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 $\delta =0.51$, 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.

6 pages, 8 figures. to appear in Physical Review E

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