Synchronizabilities of Networks: A New index

The random matrix theory is used to bridge the network structures and the dynamical processes defined on them. We propose a possible dynamical mechanism for the enhancement effect of network structures on synchronization processes, based upon which a dynamic-based index of the synchronizability is introduced in the present paper.Comment: 4pages, 2figure

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

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.Comment: 6 pages, 8 figures. to appear in Physical Review

Temporal Series Analysis Approach to Spectra of Complex Networks

The spacing of nearest levels of the spectrum of a complex network can be regarded as a time series. Joint use of Multi-fractal Detrended Fluctuation Approach (MF-DFA) and Diffusion Entropy (DE) is employed to extract characteristics from this time series. For the WS (Watts and Strogatz) small-world model, there exist a critical point at rewiring probability . For a network generated in the range, the correlation exponent is in the range of . Above this critical point, all the networks behave similar with that at . For the ER model, the time series behaves like FBM (fractional Brownian motion) noise at . For the GRN (growing random network) model, the values of the long-range correlation exponent are in the range of . For most of the GRN networks the PDF of a constructed time series obeys a Gaussian form. In the joint use of MF-DFA and DE, the shuffling procedure in DE is essential to obtain a reliable result. PACS number(s): 89.75.-k, 05.45.-a, 02.60.-xComment: 10 pages, 9 figures, to appear in PR