AvgNet: Adaptive Visibility Graph Neural Network and Its Application in Modulation Classification

“In the simulations, we use cross-entropy loss function and Adam [44] optimizer.”

“The former is used to describe the topological structure of many complex systems, where nodes represent subsystems and links capture the relationship between them, leading to the emergence of complex networks [1], [2]; while the latter is used to capture the temporal dynamics of these systems.”

“These graphs are drawn by Gephi [44], and the layout is OpenOrd [44].”

“It is worth noting that, due to the great success of GNN [34], [35], [36], [37] and effective graph mapping methods, it would be also an exciting direction to first map time series into graphs and then utilize powerful GNN to classify them.”

“Fortunately, by introducing graph data mining technologies, we can detect and filter the false information and purify the Internet environment to certain extent [8], [9].”

“What’s more, we use PyTorch Geometric [43], a geometric deep learning extension library for PyTorch, to build the neural network layers of DiffPool in our AVGNet model.”

“It is worth noting that, due to the great success of GNN [34], [35], [36], [37] and effective graph mapping methods, it would be also an exciting direction to first map time series into graphs and then utilize powerful GNN to classify them.”

“[38] proposed the natural Visibility Graph (VG) algorithm for mapping time series into graphs to analyze time series from a new point of view, whose process is shown in Fig.”

“Besides, O’Shea et al. [30] also used a residual neural network for modulation classification, namely CNN1D.”

“In the simulation, we mainly use GAF, MTF, CD, CNN2D and CNN1D mentioned in Sec.”

“• X. Yang is with the Institute of Cyberspace Security, Zhejiang University of Technology, Hangzhou 310023, China, and also with the Science and Technology on Communication Information Security Control Laboratory, Jiaxing 314033, China (E-mail: yxn2117@126.com). the graph kernel methods, which are mainly based on rand…”

“[26] proposed a two-layer Long Short Term Memory (LSTM) model for automatic modulation classification, which has considerable accuracy but is relatively time-consuming.”

“…Security Control Laboratory, Jiaxing 314033, China (E-mail: yxn2117@126.com). the graph kernel methods, which are mainly based on random walk [10], path [11] and subtree [12]; The second one is graph embedding methods, such as subgraph2vec [13] and graph2vec [14]; And the last one is deep…”

“• X. Yang is with the Institute of Cyberspace Security, Zhejiang University of Technology, Hangzhou 310023, China, and also with the Science and Technology on Communication Information Security Control Laboratory, Jiaxing 314033, China (E-mail: yxn2117@126.com). the graph kernel methods, which are mainly based on rand…”

“[39] proposed the improved mapping algorithm, Horizontal Visibility Graph (HVG), and Gao et al.”

“It is worth noting that, due to the great success of GNN [34], [35], [36], [37] and effective graph mapping methods, it would be also an exciting direction to first map time series into graphs and then utilize powerful GNN to classify them.”

“The representative methods of graph classification can be divided into three types, the first one is the graph kernel methods, which are mainly based on random walk [10], path [11] and subtree [12]; The second one is graph embedding methods, such as subgraph2vec [13] and graph2vec [14]; And the last one is deep learni…”

“In the simulation, we mainly use GAF, MTF [28], CD [27], CNN2D [29], CNN1D [30], MCLDNN [31] and SigNet [32] mentioned in Sec.”

“In the simulation, we mainly use GAF, MTF [28], CD [27], CNN2D [29], CNN1D [30], MCLDNN [31] and SigNet [32] mentioned in Sec.”

“At present, biology, social science, computer science and other disciplines have a wealth of graph data [3], [4], [5], so that researchers can make full use of network analysis technologies to solve related problems in the corresponding fields.”

“Fortunately, by introducing graph data mining technologies, we can detect and filter the false information and purify the Internet environment to certain extent [8], [9].”

“For example, in the field of life science, there are interactions between biological molecules, based on which the complex biological networks can be established and the molecular properties can thus be predicted [6], [7].”

“The first is to use the RNN model to classify the signals, specifically, we use GRU and LSTM mentioned in Sec.”

“[33] used the spectrogram in the frequency domain feature to help classify.”

