Horizontal Visibility Graph Research Articles

Statistical Exploration of the Salar de Atacama’s: Brine Measurements of the Basin Wells

Abstract In this study, we conduct a statistical analysis using the 4-plot, visibility graph, and horizontal visibility graph methods on brine extraction data from the Salar de Atacama basin in Chile. The 4-plot reveals real trends in the data that could lead model proposals. Conversely, complex networks analysis yields no significant findings, suggesting that the data lacks internal structural features and appears random. This randomness underscores the decision-making processes and highlights areas for potential optimization.

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.

Earlier smart prediction of diabetic retinopathy from fundus image under innovative ResNet optimization maneuver

AbstractDiabetic retinopathy (DR) is a complication of diabetes that causes blindness and the early detection of diabetics using retinopathy images remains a challenging task. Hence, a novel, earlier smart prediction of diabetic retinopathy from fundus image under Innovative ResNet Optimization is introduced to effectively detect the earlier stage of DR from Fundus image. Initially, the fundus image is scaled during preprocessing and converted into a grayscale format. As the existing studies neglect some deserving unique features that are crucial for predicting the earliest signs of DR, a novel Fractional Radon Transform with Visibility Graph is introduced for extracting the novel features such as microaneurysms count, dot and blot hemorrhages count, statistical measures, and retinal layer thickness, in which a Generalized Cosine Fractional Radon Transform is used to capture the image’s fine-scale texture information thereby effectively capturing the statistical measures, while a weighted Horizontal Visibility Graph is made to examine the apparent spatial relationships between pixel pairs in the image based on the values of the pixels’ gray levels. Further, the existing works failed to identify the small fine dark areas that were ignored throughout the morphological opening process. In order to overcome this issue, a Morphological Black Hat Transform with Optimized ResNet Algorithm is implemented, where segmentation is made through Enriched Black Hat Transform-based Morphological operation to identify fine dark regions among the pixels inside the eye samples, and the classification is done by using ResNet-driven S-GOA (Socio Grasshopper Optimization Algorithm), to optimally predict the stages of DR. The result obtained showed that the proposed model outperforms existing techniques with high performance and accuracy.

MDDBranchNet: A Deep Learning Model for Detecting Major Depressive Disorder Using ECG Signal.

Major depressive disorder (MDD) is a chronic mental illness which affects people's well-being and is often detected at a later stage of depression with a likelihood of suicidal ideation. Early detection of MDD is thus necessary to reduce the impact, however, it requires monitoring vitals in daily living conditions. EEG is generally multi-channel and due to difficulty in signal acquisition, it is unsuitable for home-based monitoring, whereas, wearable sensors can collect single-channel ECG. Classical machine-learning based MDD detection studies commonly use various heart rate variability features. Feature generation, which requires domain knowledge, is often challenging, and requires computation power, often unsuitable for real time processing, MDDBranchNet is a proposed parallel-branch deep learning model for MDD binary classification from a single channel ECG which uses additional ECG-derived signals such as R-R signal and degree distribution time series of horizontal visibility graph. The use of derived branches was able to increase the model's accuracy by around 7%. An optimal 20-second overlapped segmentation of ECG recording was found to be beneficial with a 70% prediction threshold for maximum MDD detection with a minimum false positive rate. The proposed model evaluated MDD prediction from signal excerpts, irrespective of location (first, middle or last one-third of the recording), instead of considering the entire ECG signal with minimal performance variation stressing the idea that MDD phenomena are likely to manifest uniformly throughout the recording.

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.

Distinction of Chaos from Randomness Is Not Possible from the Degree Distribution of the Visibility and Phase Space Reconstruction Graphs.

We investigate whether it is possible to distinguish chaotic time series from random time series using network theory. In this perspective, we selected four methods to generate graphs from time series: the natural, the horizontal, the limited penetrable horizontal visibility graph, and the phase space reconstruction method. These methods claim that the distinction of chaos from randomness is possible by studying the degree distribution of the generated graphs. We evaluated these methods by computing the results for chaotic time series from the 2D Torus Automorphisms, the chaotic Lorenz system, and a random sequence derived from the normal distribution. Although the results confirm previous studies, we found that the distinction of chaos from randomness is not generally possible in the context of the above methodologies.

