Visibility Graph Algorithm Research Articles

Time Irreversibility of Complex Time Series with Detrended Cross-Correlation Analysis Based on Visibility Graph

In this paper, we introduce detrended cross-correlation analysis (DCCA) method into visibility graph (VG) algorithm and propose VGDCCA method to reflect the time series irreversibility from a new perspective. The validity and reliability of the proposed VGDCCA method is supported by numerical simulations on synthesized short-term correlated chaotic systems and long-term correlated fractal processes, and by the empirical analysis on stock indices and traffic parameter. The VGDCCA planes suggest that autoregressive fractionally integrated moving average (ARFIMA) and fractional Gaussian noise (FGN) series show time reversible behavior, whatever the long-term correlation of the series is or however strong the persistence or anti-persistence of the series is. Meanwhile, Logistic map with [Formula: see text] and Hénon map show more time irreversible behaviors than those of ARFIMA, FGN series and other Logistic maps. It can be found that the relationship between cross-correlation of ingoing degree sequence and outgoing degree sequence for the simulated series with different parameters is consistent with the complexity and autocorrelation behavior in the corresponding definition of the time series. For the empirical analysis, VGDCCA method declares the similarity and dissimilarity between stock indices on time series irreversibility and captures the time irreversibility of traffic time series recorded by the detectors in different locations.

TVGeAN: Tensor Visibility Graph-Enhanced Attention Network for Versatile Multivariant Time Series Learning Tasks

This paper introduces Tensor Visibility Graph-enhanced Attention Networks (TVGeAN), a novel graph autoencoder model specifically designed for MTS learning tasks. The underlying approach of TVGeAN is to combine the power of complex networks in representing time series as graphs with the strengths of Graph Neural Networks (GNNs) in learning from graph data. TVGeAN consists of two new main components: TVG which extend the capabilities of visibility graph algorithms in representing MTSs by converting them into weighted temporal graphs where both the nodes and the edges are tensors. Each node in the TVG represents the MTS observations at a particular time, while the weights of the edges are defined based on the visibility angle algorithm. The second main component of the proposed model is GeAN, a novel graph attention mechanism developed to seamlessly integrate the temporal interactions represented in the nodes and edges of the graphs into the core learning process. GeAN achieves this by using the outer product to quantify the pairwise interactions of nodes and edges at a fine-grained level and a bilinear model to effectively distil the knowledge interwoven in these representations. From an architectural point of view, TVGeAN builds on the autoencoder approach complemented by sparse and variational learning units. The sparse learning unit is used to promote inductive learning in TVGeAN, and the variational learning unit is used to endow TVGeAN with generative capabilities. The performance of the TVGeAN model is extensively evaluated against four widely cited MTS benchmarks for both supervised and unsupervised learning tasks. The results of these evaluations show the high performance of TVGeAN for various MTS learning tasks. In particular, TVGeAN can achieve an average root mean square error of 6.8 for the C-MPASS dataset (i.e., regression learning tasks) and a precision close to one for the SMD, MSL, and SMAP datasets (i.e., anomaly detection learning tasks), which are better results than most published works.

Adaptive Convergent Visibility Graph Network: An interpretable method for intelligent rolling bearing diagnosis

In the domain of mechanical equipment maintenance, the necessity for efficient and accurate fault diagnosis is critical. Traditional Graph Neural Network (GNN) methods, which employ time-series data for fault diagnosis, have proven effective but are far from perfect. Their common pitfall lies in mapping time-series data into graph data, often leading to loss of crucial temporal information and computational inefficiencies. These limitations could result in suboptimal diagnosis, potentially compromising the longevity and performance of mechanical systems. Driven by the need to improve on these limitations, we introduce ACVGN, a novel end-to-end intelligent diagnostic framework that unites the strengths of the Adaptive Convergent Visibility Graph (ACVG) algorithm and an enhanced DiffPool model. The cornerstone of our approach, the ACVG algorithm, adeptly transforms time-series data into graph format, thereby preserving both local and global dynamics from the original data. This rich representation is then processed by our improved DiffPool model, a powerful GNN model purposefully designed for high-accuracy classification tasks. The effectiveness of the proposed ACVGN framework is substantiated by its performance on the widely-used rolling bearing dataset, where it outshines existing methods in terms of both mapping efficiency and fault diagnosis accuracy. These promising results not only reinforce the effectiveness of our proposed method but also highlight the potential for its wider applicability in other scenarios involving time-series data analysis and graph-based machine learning tasks. In conclusion, our study advances the development of more intelligent, efficient, and precise diagnostic tools for mechanical devices, ensuring more effective fault detection and diagnosis, and thereby potentially improving the lifespan and functionality of mechanical systems.

