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

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 $δ= 0.51$, 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 $θ\in[0.33,049]$, we have $δ=0.6\pm 0.1$. And the value of $δ$ oscillates around $δ=0.6$ abruptly. In the range of $θ\in[0.54,1]$, we have basically $δ>0.7$. 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.