Superfamily phenomena and motifs of networks induced from time series

Abstract

We introduce a transformation from time series to complex networks and then study the relative frequency of different subgraphs within that network. The distribution of subgraphs can be used to distinguish between and to characterize different types of continuous dynamics: periodic, chaotic, and periodic with noise. Moreover, although the general types of dynamics generate networks belonging to the same superfamily of networks, specific dynamical systems generate characteristic dynamics. When applied to discrete (map-like) data this technique distinguishes chaotic maps, hyperchaotic maps, and noise data.chaos ͉ complex networks ͉ subgraphs ͉ embedding ͉ dimension R ecently, a bridge between time series analysis and complex networks has emerged (1, 2). Zhang and Small (1) first introduced a transformation from pseudoperiodic (that is, oscillatory) time series to complex networks. By connecting those nodes whose corresponding cycles are morphologically similar, the dynamics of time series are encoded into the topology of the corresponding network. Lacasa et al. (2) have proposed an alternative algorithm to characterize periodic, random, and fractal time series based on a similar philosophy. In their scheme, successive scalar time series points are mapped to nodes of the network with links between nodes for which the corresponding points satisfy a condition on their relative magnitudes. By exploiting the fundamental properties of time series that manifest clearly in the corresponding networks, they are able to distinguish between broad classes of dynamical systems.Although both these schemes have been successfully applied to generate complex networks from time series, the authors of each algorithm have only explored the basic global statistics of the network, such as degree distribution and average path length (3, 4). We note that many networks that have the same basic global properties, such as small-world character (5) and scale-free distribution (6), may have wildly different local structures (7). Conversely, networks with different global properties may demonstrate similar local structures (8). Actually, mounting evidence suggests that there might be strong ties between the global topological properties and key local patterns of networks (9).In contrast to the degree distribution and the clustering coefficient, the relative frequency of small subgraphs (or motifs) can describe the local characteristics of complex networks (7). The rank distributions of these motifs can reflect the local structural properties and thus can be used to classify networks (8). To understand the transformation mechanism between time series and complex networks, it is important to make a comparison between the local structures of networks from different time series. Whereas the previous works (1, 2) focused on macroscopic properties of the dynamics evident in the network, we turn our attention to the fine features of the dynamics that are only evident on examination of the corresponding network.In this article, we will discuss complex...

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