Visibility graphs and symbolic dynamics

Abstract

Visibility algorithms are a family of geometric and ordering criteria by which a real-valued time series of N data is mapped into a graph of N nodes. This graph has been shown to often inherit in its topology nontrivial properties of the series structure, and can thus be seen as a combinatorial representation of a dynamical system. Here we explore in some detail the relation between visibility graphs and symbolic dynamics. To do that, we consider the degree sequence of horizontal visibility graphs generated by the one-parameter logistic map, for a range of values of the parameter for which the map shows chaotic behaviour. Numerically, we observe that in the chaotic region the block entropies of these sequences systematically converge to the Lyapunov exponent of the system. Via Pesin identity, this in turn suggests that these block entropies are converging to the Kolmogorov-Sinai entropy of the map, which ultimately suggests that the algorithm is implicitly and adaptively constructing phase space partitions which might have the generating property. To give analytical insight, we explore the relation k(x), x ∈ [0, 1] that, for a given datum with value x, assigns in graph space a node with degree k. In the case of the out-degree sequence, such relation is indeed a piece-wise constant function. By making use of explicit methods and tools from symbolic dynamics we are able to analytically show that the algorithm indeed performs an effective partition of the phase space and that such partition is naturally expressed as a countable union of subintervals, where the endpoints of each subinterval are related to the fixed point structure of the iterates of the map and the subinterval enumeration is associated with particular ordering structures that we called motifs.

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“…As noted above, visibility graphs and horizontal visibility graphs were introduced with the aims of using the tools of Graph Theory and Network Science 24 to describe the structure of time series and their underlying dynamics from a combinatorial perspective (other proposals for graphtheoretical time series analysis can be found in 25,26 and references therein and the extension of visibility graphs to image processing can be found in 27 ). Research on this methodology has been primarily theoretical, elaborating on mathematical methods [28][29][30][31] to extract rigorous results on the properties of these graphs when associated to canonical models of complex dynamics, including stochastic processes with and without correlations or chaotic processes [32][33][34][35] . In practice, this method can be used as a feature extraction procedure for constructing feature vectors for statistical learning purposes and has been widely applied across the disciplines (see, for instance, [36][37][38][39][40] for just a few examples).…”

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“…After converting the original time series into network domain, an important metric of the obtained network, E DD , seems promising in capturing the alterations of cardiac dynamics under different physiological and pathological conditions [ 14 , 15 ]. As proposed in the work of Lacasa et al [ 31 ], a time series with N data points, denoted as , can be converted to a network through the visibility graph (VG) algorithm. To carry out the VG algorithm, first, take each data point as a node in the network sequentially.…”

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“…In recent years, graph network approaches have been proposed as efficient tools to analyze time series data. In this fashion, a time series is transformed into an equivalent graph network, which further allows information embedded in the time series to be extracted and characterized [ 65 , 66 , 67 ]. The visibility graph technique proposed by Lacasa et al [ 58 ] is one of the most widely utilized approaches to map a time series into a graph network.…”