How to calculate the main characteristics of random uncorrelated networks

TL;DR

This paper introduces an analytical method to determine key structural properties of random uncorrelated networks with various degree distributions, enabling precise predictions of phase transitions, component sizes, and path lengths.

Contribution

The paper develops a formalism that accurately predicts structural characteristics of diverse random networks, validated by numerical simulations.

Findings

Accurately predicts phase transition points for giant component formation.

Provides analytical estimates for mean component size below transition.

Determines the size of the giant component and average path length above transition.

Abstract

We present an analytic formalism describing structural properties of random uncorrelated networks with arbitrary degree distributions. The formalism allows to calculate the main network characteristics like: the position of the phase transition at which a giant component first forms, the mean component size below the phase transition, the size of the giant component and the average path length above the phase transition. We apply the approach to classical random graphs of Erdos and Renyi, single-scale networks with exponential degree distributions and scale-free networks with arbitrary scaling exponents and structural cut-offs. In all the cases we obtain a very good agreement between results of numerical simulations and our analytical predictions.