in scale-free networks.

Findings

Degree distribution cutoff scales as N^γ with γ = min[1/2, 1/(β-1)]

Finite-size effects limit maximum degree in scale-free graphs

Inter-node correlations are essential in real scale-free networks

Abstract

We define a statistical ensemble of non-degenerate graphs, i.e. graphs without multiple- and self-connections between nodes. The node degree distribution is arbitrary, but the nodes are assumed to be uncorrelated. This completes our earlier publication \cite{bck}, where trees and degenerate graphs were considered. An efficient algorithm generating non-degenerate graphs is constructed. The corresponding computer code is available on request. Finite-size effects in scale-free graphs, i.e. those where the tail of the degree distribution falls like $n^{-\beta}$, are carefully studied. We find that in the absence of dynamical internode correlations the degree distribution is cut at a degree value scaling like $N^{\gamma}$, with $\gamma = \min[1/2, 1/(\beta-1)]$, where $N$ is the total number of nodes. The consequence is that, independently of any specific model, the inter-node correlations seem to be a necessary ingredient of the physics of scale-free networks observed in nature.