“For time series, here we mainly focus on modulation classification of radio signals [18], [19], [20], [21].”

“The representative methods of graph classification can be divided into three types, the first one is the graph kernel methods, which are mainly based on random walk [10], path [11] and subtree [12]; The second one is graph embedding methods, such as subgraph2vec [13] and graph2vec [14]; And the last one is deep learni…”

“[40] further introduced Limited Penetrable Visibility Graph (LPVG).”

“In the simulation, we mainly use GAF, MTF [28], CD [27], CNN2D [29], CNN1D [30], MCLDNN [31] and SigNet [32] mentioned in Sec.”

“For example, in the field of life science, there are interactions between biological molecules, based on which the complex biological networks can be established and the molecular properties can thus be predicted [6], [7].”

“In recent years, with the rapid development of deep learning, a number of neural network models have been introduced into this area [22], [23], [24].”

“For time series, here we mainly focus on modulation classification of radio signals [18], [19], [20], [21].”

“the graph kernel methods, which are mainly based on random walk [10], path [11] and subtree [12]; The second one is graph embedding methods, such as subgraph2vec [13] and graph2vec [14]; And the last one is deep learning methods, i.”

“In the future work, we may try to combine other graph classification models with AVG, or combine the graph data enhancement methods such as subgraph networks [45] in the field of network science, to further improve the classification performance of AVGNet.”

“For time series, here we mainly focus on modulation classification of radio signals [18], [19], [20], [21].”

“In recent years, with the rapid development of deep learning, a number of neural network models have been introduced into this area [22], [23], [24].”

“[41] further introduced circle system for the first time in the construction of VG, proposed Circular Limited Penetrable Visibility Graph (CLPVG), and realized higher accuracy in radio modulation classification with graph embedding technology than LPVG.”

“The representative methods of graph classification can be divided into three types, the first one is the graph kernel methods, which are mainly based on random walk [10], path [11] and subtree [12]; The second one is graph embedding methods, such as subgraph2vec [13] and graph2vec [14]; And the last one is deep learni…”

“For time series, here we mainly focus on modulation classification of radio signals [18], [19], [20], [21].”

“…deep neural networks (e.g., convolutional neural networks (CNNs) [40–44], recurrent neural networks (RNNs) [45–49], graph neural networks (GNNs) [50–52], transformers [53–55], and hybrid neural networks [56–59]) to perform the end-to-end feature 2 International Journal of Intelligent Systems…”

“…of DL in AMC, particularly through models, such as convolutional neural network (CNN) [7], recurrent neural networks (RNNs) [14], and graph neural network (GNN) [15], has facilitated the automated extraction of salient features from extensive data sets, substantially boosting performance.”

“These models include AvgNet [70], MCLDNN [39], Resnet1d, VGG [28], CNN2d [27], GRU [29], LSTM [8], GAF [59], and ConsCNN [62].”

“II-B and AVG [50] mentioned in Sec.”

“Recently, with the development of deep learning (DL) technology, DL-based models have made significant progress in AMR task [6], [7], [8], [9], [10], [11].”

“Traditional approaches exhibit limited performance when dealing with complex nonlinear signals, conversely, deep learning provides new possibilities for AMC by leveraging its robust capabilities for automatic feature extraction and hierarchical abstract representation [6], [7], [8].”

“Wireless signals require modulation of the baseband signal before transmission, with common modulation methods including amplitude modulation (AM), phase modulation (PM), and frequency modulation (FM) [16].”

“In addition, Xuan et al. [37] proposed an Adaptive Visibility Graph (AVG) algorithm that can adaptively map time series into graphs.”

“The key notion is developing a multiscale feature extraction pipeline to leverage different levels of topological information in graphs [37], [38], [39].”

“In recent years, deep learning (DL) has achieved remarkable advancements in various domains, including multimedia learning, machine translation, data mining, search technologies, natural language processing, and other related fields [10], [11], [12], [13], [14].”

“Furthermore, Recurrent Neural Networks (RNNs), such as Long Short Term Memory (LSTM) [44] and Gate Recurrent Unit (GRU) [45], are commonly used in wireless communication as they can effectively handle the time-varying nature of communication signals.”