Exploring the Entropy-Based Classification of Time Series Using Visibility Graphs from Chaotic Maps

The classification of time series using machine learning (ML) analysis and entropy-based features is an urgent task for the study of nonlinear signals in the fields of finance, biology and medicine, including EEG analysis and Brain–Computer Interfacing. As several entropy measures exist, the problem is assessing the effectiveness of entropies used as features for the ML classification of nonlinear dynamics of time series. We propose a method, called global efficiency (GEFMCC), for assessing the effectiveness of entropy features using several chaotic mappings. GEFMCC is a fitness function for optimizing the type and parameters of entropies for time series classification problems. We analyze fuzzy entropy (FuzzyEn) and neural network entropy (NNetEn) for four discrete mappings, the logistic map, the sine map, the Planck map, and the two-memristor-based map, with a base length time series of 300 elements. FuzzyEn has greater GEFMCC in the classification task compared to NNetEn. However, NNetEn classification efficiency is higher than FuzzyEn for some local areas of the time series dynamics. The results of using horizontal visibility graphs (HVG) instead of the raw time series demonstrate the GEFMCC decrease after HVG time series transformation. However, the GEFMCC increases after applying the HVG for some local areas of time series dynamics. The scientific community can use the results to explore the efficiency of the entropy-based classification of time series in “The Entropy Universe”. An implementation of the algorithms in Python is presented.

Crack pattern identification in cementitious materials based on acoustic emission and machine learning

The cracking patterns in cementitious materials before failure are closely related to acoustic emission (AE) monitoring signals. Traditional rise angle-average frequency analysis methods rely on empirical judgment for boundary line determination, lacking effective automated recognition methods to distinguish between tensile, shear, and mixed crack types. This paper presents an unsupervised crack-type identification model based on ensembled clustering with support vector machine (SVM) and a supervised crack-type classification model using graph neural networks (GNN). The study achieves automatic annotation of four crack signal patterns by ensembling the results of k-means++ and GMMSEQ clustering and combining them with SVM. Subsequently, with labeled data obtained, the study encodes AE signals into graphs and uses a GNN for automatic crack-type classification. The effectiveness of these methods is validated on AE test data from cementitious materials. The findings show that the mutation values of AE event interval functions can accurately pinpoint the material fracture moment, and the AE spectrum’s mid-frequency band is associated with cement–river sand interface damage. Different compositions and loading methods in cementitious materials complicate the association between signals and damage. The ensembled clustering and SVM methods can automatically distinguish different types of cracks and provide quantifiable boundary lines. With horizontal visibility graph (HVG) encoding, the supervised GraphSAGE model accurately identifies various crack types, achieving an average accuracy of up to 97.5% on the test dataset. The proposed AE time–frequency domain characterization method offers new analytical tools for understanding the damage evolution process in cementitious materials, promoting the application of machine learning in the recognition of AE signal patterns.

Power transformer fault diagnosis using dynamic multiscale graph modeling and M2SGCN network based on statistical fusion

Power equipment fault diagnostics hold significant importance for the stability of power grid systems. In pursuit of this objective, this paper proposes a fault diagnosis method that utilizes dynamic multiscale graph (DMG) modeling and the multiscale multi-stream GCN(M2SGCN) network, incorporating statistical fusion. Specifically, a novel DMG modeling method is proposed to derive visibility graph data and horizontal visibility graph data from vibration signals across multiple scales. Next, a comprehensive neural network architecture named M2SGCN is established to learn global and local features simultaneously, providing a more precise representation. Subsequently, a Dempster Shafer evidence theory statistical fusion technique combined with an adaptive threshold model (DSTFusion) is utilized to integrate primary decision results for enhanced fault diagnosis accuracy. In addition, two datasets obtained from single-phase and three-phase power transformers are analyzed to demonstrate the evolution process. When compared to state-of-the-art indicators such as accuracy, precision, recall, and F1 scores, the method proposed excels in multiple aspects, successfully detecting fault states before their occurrence and achieving outstanding performance.

Direct lingam and visibility graphs for analyzing brain connectivity in BCI.

The brain-computer interface (BCI) is a direct pathway of communication between the electrical activity of the brain and an external device. The present paper was aimed to investigate directed connectivity between different areas of the brain during motor imagery (MI)-based BCI. For this purpose, two methods were implemented including, Limited Penetrable Horizontal Visibility Graph (LPHVG) and Direct Lingam. The visibility graph (VG) is a robust algorithm for analyzing complex systems such as the brain. Direct Lingam uses a non-Gaussian model to extract causal links which is appropriate for analyzing large-scale connectivity. First, LPHVG map MI-EEG (electroencephalogram) signals into networks. After extracting the topological features of the networks, a support vector machine classifier was applied to categorize multi-classes MI. The network of all classes was found to be different from one another, and the kappa value of classification was 0.68. The degree sequence of LPHVG was calculated for each channel in order to obtain the direction of brain information flow. Transfer entropy (TE) is used to compute the relations of the channel degree sequence. Therefore, the directed graph between channels was formed. This method is called LPHVG_TE directed graph. The Bayesian network, also known as the Direct LiNGAM model, was implemented for the second method. Finally, images of the LPHVG and Direct Lingam were classified by convolutional neural network (CNN). In this study, Data sets 2a of BCI competition IV was used. The outcomes reveal that the brain network developed by LPHVG (92.7%) might be more effective to distinguish 4 classes of MI than the Direct Lingam (90.6%) and it was shown that graph theory has the potential to get better efficiency of BCI.

