We construct complex networks from pseudoperiodic time series, with each cycle represented by a single node in the network. We investigate the statistical properties of these networks for various time series and find that time series with different dynamics exhibit distinct topological structures. Specifically, noisy periodic signals correspond to random networks, and chaotic time series generate networks that exhibit small world and scale free features. We show that this distinction in topological structure results from the hierarchy of unstable periodic orbits embedded in the chaotic attractor. Standard measures of structure in complex networks can therefore be applied to distinguish different dynamic regimes in time series. Application to human electrocardiograms shows that such statistical properties are able to differentiate between the sinus rhythm cardiograms of healthy volunteers and those of coronary care patients.

Complex networks are an important paradigm of modern complex systems sciences which allows quantitatively assessing the structural properties of systems composed of different interacting entities. During the last years, intensive efforts have been spent on applying network-based concepts also for the analysis of dynamically relevant higher-order statistical properties of time series. Notably, many corresponding approaches are closely related to the concept of recurrence in phase space. In this paper, we review recent methodological advances in time series analysis based on complex networks, with a special emphasis on methods founded on recurrence plots. The potentials and limitations of the individual methods are discussed and illustrated for paradigmatic examples of dynamical systems as well as for real-world time series. Complex network measures are shown to provide information about structural features of dynamical systems that are complementary to those characterized by other methods of time series analysis and, hence, substantially enrich the knowledge gathered from other existing (linear as well as nonlinear) approaches.

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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Recently, there has been a coordinated effort from academic institutions and the pharmaceutical industry to identify biomarkers that can predict responses to immune checkpoint blockade in cancer. Several biomarkers have been identified; however, none has reliably predicted response in a sufficiently rigorous manner for routine use. Here, we argue that the therapeutic response to immune checkpoint blockade is a critical state transition of a complex system. Such systems are highly sensitive to initial conditions, and critical transitions are notoriously difficult to predict far in advance. Nevertheless, warning signals can be detected closer to the tipping point. Advances in mathematics and network biology are starting to make it possible to identify such warning signals. We propose that these dynamic biomarkers could prove to be useful in distinguishing responding from non-responding patients, as well as facilitate the identification of new therapeutic targets for combination therapy.