Scaling Invariance in Spectra of Complex Networks: A Diffusion Factorial Moment Approach

TL;DR

This paper introduces the diffusion factorial moment (DFM) method to analyze spectral scaling invariance in various complex networks, revealing universal features and differences across network types.

Contribution

The study presents a novel DFM approach to identify and compare scale invariance in spectra of different complex networks, unifying their analysis.

Findings

Erdos-Renyi networks show a scaling parameter of 0.51 for p_{ER}<1/N.

Small-world networks exhibit scale invariance in a specific p_r range.

GRN networks have a fluctuating scaling parameter around 0.6.

Abstract

A new method called diffusion factorial moment (DFM) is used to obtain scaling features embedded in spectra of complex networks. For an Erdos-Renyi network with connecting probability $p_{ER} < \frac{1}{N}$, the scaling parameter is $\delta = 0.51$, while for $p_{ER} \ge \frac{1}{N}$ the scaling parameter deviates from it significantly. For WS small-world networks, in the special region $p_r \in [0.05,0.2]$, typical scale invariance is found. For GRN networks, in the range of $\theta\in[0.33,049]$, we have $\delta=0.6\pm 0.1$. And the value of $\delta$ oscillates around $\delta=0.6$ abruptly. In the range of $\theta\in[0.54,1]$, we have basically $\delta>0.7$. Scale invariance is one of the common features of the three kinds of networks, which can be employed as a global measurement of complex networks in a unified way.