Graph Signal Research Articles

Leveraging Multi-level Correlations for Imputing Monitoring Data in Water Supply Systems Using Graph Signal Sampling Theory

Leveraging Multi-level Correlations for Imputing Monitoring Data in Water Supply Systems Using Graph Signal Sampling Theory

DFC-Igloo: A dynamic functional connectome learning framework for identifying neurodevelopmental biomarkers in very preterm infants

Background and ObjectiveVery preterm infants are susceptible to neurodevelopmental impairments, necessitating early detection of prognostic biomarkers for timely intervention. The study aims to explore possible functional biomarkers for very preterm infants at born that relate to their future cognitive and motor development using resting-state fMRI. Prior studies are limited by the sample size and suffer from efficient functional connectome (FC) construction algorithms that can handle the noisy data contained in neonatal time series, leading to equivocal findings. Therefore, we first propose an enhanced functional connectome construction algorithm as a prerequisite step. We then apply the new FC construction algorithm to our large prospective very preterm cohort to explore multi-level neurodevelopmental biomarkers. MethodsThere exists an intrinsic relationship between the structural connectome (SC) and FC, with a notable coupling between the two. This observation implies a putative property of graph signal smoothness on the SC as well. Yet, this property has not been fully exploited for constructing intrinsic dFC. In this study, we proposed an advanced dynamic FC (dFC) learning model, dFC-Igloo, which leveraged SC information to iteratively refine dFC estimations by applying graph signal smoothness to both FC and SC. The model was evaluated on artificial small-world graphs and simulated graph signals. ResultsThe proposed model achieved the best and most robust recovery of the ground truth graph across different noise levels and simulated SC pairs from the simulation. The model was further applied to a cohort of very preterm infants from five Neonatal Intensive Care Units, where an enhanced dFC was obtained for each infant. Based on the improved dFC, we identified neurodevelopmental biomarkers for neonates across connectome-wide, regional, and subnetwork scales. ConclusionThe identified markers correlate with cognitive and motor developmental outcomes, offering insights into early brain development and potential neurodevelopmental challenges.

Graph total variation and low-rank regularization for heterogeneous change detection

Heterogeneous change detection (HCD) is challenging because different imaging mechanisms for various sensors make images difficult to compare directly. To address this problem, a graph-based regression algorithm is proposed for HCD, by leveraging the Graph Total Variation regularization and Low-Rank matrices decomposition (GTVLR). Utilizing graph signal processing (GSP) theory, a directed graph (digraph) model is employed to effectively represent the orientation and correlation information of images, thereby enabling direct comparison of heterogeneous data within the same domain after graph filtering. The GTVLR framework facilitates the decomposition of post-event images into regression and changed images. This decomposition ensures that the regression image mirrors the structure similarity of the pre-event image, while the changed image highlights areas of alteration, aiding in change detection. The model characterizes the piecewise smoothness and Low-Rank properties of data through GTV regularization and Low-Rank penalty, respectively. Moreover, by integrating the higher-order neighboring information within the digraph to refine the model. Experiments conducted on three real-world datasets and comparison with several state-of-the-art methods demonstrate the effectiveness of the proposed algorithm.

Multi-level attention graph feature fusion smooth prognostics approach for aircraft engines remaining useful life prediction

In the development of a prognostic and health management system, it is a challenge to mine the degradation characteristics, estimate the RUL reliably and smoothly, and explain the RUL prediction results from the multi-sensor condition monitoring data collected under the non-fixed-length equipment operation cycle and multi-operating conditions. In this paper, a Multi-level Attention Graph feature Fusion Smooth Prognostics Approach (MAGF-SPA) is proposed to achieve accurate, smooth, and interpretable RUL prediction results. In this method, firstly, multivariate time series (MTS) are transformed into graph signals from a purely data-driven perspective. Secondly, the four-level feature extraction strategy, which includes node-level, subgraph-level, cyclical-level, and temporal-level feature extraction, is designed to extract the inherent hidden degradation characteristics of MTS. Finally, a smoothing trick proposed in this paper can optimise the RUL prediction value corresponding to the degradation characteristics in real time. Experimental results show that the proposed method achieves state-of-the-art on the N-CMAPSS dataset.

