How much random a random network is : a random matrix analysis

TL;DR

This paper uses random matrix theory to analyze the degree of randomness in complex networks, showing that the Δ3 statistic effectively measures how network eigenvalues align with random matrix predictions.

Contribution

It introduces the Δ3 statistic as a qualitative measure of network randomness, linking eigenvalue correlations to network structure deviations.

Findings

Δ3 statistic follows linear behavior in semi-log scale for random networks.

Eigenvalue correlations reflect the degree of network randomness.

Long-range eigenvalue correlations indicate deviation from regularity.

Abstract

We analyze complex networks under random matrix theory framework. Particularly, we show that $\Delta_3$ statistic, which gives information about the long range correlations among eigenvalues, provides a qualitative measure of randomness in networks. As networks deviate from the regular structure, $\Delta_3$ follows random matrix prediction of linear behavior, in semi-logarithmic scale with the slope of $1/\pi^2$, for the longer scale.