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.

Multivariate emotional EEG signal recognition based on multivariate joint motif entropy of a horizontal visibility graph

Multivariate emotional EEG signal recognition based on multivariate joint motif entropy of a horizontal visibility graph

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.

Graph- and Machine-Learning-Based Texture Classification

The analysis of textures is an important task in image processing and computer vision because it provides significant data for image retrieval, synthesis, segmentation, and classification. Automatic texture recognition is difficult, however, and necessitates advanced computational techniques due to the complexity and diversity of natural textures. This paper presents a method for classifying textures using graphs; specifically, natural and horizontal visibility graphs. The related image natural visibility graph (INVG) and image horizontal visibility graph (IHVG) are used to obtain features for classifying textures. These features are the clustering coefficient and the degree distribution. The suggested outcomes show that the aforementioned technique outperforms traditional ones and even comes close to matching the performance of convolutional neural networks (CNNs). Classifiers such as the support vector machine (SVM), K-nearest neighbor (KNN), decision tree (DT), and random forest (RF) are utilized for the categorization. The suggested method is tested on well-known image datasets like the Brodatz texture and the Salzburg texture image (STex) datasets. The results are positive, showing the potential of graph methods for texture classification.

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.

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.

Denoising graph neural network based hydraulic component fault diagnosis method

Accurately identifying faults in hydraulic components is essential for improving the reliability and safety of hydraulic systems. Since the hydraulic components of heavy-duty equipment usually work in complex and variable environments and are subject to interference from multiple sources of noise. As a result, fault diagnosis of hydraulic components is challenging, leading to the development of various fault diagnosis models. However, current hydraulic component fault diagnosis models have the following limitations: (1) only able to handle Euclidean space data and ignore the topological relationships within the signals that could provide additional useful information for fault diagnosis, (2) only diagnose faults using single-dimensional acceleration data and fail to incorporate more equipment working conditions. This study focuses on the hydraulic directional valve and proposes a fault diagnosis method based on denoising graph neural network. Firstly, computational fluid dynamics(CFD) is used to analyze the flow field of the spool and sleeve of the valve under different wear states and working conditions. It is found that different degrees of wear have similarities under different working conditions, and there is a regular correlation between the acceleration signal of the valve body and the fault mode under the same working condition. Therefore, acceleration and working condition information are selected as the raw information for fault diagnosis. Then, horizontal visibility graph(HVG) was utilized to transform acceleration signal to graph, and the graph nodes, which can incorporate multidimensional information, were set to be the equipment working conditions and acceleration values. However, the noise in the raw signal could seriously affect the quality of the HVG. To solve this problem, an adaptive threshold denoising algorithm was proposed to reduce the interference noise in the acceleration signal. The denoising threshold parameter was learned directly from the raw data. Finally, graph neural network(GNN) is applied to fault diagnosis of hydraulic components from non-Euclidean multidimensional information. Experimental results demonstrate the superior robustness and accuracy of this method in handling strong noise and multiple working conditions compared to several classical fault diagnosis methods. This research provides an effective approach for hydraulic component fault diagnosis.

Horizontal visibility graphs mapped from multifractal trinomial measures

Horizontal visibility graphs mapped from multifractal trinomial measures

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.

A review of symbolic dynamics and symbolic reconstruction of dynamical systems

Discretizing a nonlinear time series enables us to calculate its statistics fast and rigorously. Before the turn of the century, the approach using partitions was dominant. In the last two decades, discretization via permutations has been developed to a powerful methodology, while recurrence plots have recently begun to be recognized as a method of discretization. In the meantime, horizontal visibility graphs have also been proposed to discretize time series. In this review, we summarize these methods and compare them from the viewpoint of symbolic dynamics, which is the right framework to study the symbolic representation of nonlinear time series and the inverse process: the symbolic reconstruction of dynamical systems. As we will show, symbolic dynamics is currently a very active research field with interesting applications.