Multiple serial correlations in global air temperature anomaly time series.

Serial correlations within temperature time series serve as indicators of the temporal consistency of climate events. This study delves into the serial correlations embedded in global surface air temperature (SAT) data. Initially, we preprocess the SAT time series to eradicate seasonal patterns and linear trends, resulting in the SAT anomaly time series, which encapsulates the inherent variability of Earth's climate system. Employing diverse statistical techniques, we identify three distinct types of serial correlations: short-term, long-term, and nonlinear. To identify short-term correlations, we utilize the first-order autoregressive model, AR(1), revealing a global pattern that can be partially attributed to atmospheric Rossby waves in extratropical regions and the Eastern Pacific warm pool. For long-term correlations, we adopt the standard detrended fluctuation analysis, finding that the global pattern aligns with long-term climate variability, such as the El Niño-Southern Oscillation (ENSO) over the Eastern Pacific. Furthermore, we apply the horizontal visibility graph (HVG) algorithm to transform the SAT anomaly time series into complex networks. The topological parameters of these networks aptly capture the long-term correlations present in the data. Additionally, we introduce a novel topological parameter, Δσ, to detect nonlinear correlations. The statistical significance of this parameter is rigorously tested using the Monte Carlo method, simulating fractional Brownian motion and fractional Gaussian noise processes with a predefined DFA exponent to estimate confidence intervals. In conclusion, serial correlations are universal in global SAT time series and the presence of these serial correlations should be considered carefully in climate sciences.

An approach to complex network analysis on pp collisions at LHC energies

We have introduced the complex network analysis techniques such as Visibility Graph (VG), Horizontal Visibility Graph (HVG) and Sandbox (SB) algorithm as a mapping between complex network and time series measurements. This study explores the application of complex network-based analysis techniques to proton–proton [Formula: see text] collisions data at [Formula: see text], 7, 5.02 and 2.76[Formula: see text][Formula: see text]TeV to unravel the underlying dynamics of the particles and their interactions. To ensure the robustness and comprehensive analysis, we have incorporated UrQMD-, PYTHIA- and AMPT-simulated data sets of [Formula: see text] interactions at LHC energies that enable a thorough investigation of the underlying phenomena of the interactions. We have analyzed the fluctuations in pseudorapidity densities of singly charged particles created in [Formula: see text] interactions at LHC energies. We have also investigated some properties like symmetry scaling, correlation, clustering and scale-freeness among the produced particles. The findings from this study contribute to our understanding of the complex dynamics within [Formula: see text] collisions and have implications for the field of high-energy physics and particle phenomenology.

Rapeseed Seed Coat Color Classification Based on the Visibility Graph Algorithm and Hyperspectral Technique

Information technology and statistical modeling have made significant contributions to smart agriculture. Machine vision and hyperspectral technologies, with their non-destructive and real-time capabilities, have been extensively utilized in the non-destructive diagnosis and quality monitoring of crops and seeds, becoming essential tools in traditional agriculture. This work applies these techniques to address the color classification of rapeseed, which is of great significance in the field of rapeseed growth diagnosis research. To bridge the gap between machine vision and hyperspectral technology, a framework is developed that includes seed color calibration, spectral feature extraction and fusion, and the recognition modeling of three seed colors using four machine learning methods. Three categories of rapeseed coat colors are calibrated based on visual perception and vector-square distance methods. A fast-weighted visibility graph method is employed to map the spectral reflectance sequences to complex networks, and five global network attributes are extracted to fuse the full-band reflectance as model input. The experimental results demonstrate that the classification recognition rate of the fused feature reaches 0.943 under the XGBoost model, confirming the effectiveness of the network features as a complement to the spectral reflectance. The high recognition accuracy and simple operation process of the framework support the further application of hyperspectral technology to analyze the quality of rapeseed.

