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 the multifractal detrended fluctuation approach (MF-DFA) and diffusion entropy (DE) is employed to extract characteristics from this time series. For the Watts-Strogatz small-world model, there exists a critical point at rewiring probability P(r) =0.32. For a network generated in the range 0< P(r) <0.32, the correlation exponent is in the range of 1.0-1.64. Above this critical point, all the networks behave similar to that at p(r) =1. For the Erdos-Renyi model, the time series behaves like fractional Brownian motion noise at p(ER) =1/N. For the growing random network (GRN) model, the values of the long-range correlation exponent are in the range of 0.74-0.83. For most of the GRN networks the probability distribution function 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.