“We also compare GIGNet with existing GNN-based models from [15], [17] named as AvgNet and GGCNN.”

“The first approach is based on the concept of Laplacian matrix [1], [2], [12] while the second one, stemming from algebraic signal processing, is based on graph adjacency matrices, which are also referred as graph shift operators [1], [3]–[11], [13].”

“In [118], the authors tackle the task of radio signal modulation classification by developing an end-to-end GNN framework, termed AvgNet, which integrates the adaptive visibility graph (AVG) algorithm and a modified DiffPool model [145].”

“AvgNet [123] is an Adaptive Visibility Graph algorithm that maps time series data into graphs and can be used to classify various signal modulations.”

“This most likely is the reason for the low number of scientific publications related to AMC using GNN [7], [8].”

“This trend of AI-enabled, data-driven channel modeling, which includes diverse applications from radio map construction to modulation classification [74], is seen as a key enabler for future communication systems, promising to deliver more accurate and adaptable models than classical methods [58].”

“Besidesthe sequencerepresentation,otherformsofsignalrepresentations, suchasimageandgraphrepresentation[52],[53],arealso exploredasinputtotheDLmodel.”

“Currently, deep learning (DL) has been applied to signal processing areas, such as channel estimation [46], signal modulation [47], and TFA due to its powerful parameter representation capabilities [48], [49], [50], [51], [52], [53].”

“In [136], time series are first mapped into graphs using adaptive visibility graph algorithm, and then an end-to-end graph neural network is used for modulation classification.”

“In the field of AMC, Xuan [38] proposed an adaptive visibility graph algorithm to map a time series Entropy 2023, 25, 1096 3 of 20 into a graph adaptively; they used the proposed method and GCN to achieve modulation classification of radio signals.”

“Furthermore, SMC has been combined with complex networks [35], [36], [37], [38], [39], [40], [41].”

“For the AMR system with complex and variable channels, DL can automatically extract the features from the signal and accurately identify the modulation mode of the signal by using its powerful nonlinear mapping ability [12], [13], [14], [15].”

“Graph Neural Network (GNN) has been widely used in the field of Multivariate Time Series (MTS) classification, with great potential for applications [19].”

“Two other works utilize VG representations of uni-variate TS, first for ECG signals [27], while [28] integrated a VG and trained it together with GNN classification model for radio signal modulation.”

“Recently, one of the first approaches to univariate TS data classification was proposed by [18].”

“Qi et al. proposed an AvgNet modulation classification framework based on the adaptive visibility graph (AVG) algorithm in [31], which enabled the adaptive mapping of signals to graphs by AVG, maximizing the preservation of local and global information of signals.”

“Deeplearninghasastrangerabilitytodealwithhigh-dimensionaldataincommunitydetec-tion(Kosasih&Brintrup,2022),suchasspectralconvolutionneuralnetworks(CNN),and recurrentneuralnetworks(RNN)(Xuanetal.,2022;Dernoncourtetal.,2017).”

“Deep learning (DL) has made significant advancements in various fields such as computer vision (CV) [1], secure communication [2], signal recognition [3], [4], [5], [6], and specific emitter identification (SEI)[7]duetoitspowerfulandefficientfeature-extractioncapabilities.”

“For instance, in [33], an adaptive visibility graph (AVG) is used to adaptively map time-series signals into graphs, and with the support of an improved GCN model, DiffPool, classification performance in radio signal modulation tasks is significantly improved.”

“Consequently, DL-based signal recognition techniques have been successfully applied to various wireless communication scenarios [7], [8], [9], [10], [11].”

“EdgePool [28] implements the pooling operation by combining the weights of edges and structural information in the graph, DiffPool [29,51] learns the soft clustering of nodes based on the output of the GNN model, and HaarPool [10] generates the pooled graph based on the compressed Haar transform.”

“Consequently, DL-based signal recognition techniques have been successfully developed in numerous applications [5]–[9].”

“Xuan et al. [26] integrated signals with graph machine learning, achieving effective results using the Adaptive Visibility Graph (AVG) algorithm, which dynamically converts signals into graph representations.”

