Transformation from Complex Networks to Time Series Using Classical Multidimensional Scaling

Abstract

Various complex phenomena exist in the real world. Then, many methods have already been proposed to analyze the complex phenomena. Recently, novel methods have been proposed to analyze the deterministic nonlinear, possibly chaotic, dynamics using the complex network theory [1, 2, 3]. These methods evaluate the chaotic dynamics by transforming an attractor of nonlinear dynamical systems to a network. In this paper, we investigate the opposite direction: we transform complex networks to a time series. To realize the transformation from complex networks to time series, we use the classical multidimensional scaling. To justify the proposed method, we reconstruct networks from the time series and compare the reconstructed network with its original network. We confirm that the time series transformed from the networks by the proposed method completely preserves the adjacency information of the networks. Then, we applied the proposed method to a mathematical model of the small-world network (the WS model). The results show that the regular network in the WS model is transformed to a periodic time series, and the random network in the WS model is transformed to a random time series. The small-world network in the WS model is transformed to a noisy periodic time series. We also applied the proposed method to the real networks - the power grid network and the neural network of C. elegans - which are recognized to have small-world property. The results indicate that these two real networks could be characterized by a hidden property that the WS model cannot reproduce.