Horizontal Visibility Algorithm Research Articles

Mapping time series into signed networks via horizontal visibility graph

Time series could be mapped into complex networks through the visibility or horizontal visibility algorithms, and the properties of the constructed network reflect the nonlinear dynamics of the time series. When horizontal visibility algorithm is directly applied to climate anomaly time series, in which both local maximum and local minimum are equally important, local minimum might be “overlooked”. In this paper, we propose a new method that maps climate anomaly time series into signed networks. Positive and negative data values of climate anomaly time series are classified into two types and mapped as nodes of signed networks. Links connecting nodes of the same type are assigned positive signs, while links connecting neighboring nodes of different types are assigned negative signs. This method is also applicable to time series those are assumed to be “stationary” or with no significant trends. Four kinds of degree as well as the degree distributions of the signed networks have been defined. Specifically, the degree and degree distribution could be partly derived analytically for periodic and uncorrelated random time series. The theoretical predictions for periodic and uncorrelated random time series have also been verified by extensive numerical simulations. Based on the entropy of the distribution of net degree, we propose a new complexity measure for chaotic time series. Compared to some previous complexity measures, the new complexity measure is an objective measure without transforming continuous values into discrete probability distributions but still has higher accuracy and sensitivity. Moreover, correlation information of stochastic time series can also be extracted via a topological parameter, the mean of ratio degree, of the signed networks. The extraction of serial correlation has been illustrated through numerical simulations and verified through an empirical climate time series.

Prediction of Membrane Protein Amphiphilic Helix Based on Horizontal Visibility Graph and Graph Convolution Network.

Membrane protein amphiphilic helices play an important role in many biological processes. Based on the graph convolution network and the horizontal visibility graph the prediction method of membrane protein amphiphilic helix structure is proposed in this paper. The new dataset of amphiphilic helix is constructed. In this paper, we propose the novel feature extraction method, which characterize the amphiphilicity of membrane protein. We also extract three commonly used protein features together with the new features as protein node features. The neighbor information and long-distance dependence information of proteins are further extracted by sliding window and bidirectional long-short term memory network respectively. From the perspective of horizontal visibility algorithm, we transform protein sequences into complex networks to obtain the graph features of proteins. Then, graph convolutional network model is employed to predict the amphiphilic helix structure of membrane protein. A rigorous ten-fold cross-validation shows that the proposed method outperforms other AH prediction methods on the newly constructed dataset.

A simple scalable linear time algorithm for horizontal visibility graphs

Horizontal Visibility Graphs establish a connection between time series and complex networks. As a feature, they have shown strong results in time series classification. For real-world applications, algorithms for computing HVGs are necessary that work efficiently on streamed data, that can be parallelized, and whose runtime is independent of the type of time series. Our proposed algorithm extends the fast horizontal visibility algorithm of Zhu et al. satisfying all these desirable properties. The extended version stays worst-case in O(n), works additionally efficiently on streamed data, and becomes parallelizable. Contrary to recent publications, it does not require a complex data structure. This approach enables the computation of HVGs with millions of vertices in a short period, opening up new application areas of HVGs for time series generated batch-wise or resulting from measurements with a high sampling rate.

Hierarchical deposition and scale-free networks: A visibility algorithm approach.

The growth of an interface formed by the hierarchical deposition of particles of unequal size is studied in the framework of a dynamical network generated by a horizontal visibility algorithm. For a deterministic model of the deposition process, the resulting network is scale free with dominant degree exponent γ_{e}=ln3/ln2 and transient exponent γ_{o}=1. An exact calculation of the network diameter and clustering coefficient reveals that the network is scale invariant and inherits the modular hierarchical nature of the deposition process. For the random process, the network remains scale free, where the degree exponent asymptotically converges to γ=3, independent of the system parameters. This result shows that the model is in the class of fractional Gaussian noise through the relation between the degree exponent and the series' Hurst exponent H. Finally, we show through the degree-dependent clustering coefficient C(k) that the modularity remains present in the system.

Complexity traits and synchrony of cryptocurrencies price dynamics

In this study, we characterized the dynamics and analyzed the degree of synchronization of the time series of daily closing prices and volumes in US$ of three cryptocurrencies, Bitcoin, Ethereum, and Litecoin, over the period September 1,2015–March 31, 2020. Time series were first mapped into a complex network by the horizontal visibility algorithm in order to revel the structure of their temporal characters and dynamics. Then, the synchrony of the time series was investigated to determine the possibility that the cryptocurrencies under study co-bubble simultaneously. Findings reveal similar complex structures for the three virtual currencies in terms of number and internal composition of communities. To the aim of our analysis, such result proves that price and volume dynamics of the cryptocurrencies were characterized by cyclical patterns of similar wavelength and amplitude over the time period considered. Yet, the value of the slope parameter associated with the exponential distributions fitted to the data suggests a higher stability and predictability for Bitcoin and Litecoin than for Ethereum. The study of synchrony between the time series investigated displayed a different degree of synchronization between the three cryptocurrencies before and after a collapse event. These results could be of interest for investors who might prefer to switch from one cryptocurrency to another to exploit the potential opportunities of profit generated by the dynamics of price and volumes in the market of virtual currencies.

