Correlations in random Apollonian network

TL;DR

This paper investigates degree correlations in random Apollonian networks using simulations and theoretical analysis, revealing that average nearest-neighbor degree is uncorrelated with degree while clustering coefficient depends on degree with a power-law decay.

Contribution

The paper provides the first analytical solutions for two and three node correlations in RAN, including a more accurate mean clustering coefficient based on degree correlations.

Findings

ANND shows no correlation with degree

Clustering coefficient depends on degree as a power-law

Analytical results match extensive numerical simulations

Abstract

In this paper, by both simulations and theoretical predictions we study two and three node (or degree) correlations in random Apollonian network (RAN), which have small-world and scale-free topologies. Using the rate equation approach under the assumption of continuous degree, we first give the analytical solution for two node correlations, expressed by average nearest-neighbor degree (ANND). Then, we revisit the degree distribution of RAN using rate equation method and get the exact connection distribution, based on which we derive a more accurate result for mean clustering coefficient as an average quantity of three degree correlations than the one previously reported. Analytical results reveal that ANND has no correlations with respect to degree, while clustering coefficient is dependent on degree, showing a power-law behavior as $C(k)\sim k^{-1}$. The obtained expressions are successfully contrasted with extensive numerical simulations.