The network structure of Sturmian sequences based on HVG

The horizontal visibility graph algorithm is a powerful tool to study time series. In this paper, we use this algorithm maps Sturmian sequences to complex networks and find that the degree sequences partly inherit the Sturmian character. Firstly, we prove that Sturmian sequences and their horizontal visibility graph (HVG) degree sequences can be generated separately by coding sequences. Then, using coding factors, we divide the Sturmian sequences of type 1 into six types and calculate the complexity functions of their HVG-degree sequences. Moreover, we show that the HVG-degree sequences of Sturmian sequences of type 0 are the same Sturmian sequence. Finally, we use the complexity functions of HVG-degree sequences to uniquely characterize the Sturmian sequences.

A generalized visibility graph algorithm for analyzing biological time series having rotation in polar plane

Time series analysis is exploited when temporal measurements about a phenomenon exist. Visibility graphs are a method for representing and analyzing time series. One of the challenges biological time series introduce is rotation in polar plane. Rotation in polar plane is a change in the angle of the time series in polar plane. Rotation in the polar plane exists in any time series that has positive and negative observations together. In this research, a visibility graph algorithm is proposed which could efficiently handle biological time series which have rotation in polar plane. In the proposed algorithm, two observation in the time series are visible to each other in the graph, if the line connecting the points doesn't cross any other point placed between them. The proposed visibility graph algorithm has been evaluated with synthetic and real world time series. The generated visibility graphs are used to evaluate the proposed visibility graph algorithm with different metrics such as pearson autocorrelation and classification accuracy. The synthetic time series are generated by a bipolar feedback process. In one of the synthetic bios series with g = 10, the pearson autocorrelation of the graph is 0.9815 whereas the baseline is 0.9211. In the main experiment with real world time series, the proposed visibility graph algorithm is compared with the state of the art algorithm. The ANOVA test is performed on the results of 60 experiments. The p-value is smaller than 0.05, which states GVG has improved CLPVG significantly.

Degree distributions and motif profiles of Thue–Morse complex network

In this paper, we study both the horizontal visibility graph (HVG) and the limited penetrable horizontal visibility graph (LPHVG) mapped from the Thue–Morse sequence. Firstly, we map the series to complex networks by using visibility graph algorithms. Then, we obtain the analytical degree distribution of the Thue–Morse HVG through an iterative method. And we also derive the degree distribution of the Thue–Morse LPHVG with a penetrable distance of ρ=1. Finally, we investigate the profile of sequential 4-node motifs for the Thue–Morse HVG. The relative frequencies of the 4-node motifs for the Thue–Morse HVG converge to a constant vector (12,16,16,16,0,0). The numerical simulations coincide excellently with theoretical results of the degree distributions and motif profiles.

Investigation of the Global Stock Trading Based on Visibility Graph and Entropy Weight Method

The increasing complexity and dynamics of the stock trading market are major challenges for the financial industry and are primary dilemmas for all countries nowadays. In addition, the stock trading market has a considerable impact on the global economy, and its importance is self-evident. To cope with the complexity and dynamics of a stock trading market, this paper applies complex network theory and model to explore the topology of the global stock trading network. First, this paper collects stock trading data from 74 countries from 1999 to 2020. It converts the collected stock trading data of these countries into a complex network using a type of algorithm based on the time series visibility graph (VG) algorithm. Then, the data are analyzed by a complex network model, and six analytical metrics are obtained. Finally, the six metrics are analyzed by the entropy weight method to identify the key nodes in the network and to obtain the ranking of each country’s stock trading data. This paper is an effective application of complex network and entropy weight method in stock trend analysis, which mainly includes two contributions. First, the VG algorithm provides a novel research perspective for modeling the global stock trading trend. Second, key nodes in the network are analyzed and identified based on the entropy weight method, and the ranking of key nodes in the stock trading network is obtained, which provides a new method for further research on the stock trading trend, investment portfolio, and stock return forecasting.