Construction and analysis of interdependent networks based on time series and horizontal visibility graphs

In this paper, the construction and statistical features of the directional complex networks with interdependencies (i.e. interdependent networks) are investigated, in which the intra-structure of each sub-network is not considered. Based on time series and horizontal visibility graph, two new visibility model methods are presented: many-to-many directional horizontal visibility algorithm and one-to-one directional horizontal visibility algorithm. Using R software, the interdependent structure of the directional interdependent network constructed by these two algorithms are analyzed and the statistical features are evaluated associated with two periodic time series, a periodic time series and a fractal time series, a periodic time series and a random series, separately. The results show that both algorithms ensure the simplicity of the interdependent structure of the networks and inherit the morphological characteristics of the time series.

A fault detection method based on horizontal visibility graph‐integrated complex networks: Application to complex chemical processes

AbstractWith the trend towards large‐scale structure and complexity in modern chemical processes, it is difficult to describe the system operating status and fault conditions via traditional approaches. In addition, there are obstacles to fault detection research in complex chemical processes due to the characteristics of the equipment hardware used in chemical systems, data collection and handling, strong internal correlations in chemical systems, factors affecting transmission, and randomness and cascade in process failures. Therefore, this paper presents a fault detection method based on horizontal visibility graph (HVG) analysis‐integrated complex networks. The data for each variable in the system are regarded as a time series, each time series is modelled into a network by the horizontal visibility algorithm, and each single‐layer network corresponding to a time series is abstracted as a node. Meanwhile, the correlation between two single‐layer networks is used to characterize the correlation between the corresponding nodes. Moreover, a complex network structure representing the chemical system can be constructed from the correlations. In addition, according to the correlation ratio matrix obtained from the fault state and the normal state, the variance of each node determines the faulty node. Finally, we verify the validity and the effectiveness of the proposed method by applying it to the Tennessee Eastman (TE) process.

Manifestations of the onset of chaos in condensed matter and complex systems

We review the occurrence of the patterns of the onset of chaos in low-dimensional nonlinear dissipative systems in leading topics of condensed matter physics and complex systems of various disciplines. We consider the dynamics associated with the attractors at period-doubling accumulation points and at tangent bifurcations to describe features of glassy dynamics, critical fluctuations and localization transitions. We recall that trajectories pertaining to the routes to chaos form families of time series that are readily transformed into networks via the Horizontal Visibility algorithm, and this in turn facilitates establish connections between entropy and Renormalization Group properties. We discretize the replicator equation of game theory to observe the onset of chaos in familiar social dilemmas, and also to mimic the evolution of high-dimensional ecological models. We describe an analytical framework of nonlinear mappings that reproduce rank distributions of large classes of data (including Zipf's law). We extend the discussion to point out a common circumstance of drastic contraction of configuration space driven by the attractors of these mappings. We mention the relation of generalized entropy expressions with the dynamics along and at the period doubling, intermittency and quasi-periodic routes to chaos. Finally, we refer to additional natural phenomena in complex systems where these conditions may manifest.

Exact results of the limited penetrable horizontal visibility graph associated to random time series and its application

The limited penetrable horizontal visibility algorithm is an analysis tool that maps time series into complex networks and is a further development of the horizontal visibility algorithm. This paper presents exact results on the topological properties of the limited penetrable horizontal visibility graph associated with independent and identically distributed (i:i:d:) random series. We show that the i.i.d: random series maps on a limited penetrable horizontal visibility graph with exponential degree distribution, independent of the probability distribution from which the series was generated. We deduce the exact expressions of mean degree and clustering coefficient, demonstrate the long distance visibility property of the graph and perform numerical simulations to test the accuracy of our theoretical results. We then use the algorithm in several deterministic chaotic series, such as the logistic map, H´enon map, Lorenz system, energy price chaotic system and the real crude oil price. Our results show that the limited penetrable horizontal visibility algorithm is efficient to discriminate chaos from uncorrelated randomness and is able to measure the global evolution characteristics of the real time series.