GraphSmart: a method for green and accurate IoT water monitoring

Water scarcity is nowadays a critical global concern and an efficient management of water resources is paramount. This paper presents an original approach for monitoring Water Distribution Systems (WDSs) through Internet of Things (IoT) that involves the integration of multiple sensors placed across the distribution network to accurately measure water flow. To enhance energy efficiency for green monitoring and communication process, we harness the power of graph theory and graph signal processing to represent in a tunable and accurate way the water flow and simultaneously minimize the number of IoT sensors communicating those measurements. We propose a graph model where water flow is represented as signal on graph and we introduce an algorithm, named GraphSmart, designed to reconstruct the graph signal when certain measurements are unknown or missing. Our framework is applied on synthetic realistic environment within the context of LoRaWAN (Long Range Wide Area Network), an infrastructure and protocol designed for ultra-low-power IoT devices. Our findings show that GraphSmart significantly reduces energy consumption while ensuring precise flow estimation. Our research demonstrates high potential for energy-efficient and accurate water flow monitoring, paving the way to improve the management of WDSs and enabling water operators to address water scarcity challenges.

AFGN: Adaptive Filtering Graph Neural Network for Few-Shot Learning

The combination of few-shot learning and graph neural networks can effectively solve the issue of extracting more useful information from limited data. However, most graph-based few-shot models only consider the global feature information extracted by the backbone during the construction process, while ignoring the dependency information hidden within the features. Additionally, the essence of graph convolution is the filtering of graph signals, and the majority of graph-based few-shot models construct fixed, single-property filters to process these graph signals. Therefore, in this paper, we propose an Adaptive Filtering Graph Convolutional Neural Network (AFGN) for few-shot classification. AFGN explores the hidden dependency information within the features, providing a new approach for constructing graph tasks in few-shot scenarios. Furthermore, we design an adaptive filter for the graph convolution of AFGN, which can adaptively adjust its strategy for acquiring high and low-frequency information from graph signals based on different few-shot episodic tasks. We conducted experiments on three standard few-shot benchmarks, including image recognition and fine-grained categorization. The experimental results demonstrate that our AFGN performs better compared to other state-of-the-art models.

Sensor placement method for water distribution networks based on sampling of non-bandlimited graph signals

Monitoring water distribution networks (WDNs) requires careful consideration of sensor placement, which is essential for obtaining comprehensive information about the network. A natural graphical structure underlies WDN, making graph sampling theory advantageous for selecting monitoring nodes. However, graph sampling theory is only applied only to restrictive band-limited signals, while the pressure data of WDN is a restrictive non-band-limited signal. To address this issue, this paper presents an approximate conversion method for transforming non-band-limited signals into band-limited signals, accompanied by an optimal spectrum threshold formula. This formula is used to perform spectral screening in the graph frequency domain, effectively converting non-band-limited signals into band-limited signals that preserve the major frequency components while ignoring smaller-value frequency components. By sampling band-limited signal, we identify sampling nodes that perfectly recover the signal. These sampling nodes act as monitoring nodes that can perform a comprehensive inspection of the WDN and accurately locate leaks. The accuracy of our method in recovering the signal and locating the leak is demonstrated by comparing it with two existing sensor placement optimization methods.

Nonintrusive load disaggregation by fusion of graph signal processing and CNN

Abstract Nonintrusive load disaggregation techniques play a pivotal role in power system planning and decision making for demand response. The sparsity of low frequency total power data features, coupled with the model’s challenge in effectively extracting and utilizing spatial and temporal correlations in diverse load data, presents obstacles to achieving high precision disaggregation in nonintrusive load disaggregation. To address these challenges, this paper proposes a disaggregation framework that integrates graph signal processing (GSP) with a convolutional neural network (CNN). In this framework, the GSP module tackles the issue of sparse load features, while the CNN disaggregation module extracts and exploits various features, such as spatial and temporal correlations between different loads. The GSP module directly provides load features to the CNN disaggregation module, assigning weights to the outputs for selection and thereby enhancing the accuracy of the disaggregation process. Our method is trained and validated using real datasets, demonstrating satisfactory performance when compared to existing methods.