Identifying nonlinear interactions between PM2.5 and O3 on small time scales with visibility graph approaches

Identifying nonlinear interactions between PM2.5 and O3 is crucial for fully understanding atmospheric pollution. These interactions between PM2.5 and O3 involve complex chemical processes and boundary layer dynamics under different meteorological drivers. Due to the deficiency of traditional linear statistical methods (e.g. Pearson correlation or Pearson partial correlation analysis), they fail in quantifying nonlinear interactions between PM2.5 and O3 on small time scales and nonlinear features in PM2.5 and O3 series. We adopt two statistical indicators based on visibility graph (VG) algorithm to analyze the PM2.5 and O3 time series: average edge overlap (AEO) as measure for nonlinear interactions and Knp as measure for cross-scale interaction by frequency modulation (FM). Detail studies show that AEO is insensitive to large-scale nonstationary variations but indeed sensitive to nonlinear coupling variations in the tailored models with known prescribed coupling. The AEO can successfully identify true both linear and nonlinear interactions in these tailored models, while Pearson correlation analysis fails. Then AEO is applied to reanalysis data of PM2.5 and O3 to identify events with specific nonlinear interactions between PM2.5 and O3 with significant cross-scale interactions quantified by calculated Knp, which exist in both within and between these pollutant time series. Further analysis shows that the possible mechanisms underlying these nonlinear interactions may be mainly local mixing (or emission) and chemical processes related to meteorological drivers such as surface air temperature, surface solar radiation and surface relative humidity.

Detection of EEG signals in normal and epileptic seizures with multiscale multifractal analysis approach via weighted horizontal visibility graph

Electroencephalogram (EEG) signals contain important information about the regulation of brain system. Thus, automatic detection of epilepsy by analyzing the characteristics obtained from EEG signals has important research implications in the field of clinical medicine. In this paper, the horizontal visibility graph (HVG) algorithm is used to map multifractal EEG signals into complex networks. Then, we study the structure of the networks and explore the nonlinear dynamics properties of the EEG signals inherited from these networks. In order to better describe complex brain behaviors, we use the angle between two connected nodes as the edge weight of the network and construct the weighted horizontal visibility graph (WHVG). In our studies, fractality and multifractality of WHVG are innovatively used to analyze the structure of related networks. However, these methods only analyze the reconstructed dynamical system in general characterizations, they are not sufficient to describe the complex behavior and cannot provide a comprehensive picture of the system. To this effect, we propose an improved multiscale multifractal analysis (MMA) for network, which extends the description of the network dynamics features by focusing on the relationship between the multifractality and the measured scale-free intervals. Furthermore, neural networks are applied to train the above-mentioned parameters for the classification and identification of three kinds of EEG signals, i.e., health, interictal phase, and ictal phase. By evaluating our experimental results, the classification accuracy is 99.0%, reflecting the effectiveness of the WHVG algorithm in extracting the potential dynamic characteristics of EEG signals.

Classification of healthy and epileptic seizure EEG signals based on different visibility graph algorithms and EEG time series

Classification of healthy and epileptic seizure EEG signals based on different visibility graph algorithms and EEG time series

Visibility Graph Analysis of Heartbeat Time Series: Comparison of Young vs. Old, Healthy vs. Diseased, Rest vs. Exercise, and Sedentary vs. Active.