Topological entropy and geometric entropy and their application to the horizontal visibility graph for financial time series

In this paper, we introduce topological entropy (TE) based on time series, which characterizes the total exponential complexity of a quantified system with a single number. Combined with multiscale theory, we propose geometric entropy (GE), aiming to examine the correlation among different time series. In order to detect the properties of TE and GE, we apply them to an original symbolic method utilized to measure time series irreversibility, namely horizontal visibility algorithm. On this basis, we propose a time series irreversibility measure, i.e., normalized index. Then, we employ TE and GE based on the horizontal visibility graph symbolic algorithm to simulated time series, which is generated by the logistic map with different parameters. Through the comparison of the results, we find out that different simulated data have the same variation tendency of TE, which means that TE is capable of reflecting the similarity among different time series. On the basic of these results, we further analyze the irreversibility of simulated data and also get some interesting findings. From the GE results comparison, we conclude that the GE method can distinguish different time series and expose their correlation efficiently. As a farther validation, we explore the effects of these methods on the analysis of different stock time series. Results show that they can reflect a large number of interrelationships, and successfully quantify the changes in the complexity of different stock market data.

Predicting catastrophes of non-autonomous networks with visibility graphs and horizontal visibility

Abstract Prediction of potential catastrophes in engineering systems is a challenging problem. We first attempt to construct a complex network to predict catastrophes of a multi-modular floating system in advance of their occurrences. Response time series of the system can be mapped into an virtual network by using visibility graph or horizontal visibility algorithm. The topology characteristics of the networks can be used to forecast catastrophes of the system. Numerical results show that there is an obvious corresponding relationship between the variation of topology characteristics and the onset of catastrophes. A Catastrophe Index (CI) is proposed as a numerical indicator to measure a qualitative change from a stable state to a catastrophic state. The two approaches, the visibility graph and horizontal visibility algorithms, are compared by using the index in the reliability analysis with different data lengths and sampling frequencies. The technique of virtual network method is potentially extendable to catastrophe predictions of other engineering systems.

Electroencephalography‐based feature extraction using complex network for automated epileptic seizure detection

AbstractElectroencephalography signals are typically used for analyzing epileptic seizures. These signals are highly nonlinear and nonstationary, and some specific patterns exist for certain disease types that are hard to develop an automatic epileptic seizure detection system. This paper discussed statistical mechanics of complex networks, which inherit the characteristic properties of electroencephalography signals, for feature extraction via a horizontal visibility algorithm in order to reduce processing time and complexity. The algorithm transforms a time series signal into a complex network, which some features are abbreviated. The statistical mechanics are calculated to capture distinctions pertaining to certain diseases to form a feature vector. The feature vector is classified by multiclass classification via a k‐nearest neighbor classifier, a multilayer perceptron neural network, and a support vector machine with a 10‐fold cross‐validation criterion. In performance evaluation of proposed method with healthy, seizure‐free interval, and seizure signals, firstly, input data length is regarded among some practical signal samples by optimizing between accuracy‐processing time, and the proposed method yields outstanding performance on the average classification accuracy for 3‐class problems mainly for detection of seizure‐free interval and seizure signals and acceptable results for 2‐class and 5‐class problems comparing with conventional methods. The proposed method is another tool that can be used for classifying signal patterns, as an alternative to time/frequency analyses.

Complex network approach to fractional time series.

In order to extract correlation information inherited in stochastic time series, the visibility graph algorithm has been recently proposed, by which a time series can be mapped onto a complex network. We demonstrate that the visibility algorithm is not an appropriate one to study the correlation aspects of a time series. We then employ the horizontal visibility algorithm, as a much simpler one, to map fractional processes onto complex networks. The degree distributions are shown to have parabolic exponential forms with Hurst dependent fitting parameter. Further, we take into account other topological properties such as maximum eigenvalue of the adjacency matrix and the degree assortativity, and show that such topological quantities can also be used to predict the Hurst exponent, with an exception for anti-persistent fractional Gaussian noises. To solve this problem, we take into account the Spearman correlation coefficient between nodes' degrees and their corresponding data values in the original time series.

Fully developed turbulence in the view of horizontal visibility graphs

We employ the horizontal visibility algorithm to map the velocity and acceleration time series in turbulent flows with different Reynolds numbers, onto complex networks. The universal nature of velocity fluctuations in high Reynolds turbulent Helium flow is found to be inherited in the corresponding network topology. The degree distributions of the acceleration series are shown to have stretched exponential forms with the Reynolds number dependent fitting parameter. Furthermore, for acceleration time series, we find a transitional behavior in terms of the Reynolds number in all network features which is in agreement with recent empirical studies.