Graph signal processing and graph learning approaches to Schizophrenia pattern identification in brain Electroencephalogram

The detection of Schizophrenia (SZ) directly from brain Electroencephalogram (EEG) signals has recently gained importance. Traditionally, EEG-based SZ detection is done by either manually extracting features from each EEG channel and using Machine Learning classifiers or by Deep Learning-based automated detection. These methods neglect the functional interactions between spatially distributed brain regions. This study proposes a graph signal processing and graph learning framework to detect SZ. Here, the individual channels’ features and the global interactions are combined into a unified graph signal (GS) representation. The proposed representation consists of the GS values and the underlying connectivity network. An SZ-specific feature, namely, the permutation entropies of each electrode’s signal, is considered as the GS values. The graph learning method learns the underlying network from a collection of GS observations. The learned network serves as the Fourier basis for the graph Fourier and wavelet transform. Utilizing these transforms, the graph spectral features are extracted from GSs. Lastly, the extracted features are classified using a set of Machine Learning models. The proposed graph signal processing and graph learning method is an effective approach to decode Schizophrenia patterns in brain EEG signals and outperform the existing approaches.

Relative entropy based uncertainty principles for graph signals

In physical quantum mechanics, the uncertainty principle in presence of quantum memory [Berta M, Christandl M, Colbeck R,et al., Nature Physics] can reach much lower bound, which has resulted in a huge breakthrough in quantum mechanics. Inspired by this idea, this paper would propose some novel uncertainty relations in terms of relative entropy for signal representation and time-frequency resolution analysis. On one hand, the relative entropy measures the distinguishability between the known (priori) basis and the client basis, which implies that we have partial “memory” of the client basis so that the uncertainty bounds become sharper in some cases. On the other hand, in some cases, if the reference basis along with nearly the same energy distribution could be given, then the uncertainty bound would tend to zero, as shows that there is no uncertainty any longer. These novel uncertainty relationships with sharper bounds would give us the potential advantages over the classical counterpart. In addition, the detailed comparison with classical Shannon entropy based uncertainty principle has been addressed as well via combined uncertainty relations. Finally, the theoretical analysis and numerical experiments on certain application over graph signals have been demonstrated to show the efficiency of these proposed relations.

Hermitian random walk graph Fourier transform for directed graphs and its applications

Signal processing on directed graphs present additional challenges since a complete set of eigenvectors is unavailable generally. To solve this problem, in this paper, a novel graph Fourier transform is constructed for representing and processing signals on directed graphs. Firstly, we introduce a Hermitian random walk Laplacian operator and derive that it is Hermitian positive semi-definite. Hence, the obtained Laplacian operator is diagonalizable and yields orthogonal eigenvectors as graph Fourier basis. Secondly, we propose the Hermitian random walk graph Fourier transform (HRWGFT) with good properties including unitary and preserving inner products. Furthermore, HRWGFT records the directionality of edges without sacrificing the information about the graph signal. Then, using these favorable properties, we derive spectral convolution to define the graph filter which is the core tool for processing graph signals. Finally, based on the proposed Laplacian matrix and HRWGFT, we present several applications on synthetic and real-world networks, including signal denoising, data classification. The rationality and validity of our work are verified by simulations.

Reconstruction of bandlimited graph signals from random local sampling

Sampling and reconstruction on the spatially distributed networks is an innovative topic in graph signal processing. Recently, it has been shown that k-bandlimited graph signals can be reconstructed from a random collection of physically constrained sampled data. In this paper, we first study the random sampling scheme of k-bandlimited signals from a general local measurement, and then an iterative reconstruction algorithm based on frame theory is proposed with exponential convergence. It can yield a distributed implementation at a vertex level, which enables the devices that are limited by storage and computing power to recover signals more effectively. Numerical experiments on synthetic and real-world data are performed to validate the effectiveness of the proposed approach.