Using the visibility graph algorithm (VGA), a complex network can be associated with a time series, such that the properties of the time series can be obtained by studying those of the network. Any value of the time series becomes a node of the network, and the number of other nodes that it is connected to can be quantified. The degree of connectivity of a node is positively correlated with its magnitude. The slope of the regression line is denoted by k-M, and, in this work, this parameter was calculated for the cardiac interbeat time series of different contrasting groups, namely: young vs. elderly; healthy subjects vs. patients with congestive heart failure (CHF); young subjects and adults at rest vs. exercising young subjects and adults; and, finally, sedentary young subjects and adults vs. active young subjects and adults. In addition, other network parameters, including the average degree and the average path length, of these time series networks were also analyzed. Significant differences were observed in the k-M parameter, average degree, and average path length for all analyzed groups. This methodology based on the analysis of the three mentioned parameters of complex networks has the advantage that such parameters are very easy to calculate, and it is useful to classify heartbeat time series of subjects with CHF vs. healthy subjects, and also for young vs. elderly subjects and sedentary vs. active subjects.

Characterizing the Solar Activity Using the Visibility Graph Method

In this paper, the Sun and its behavior are studied by means of complex networks. The complex network was built using the Visibility Graph algorithm. This method maps time series into graphs in which every element of the time series is considered as a node and a visibility criterion is defined in order to connect them. Using this method, we construct complex networks for magnetic field and sunspots time series encompassing four solar cycles, and various measures such as degree, clustering coefficient, mean path length, betweenness centrality, eigenvector centrality and decay exponents were calculated. In order to study the system in several time scales, we perform both a global, where the network contains information on the four solar cycles, and a local analysis, involving moving windows. Some metrics correlate with solar activity, while others do not. Interestingly, those metric which seem to respond to varying levels of solar activity in the global analysis, also do in the moving windows analysis. Our results suggest that complex networks can provide a useful way to follow solar activity, and reveal new features on solar cycles.

Visibility graph for time series prediction and image classification: a review.

The analysis of time series and images is significant across different fields due to their widespread applications. In the past few decades, many approaches have been developed, including data-driven artificial intelligence methods, mechanism-driven physical methods, and hybrid mechanism and data-driven models. Complex networks have been used to model numerous complex systems due to its characteristics, including time series prediction and image classification. In order to map time series and images into complex networks, many visibility graph algorithms have been developed, such as horizontal visibility graph, limited penetrable visibility graph, multiplex visibility graph, and image visibility graph. The family of visibility graph algorithms will construct different types of complex networks, including (un-) weighted, (un-) directed, and (single-) multi-layered networks, thereby focusing on different kinds of properties. Different types of visibility graph algorithms will be reviewed in this paper. Through exploring the topological structure and information in the network based on statistical physics, the property of time series and images can be discovered. In order to forecast (multivariate) time series, several variations of local random walk algorithms and different information fusion approaches are applied to measure the similarity between nodes in the network. Different forecasting frameworks are also proposed to consider the information in the time series based on the similarity. In order to classify the image, several machine learning models (such as support vector machine and linear discriminant) are used to classify images based on global features, local features, and multiplex features. Through various simulations on a variety of datasets, researchers have found that the visibility graph algorithm outperformed existing algorithms, both in time series prediction and image classification. Clearly, complex networks are closely connected with time series and images by visibility graph algorithms, rendering complex networks to be an important tool for understanding the characteristics of time series and images. Finally, we conclude in the last section with future outlooks for the visibility graph.

A graph structure feature-based framework for the pattern recognition of the operational states of integrated energy systems

The recognition of operational state patterns is crucial for the safe, reliable and profitable operational management of integrated energy systems (IES). However, considering that the monitored operational data are often not labelled, supervised machine learning classification methods cannot be directly applied. In this study, based on the idea of graph theory, a framework of interpretable time series pattern recognition methods based on graph structural features is proposed. Firstly, the time series data is segmented into local subsequences by a shapelet-based time series classification method. Secondly, time-graphlets are constructed by means of a visibility graph algorithm and an original graph structure feature-based representation method is proposed. Then, pattern recognition of the different operational states of IES is performed in the form of multidimensional feature vectors. Thirdly, an association rule model is designed to analyze the supply and demand fluctuations, price fluctuations and couplings between multiple energy subsystems, based on which the pattern recognition results are annotated to match the operational characteristics of the IES. Fourthly, a shapelet-based state transfer network graph is constructed, and the complex network centrality metrics are used to analyze the operating characteristics and key operating states of the system over time. The new method is applied for pattern recognition of operational states in a Spanish IES. The results show that the proposed method can increase the interpretability of the recognition results and provide explanatory labels for the operational data. Furthermore, the vulnerable operational states of the system can be revealed, which allows managers to monitor the operational state of the system and take early defensive measures against the more threatening operational patterns. This suggests that the methodological framework presented in this paper is a promising approach to pattern recognition that can provide interpretable annotations for unsupervised operational data of IES and their safe management.