Epileptic seizure detection in EEGs signals using a fast weighted horizontal visibility algorithm

This paper proposes a fast weighted horizontal visibility graph constructing algorithm (FWHVA) to identify seizure from EEG signals. The performance of the FWHVA is evaluated by comparing with Fast Fourier Transform (FFT) and sample entropy (SampEn) method. Two noise-robustness graph features based on the FWHVA, mean degree and mean strength, are investigated using two chaos signals and five groups of EEG signals. Experimental results show that feature extraction using the FWHVA is faster than that of SampEn and FFT. And mean strength feature associated with ictal EEG is significant higher than that of healthy and inter-ictal EEGs. In addition, an 100% classification accuracy for identifying seizure from healthy shows that the features based on the FWHVA are more promising than the frequency features based on FFT and entropy indices based on SampEn for time series classification.

Quasiperiodic graphs at the onset of chaos

We examine the connectivity fluctuations across networks obtained when the horizontal visibility (HV) algorithm is used on trajectories generated by nonlinear circle maps at the quasiperiodic transition to chaos. The resultant HV graph is highly anomalous as the degrees fluctuate at all scales with amplitude that increases with the size of the network. We determine families of Pesin-like identities between entropy growth rates and generalized graph-theoretical Lyapunov exponents. An irrational winding number with pure periodic continued fraction characterizes each family. We illustrate our results for the so-called golden, silver, and bronze numbers.

Horizontal visibility graphs generated by type-I intermittency

The type-I intermittency route to (or out of) chaos is investigated within the horizontal visibility (HV) graph theory. For that purpose, we address the trajectories generated by unimodal maps close to an inverse tangent bifurcation and construct their associated HV graphs. We show how the alternation of laminar episodes and chaotic bursts imprints a fingerprint in the resulting graph structure. Accordingly, we derive a phenomenological theory that predicts quantitative values for several network parameters. In particular, we predict that the characteristic power-law scaling of the mean length of laminar trend sizes is fully inherited by the variance of the graph degree distribution, in good agreement with the numerics. We also report numerical evidence on how the characteristic power-law scaling of the Lyapunov exponent as a function of the distance to the tangent bifurcation is inherited in the graph by an analogous scaling of block entropy functionals defined on the graph. Furthermore, we are able to recast the full set of HV graphs generated by intermittent dynamics into a renormalization-group framework, where the fixed points of its graph-theoretical renormalization-group flow account for the different types of dynamics. We also establish that the nontrivial fixed point of this flow coincides with the tangency condition and that the corresponding invariant graph exhibits extremal entropic properties.

Quasiperiodic Graphs: Structural Design, Scaling and Entropic Properties

A novel class of graphs, here named quasiperiodic, are constructed via application of the Horizontal Visibility algorithm to the time series generated along the quasiperiodic route to chaos. We show how the hierarchy of mode-locked regions represented by the Farey tree is inherited by their associated graphs. We are able to establish, via Renormalization Group (RG) theory, the architecture of the quasiperiodic graphs produced by irrational winding numbers with pure periodic continued fraction. And finally, we demonstrate that the RG fixed-point degree distributions are recovered via optimization of a suitably defined graph entropy.

Feigenbaum graphs at the onset of chaos

We analyze the properties of networks obtained from the trajectories of unimodal maps at the transition to chaos via the horizontal visibility (HV) algorithm. We find that the network degrees fluctuate at all scales with amplitude that increases as the size of the network grows, and can be described by a spectrum of graph-theoretical generalized Lyapunov exponents. We further define an entropy growth rate that describes the amount of information created along paths in network space, and find that such entropy growth rate coincides with the spectrum of generalized graph-theoretical exponents, constituting a set of Pesin-like identities for the network.

DETECTING SERIES PERIODICITY WITH HORIZONTAL VISIBILITY GRAPHS

The horizontal visibility algorithm was recently introduced as a mapping between time series and networks. The challenge lies in characterizing the structure of time series (and the processes that generated those series) using the powerful tools of graph theory. Recent works have shown that the visibility graphs inherit several degrees of correlations from their associated series, and therefore such graph theoretical characterization is in principle possible. However, both the mathematical grounding of this promising theory and its applications are in its infancy. Following this line, here we address the question of detecting hidden periodicity in series polluted with a certain amount of noise. We first put forward some generic properties of horizontal visibility graphs which allow us to define a (graph theoretical) noise reduction filter. Accordingly, we evaluate its performance for the task of calculating the period of noisy periodic signals, and compare our results with standard time domain (autocorrelation) methods. Finally, potentials, limitations and applications are discussed.