Detection of Evasive Android Malware Using EigenGCN

Recently there is an upsurge in Android malware that use obfuscation and repackaging techniques for evasion. Malware may also combine both these techniques to create stealthy adversarial mimicry samples to launch mimicry attacks. In mimicry attacks, the adversary makes sure that the static and dynamic features present in the crafted malware mimics the features present in the legitimate applications. In such cases, the existing detection mechanisms may become less effective. We found that the malicious nature of Android applications can be determined by identifying certain subgraphs that appear in their system call graphs. These subgraphs can be determined with the help of spectral clustering mechanism present in EigenGCN. With this, the system call graph G will be partitioned into two subgraphs G1 and G2, in which the malicious functionality if any will be present in the subgraph G1. The graph Fourier transform based pooling technique in EigenGCN then computes the features of the subgraphs in the form of graph signals. This graph signals serve as a robust signature to detect malware. The proposed mechanism gave an accuracy of 98.7% on common malware, 97.3% on obfuscated malware, 97.8% on repackaged malware, and 90% on adversarial mimicry malware datasets. As far as we know, this is the first work that proposes a malware detection mechanism, that can detect common as well as obfuscated, repackaged, and mimicry malware in Android.

Gegenbauer Graph Neural Networks for Time-Varying Signal Reconstruction.

Reconstructing time-varying graph signals (or graph time-series imputation) is a critical problem in machine learning and signal processing with broad applications, ranging from missing data imputation in sensor networks to time-series forecasting. Accurately capturing the spatio-temporal information inherent in these signals is crucial for effectively addressing these tasks. However, existing approaches relying on smoothness assumptions of temporal differences and simple convex optimization techniques that have inherent limitations. To address these challenges, we propose a novel approach that incorporates a learning module to enhance the accuracy of the downstream task. To this end, we introduce the Gegenbauer-based graph convolutional (GegenConv) operator, which is a generalization of the conventional Chebyshev graph convolution by leveraging the theory of Gegenbauer polynomials. By deviating from traditional convex problems, we expand the complexity of the model and offer a more accurate solution for recovering time-varying graph signals. Building upon GegenConv, we design the Gegenbauer-based time graph neural network (GegenGNN) architecture, which adopts an encoder-decoder structure. Likewise, our approach also uses a dedicated loss function that incorporates a mean squared error (MSE) component alongside Sobolev smoothness regularization. This combination enables GegenGNN to capture both the fidelity to ground truth and the underlying smoothness properties of the signals, enhancing the reconstruction performance. We conduct extensive experiments on real datasets to evaluate the effectiveness of our proposed approach. The experimental results demonstrate that GegenGNN outperforms state-of-the-art methods, showcasing its superior capability in recovering time-varying graph signals.

Graph Fourier transform for spatial omics representation and analyses of complex organs

Spatial omics technologies decipher functional components of complex organs at cellular and subcellular resolutions. We introduce Spatial Graph Fourier Transform (SpaGFT) and apply graph signal processing to a wide range of spatial omics profiling platforms to generate their interpretable representations. This representation supports spatially variable gene identification and improves gene expression imputation, outperforming existing tools in analyzing human and mouse spatial transcriptomics data. SpaGFT can identify immunological regions for B cell maturation in human lymph nodes Visium data and characterize variations in secondary follicles using in-house human tonsil CODEX data. Furthermore, it can be integrated seamlessly into other machine learning frameworks, enhancing accuracy in spatial domain identification, cell type annotation, and subcellular feature inference by up to 40%. Notably, SpaGFT detects rare subcellular organelles, such as Cajal bodies and Set1/COMPASS complexes, in high-resolution spatial proteomics data. This approach provides an explainable graph representation method for exploring tissue biology and function.

An all-purpose method for optimal pressure sensor placement in water distribution networks based on graph signal analysis

Many researchers have addressed the challenge of optimal pressure sensor placement for different purposes, such as leakage detection, model calibration, state estimation, etc. However, pressure data often need to serve multiple purposes, and a method to optimize sensor locations with versatility for various objectives is still lacking. In this paper, a graph-based optimal sensor placement (GOSP) framework is proposed, which aims to provide a robust and all-purpose approach to identify critical points for pressure monitoring. By analysing the spatial variation frequencies of WDN pressures, the relationship between measurements and the global variation of original pressures is established. On this basis, the D-optimality criterion is adopted to formulate the objective of GOSP, which aims to maximize the information on the spatial distribution of pressures that can be obtained from measurements. The new-proposed objective ensures that the sensor locations are compatible with various application scenarios. The proposed method was applied to a real-life distribution network, and was compared with other optimal sensor placement methods oriented towards burst detection and pipe roughness calibration. Based on comparative studies in different scenarios including unknown pressure estimation, burst detection, and model calibration, the effectiveness and robustness of the proposed method have been proved.

