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. DOI: 10.1103/PhysRevLett.96.238701 PACS numbers: 89.75.Hc, 05.45.Tp, 87.19.Hh Characterizing complicated dynamics from experimental time series is a fundamental problem of continuing interest in a wide variety of fields. Different measures have been proposed to analyze these dynamics: for example, Lyapunov exponent, entropies, and correlation dimension [1] . In this Letter, we focus on time series with strong pseudoperiodic behavior such as the human electrocardiogram (ECG), human speech, laser output, and annual sunspot numbers. These time series have aroused great interest due to their close relation to some important physiological and natural systems [2, 3] . Meanwhile, the past few years have witnessed dramatic advances in the field of complex networks [4] and complex networks have been observed to arise naturally in a vast range of physical phenomena. In this Letter, we show that pseudoperiodic time series can also be investigated from the complex network perspective: the nodes of the network correspond directly to cycles in the time series, and network connectivity is determined by the strength of temporal correlation between cycles. This representation encodes the underlying time series dynamics in the network topology, which may then be quantified via the usual statistical properties of the network. We study noisy periodic and chaotic time series within the above framework. In particular, we seek to characterize chaotic dynamics through the basic statistical properties of the network, such as the degree distribution, average path length, and clustering coefficient. These statistical properties actually reflect and quantify the hierarchy of unstable periodic orbits embedded in the chaotic attractor which leads to small world characteristics. Therefore, this approach provides information that is not available from classical nonlinear time series analysis. The approach we present in this Letter is a transformation from time domain dynamics to complex network topology. We find that a complex network with small world and scale free characteristics can essentially be considered as the dual of a time series exhibiting chaotic dynamics. Chaotic dynamical systems therefore have a corresponding complex network topological signature. We start from the construction of the network given a pseudoperiodic time series fx i g n 1 of n observations. First, the pseudoperiodic time series is divided into m disjoint cycles according to the local minimum (or maximum), denoted as fC 1 ; C 2 ; . . . ; C m g. By considering each cycle as a basic node of a graph, a network representation of a given time series is achieved. Next we determine the connection between nodes. A natural idea is that two nodes are deemed to be connected if the phase space distance between the corresponding cycles is less than a predetermined value D. The phase space distance between C i and C j is defined as [5] D ij min l0;1;...;l j ÿl i where X k and Y k is the kth point of C i and C j (with length l i and l j , respectively, and l i < l j ) in the reconstructed phase space. Alternatively, we can also use the linear correlation coefficient between two cycles [5] . The correlation coefficient characterizes the similarity between cycles, since two cycles with a larger temporal correlation will be close in phase space, these two measures are, in fact, inversely proportioned to each other and can be used equivalently. Because the phase space reconstruction is sometimes not reliable for noisy and nonstationary time series, we will use D for the toy models and (which does not need phase space reconstruction) for the experimental time series. After constructing the complex network from the time series, we investigate its basic statistical properties [4] including: (1) the degree distribution pk-the of the degree of the nodes in a graph; (2) the average path length L-the average of the minimum number of links necessary to connect all pairs of nodes; (3) the clustering coefficient C-the fraction of connections between the topological neighbors of a vertex with respect to the maximum possible. We will use two typical pseudoperiodic time series with 4000 cycles for comparison and analysis. The first is a PRL 96,
In 2006, Zhang and Small REF introduced the concept of mapping time series to complex network and discovered that complex network is an alternative approach to visualize the underlying hidden patterns of the time series.
Complex network from pseudoperiodic time series: topology versus dynamics.
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