“Xuan et al. [13] proposed an Adaptive Visibility Graph (AVG) algorithm that can adaptively map time series into a graph.”

“[24] and the Avg method in AvgNet [12], our method improves the recognition accuracy with low computational complexity.”

“To evaluate performance, the proposed DCT-OCC model is compared with several representative AMR models, including CLDNN [15], AvgNet [16], AWN [17], MCDformer [18], and PETCGDNN [19].”

“Representative models in this category include GCN [28], AvgNet [27], DGFDNN [37], and KG-MTSNet [38].”

“Emerging deep learning methods, including graph neural networks (GCN) [7] and complex-valued neural networks (CVNN) [8] are gradually being adopted for SMC tasks.”

“Moreover, these models utilized CNN as the main architecture, yet recent experimental results in[11, 12] indicated that GNN was more effective in extracting intrinsic features of signals, thereby leading to enhanced recognition performance compared to CNN.”

“However, most existing GNN [59], [60] [61] methods are primarily designed for undirected graphs, often overlooking the critical directional information inherent in directed graphs.”

“Furthermore, DWMTN was compared with PGNet [16], CIST [3], PCTNet [11], VIG [17], PET-CGDNN [18], and AVGNet [19] to evaluate the effectiveness of the proposed method.”

“…a variety of signal preprocessing techniques were introduced to facilitate classification, including the Amplitude and Phase (AP) [4], Adaptive Visibility Graph (AVG) [10], Wavelet Transform (WT) [11], Pseudo-Wigner Distribution (PWVD) [12], Con-stellation Diagram (CD) [13], Eye Diagrams [14].”

“This section presents the proposed EAGCN network and the GDIR regularization method, where EAGCN mainly consists of three components: AVG generation [21], GNN training, and ECA-based feature fusion.”

“For instance, [13] proposed an approach that combined a visibility graph trained concurrently with a radio signal modulation classification model.”

“In 2021, Xuan et al. [22] extracted signal information in a time series using an Adaptive Visibility Graph (AVG) proposed to map each signal into a graph, and used the GNN model for feature extraction to achieve end-to-end signal recognition.”

“Moreover, [15]– [17] transform the received signal into a graph to utilize the enormous potential of graph neural networks (GNNs).”

“This algorithm has found applications in solving various engineering problems [4], [11], with one typical application being time series forecasting, as similar data points tend to exhibit similar tendencies [12], [13].”

“Except these, there are many other novel architectures, such as graph neural network [22], Transformer [23] and Mamba [24], which have also been applied to AMC and achieved advanced performance.”

“The advantages over other methods of building networks are as follow [15] : (1) No need to go through the complicated process of phase space division, which is suitable for the analysis of large, heterogeneous, non-stationary time series; (2) The method has no parameters, directionality or weights,…”

“Significant advancements have been made in re-representing signals into network structures, driven by the remarkable success of Graph Neural Networks (GNNs) [22], [23] and effective graph mapping techniques [24], [25], [26], [27], [28].”

“Typically,GCNinvolvestheproblemofrelation-shipconstructioninnon-graphdata,includingAMR[16],[17].”

“At present, deep learning technology has been widely applied in many fields due to its powerful feature learning ability, including the wireless communication field [6] [7] [8].”

“Updated approaches like AvgNet [18] and GIGNet [19] further enhance flexibility and feature integration, providing more robust frameworks for AMR tasks.”

“The growing demand for high-speed data transmission, effectiveness of spectrum and real time adaptability form automatic and adaptive signal modulation a crucial part of next-generation wireless networks like 5G and beyond [2].”

“…neural networks (GNNs) have emerged as strong candidates for learning on non-Euclidean or irregular structures and are increasingly applied to wireless problems by casting signals/devices (or time samples via graph mappings) into graphs for tasks such as modulation or waveform recognition [2], [3].”

“For instance, Xuan et al introduced an adaptive visibility graph algorithm, which adaptively transforms time series into networks [30].”

“We compare the proposed method against six SOTA DL-AMR models, including GIGNet [39], AVGNet [40], FE-SKViT [41], MCLDNN [42], LSTM [13], and GRU [14], all of which are retrained and evaluated under the same data split and protocol.”