Multifractal detrended fluctuation and rank-based analyses of time series generated by unbiased random walks on complex networks

We simultaneously generate the vertex degree (VD) series and the vertex closeness centrality (VCC) series using unbiased random walks on three typical model networks, namely Erdös–Rényi (ER) random networks, Watts–Strogatz (WS) small-world networks, and Barabási–Albert (BA) scale-free networks, and then focus on distribution and correlation of these resultant series. First, distributions of the VD time series are found to inherit information about degree distributions of original networks so that benefit distinguishing the BA networks from the ER and WS networks. Second, for the ER and WS networks with the similar degree distributions, the multifractal detrended fluctuation analysis (MF-DFA) is applied to series of the ratio of their VCC to VD series, and reveals obviously different multifractal correlations. Therefore, the MF-DFA results can discriminate the ER from WS networks. Third, we employ the rank-based analysis to further investigate the VD and VCC series of three model networks and their corresponding reshuffling series. The resultant standard deviation series ${\sigma }_{rank}$ of reshuffling series for all three networks approximate a constant very close to that of the strict Gaussian white noise series; whereas the ${\sigma }_{rank}$ series of their VD and VCC series deviate from those of the reshuffling series in different patterns, and therefore can capture the intrinsic distinctions among these networks. Consequently, the proposed analyses of time series generated by unbiased random walks can effectively capture the inherent nature of original networks, which is further supported by analyses of a visibility network constructed from fractional Brownian motion and a real-world biological network.