The visibility graph of n-bonacci sequence

In this paper, we study both the visibility graph and horizontal visibility graph of n-bonacci sequence. Firstly, we map Fibonacci sequence to complex network by using visibility graph algorithm, and its degree sequence is related to some combinatorial properties of words. Then, we study the visibility graph degree distribution of n-bonacci sequence by coding words. We obtain that its degree distribution is between the exponential and power-law distributions. On the other hand, the horizontal visibility graph sequences of n-bonacci sequence over different alphabets correspond to the same n-bonacci sequence, and its degree distribution tends to exponential distribution as n→∞. Finally, we explain the reason why fractal sequences are mapped into scale-free networks.

Network Theory Based EHG Signal Analysis and its Application in Preterm Prediction.

Preterm birth is the leading cause of neonatal morbidity and mortality. Early identification of high-risk patients followed by medical interventions is essential to the prevention of preterm birth. Based on the relationship between uterine contraction and the fundamental electrical activities of muscles, we extracted effective features from EHG signals recorded from pregnant women, and use them to train classifiers with the purpose of providing high precision in classifying term and preterm pregnancies. To characterize changes from irregularity to coherence of the uterine activity during the whole pregnancy, network representations of the original electrohysterogram (EHG) signals are established by applying the Horizontal Visibility Graph (HVG) algorithm, from which we extract network degree density and distribution, clustering coefficient and assortativity coefficient. Concerns on the interferences of different noise sources embedded in the EHG signal, we apply Short-Time Fourier Transform (STFT) to expand the original signal in the time-frequency domain. This allows a network representation and the extraction of related features on each frequency component. Feature selection algorithms are then used to filter out unrelated frequency components. We further apply the proposed feature extraction method to EHG signals available in the Term-Preterm EHG database (TPEHG), and use them to train classifiers. We adopt the Partition-Synthesis scheme which splits the original imbalanced dataset into two sets, and synthesizes artificial samples separately within each subset to solve the problem of dataset imbalance. The optimally selected network-based features, not only contribute to the identification of the essential frequency components of uterine activities related to preterm birth, but also to improved performance in classifying term/preterm pregnancies, i.e., the SVM (Support Vector Machine) classifier trained with the available samples in the TPEHG gives sensitivity, specificity, overall accuracy, and auc values as high as 0.89, 0.93, 0.91, and 0.97, respectively.

Global competitiveness analysis of industrial robot technology innovations market layout using visibility graph

The competition degree and characteristics of the target markets based on a global vision provide references for market entrants and policymakers. This study describes a method for identifying the global technology innovations market layout competitiveness based on time series data analysis. We map the time series data from 30 countries or areas of the global industrial robot technology innovations market through retrieval strategy and data mining onto complex networks by visibility graph algorithm. We analyze the dynamic characteristics by the VGNs’ topological measures after the development trend overview and stage division analyses. We use cosine similarity to evaluate the differences and similarities between countries or areas. To further uncover the relationships among them, a similarity complex network is constructed by setting a link threshold. Seven community categories as sub-markets are found through community division. CN and US rank as the top two largest industrial robot innovation markets. Most European countries share the same community because of their similarity of economic development brought by geographical proximity except for several earlier developed economies such as DE, GB, FR, and IT. Some catching-up countries, for example, IN and PH, show potential similar dynamic trends respectively in their group for the sharing characteristics probably with the similar economic development type as the reason behind, whereas RU is distinct for its unique economy type.