A Self-Attention Legendre Graph Convolution Network for Rotating Machinery Fault Diagnosis.

Rotating machinery is widely used in modern industrial systems, and its health status can directly impact the operation of the entire system. Timely and accurate diagnosis of rotating machinery faults is crucial for ensuring production safety, reducing economic losses, and improving efficiency. Traditional deep learning methods can only extract features from the vertices of the input data, thereby overlooking the information contained in the relationships between vertices. This paper proposes a Legendre graph convolutional network (LGCN) integrated with a self-attention graph pooling method, which is applied to fault diagnosis of rotating machinery. The SA-LGCN model converts vibration signals from Euclidean space into graph signals in non-Euclidean space, employing a fast local spectral filter based on Legendre polynomials and a self-attention graph pooling method, significantly improving the model's stability and computational efficiency. By applying the proposed method to 10 different planetary gearbox fault tasks, we verify that it offers significant advantages in fault diagnosis accuracy and load adaptability under various working conditions.

Discrete linear canonical transform on graphs: Uncertainty principle and sampling

With an increasing influx of classical signal processing methodologies into the field of graph signal processing, approaches grounded in discrete linear canonical transform have found application in graph signals. In this paper, we initially propose the uncertainty principle of the graph linear canonical transform (GLCT), which is based on a class of graph signals maximally concentrated in both vertex and graph spectral domains. Subsequently, leveraging the uncertainty principle, we establish conditions for recovering bandlimited signals of the GLCT from a subset of samples, thereby formulating the sampling theory for the GLCT. We elucidate interesting connections between the uncertainty principle and sampling. Further, by employing sampling set selection and experimental design sampling strategies, we introduce optimal sampling operators in the GLCT domain. Finally, we evaluate the performance of our methods through simulations and numerical experiments across applications.

TaylorNet: A novel approach for spectral filter learning on graph data

Graphs are indispensable data structures used to model interconnected nodes and their relationships. However, devising filters for graph data is challenging. Previous approaches, including BernNet, utilized Bernstein polynomials to approximate spectral graph convolutions but suffered from lengthy training times compared to alternatives. To address this, we propose TaylorNet, a novel graph neural network built upon rigorous mathematical principles and theoretical foundations. TaylorNet provides an efficient approach for learning arbitrary graph spectral filters. Our methodology revolves around utilizing Kth order Taylor polynomials to estimate filters on the normalized Laplacian operator of graphs. By carefully setting the coefficients of these polynomials, we achieve effective filter design. Additionally, TaylorNet personalizes filter learning for specific graphs by leveraging observed graph data and signals to determine these coefficients. Extensive experimentation demonstrates TaylorNet’s ability to learn arbitrary spectral filters, surpassing current state-of-the-art methods, particularly for handling large graphs. Its exceptional performance extends to real-world graph modeling tasks, highlighting its versatility and applicability. For interested researchers, the complete implementation of TaylorNet is available at https://github.com/12chen20/TaylorNet.

Optimal sensor placement for leak location in water distribution networks based on graph learning algorithm

Abstract The effective monitoring of urban water distribution networks (WDNs) relies heavily on pressure sensor placement. Nevertheless, a WDN may consist of hundreds of nodes, and it is not economically feasible to install sensors at each node. Therefore, how to identify an optimal location for sensor placement becomes a crucial issue. We use graph signal processing to analyze the pressure signals and introduce a framework for optimal sensor placement specifically designed for pressure signals. To address the limitation that pressure signals cannot be sampled directly, we propose a method to convert the signal into a band-limited signal that meets the requirements. Central to the method is learning a graph Fourier operator, and the effectiveness of the proposed method is proved theoretically. The graph Fourier operator enables the pressure data to become a smooth graph signal with variations in its topology. In addition, we design a graph filter based on the energy of the signal and obtain a band-limited signal that meets the requirements. To ensure the selection of representative nodes, we use a noise-robust graph sampling method to obtain the sensor node. Our method is further evaluated using the pressure data from Anytown versus Net3, showing strong performance in leak identification and signal reconstruction capabilities.