Signal Processing over Time-Varying Graphs: A Systematic Review

As irregularly structured data representations, graphs have received a large amount of attention in recent years and have been widely applied to various real-world scenarios such as social, traffic, and energy settings. Compared to non-graph algorithms, numerous graph-based methodologies benefited from the strong power of graphs for representing high-dimensional and non-Euclidean data. In the field of Graph Signal Processing (GSP), analogies of classical signal processing concepts, such as shifting, convolution, filtering, and transformations are developed. However, many GSP techniques usually postulate the graph is static in both signal and typology. This assumption hinders the effectiveness of GSP methodologies as the assumption ignores the time-varying properties in numerous real-world systems. For example, in the traffic network, the signal on each node varies over time and contains underlying temporal correlation and patterns worthy of analysis. To tackle this challenge, more and more work are being done recently to investigate the processing of time-varying graph signals. They cope with time-varying challenges from three main directions: 1) graph time-spectral filtering, 2) multi-variate time-series forecasting, and 3) spatiotemporal graph data mining by neural networks, where non-negligible progress has been achieved. Despite the success of signal processing and learning over time-varying graphs, there is no survey to compare and conclude the current methodology for GSP and graph learning. To compensate for this, in this paper, we aim to review the development and recent progress on signal processing and learning over time-varying graphs, and compare their advantages and disadvantages from both the methodological and experimental side, to outline the challenges and potential research directions for future research.

LLM Online Spatial-temporal Signal Reconstruction Under Noise

This work introduces the LLM Online Spatial-temporal Reconstruction (LLM-OSR) framework, which integrates Graph Signal Processing (GSP) and Large Language Models (LLMs) for online spatial-temporal signal reconstruction. The LLM-OSR utilizes a GSP-based spatial-temporal signal handler to enhance graph signals and employs LLMs to predict missing values based on spatiotemporal patterns. The performance of LLM-OSR is evaluated on traffic and meteorological datasets under varying Gaussian noise levels. Experimental results demonstrate that utilizing GPT-4-o mini within the LLM-OSR is accurate and robust under Gaussian noise conditions. The limitations are discussed along with future research insights, emphasizing the potential of combining GSP techniques with LLMs for solving spatiotemporal prediction tasks.

LLM-based Online Prediction of Time-varying Graph Signals

In this paper, we propose a novel framework that leverages large language models (LLMs) for predicting missing values in time-varying graph signals by exploiting spatial and temporal smoothness. We leverage the power of LLM to achieve a message-passing scheme. For each missing node, its neighbors and previous estimates are fed into and processed by LLM to infer the missing observations. Tested on the task of the online prediction of wind-speed graph signals, our model outperforms online graph filtering algorithms in terms of accuracy, demonstrating the potential of LLMs in effectively addressing partially observed signals in graphs.

Graph Signal Adaptive Message Passing

This paper proposes Graph Signal Adaptive Message Passing (GSAMP), a novel message passing method that simultaneously conducts online prediction, missing data imputation, and noise removal on time-varying graph signals. Unlike conventional Graph Signal Processing methods that apply the same filter to the entire graph, the spatiotemporal updates of GSAMP employ a distinct approach that utilizes localized computations at each node. This update is based on an adaptive solution obtained from an optimization problem designed to minimize the discrepancy between observed and estimated values. GSAMP effectively processes real-world, time-varying graph signals under Gaussian and impulsive noise conditions.

Time-varying Graph Signal Estimation via Dynamic Multi-hop Topologies

The assumption of using a static graph to represent multivariate time-varying signals oversimplifies the complexity of modeling their interactions over time. We propose a Dynamic Multi-hop model that captures dynamic interactions among time-varying node signals, while also accounting for time-varying edge signals, by extracting latent edges through topological diffusion and edge pruning. The resulting graphs are time-varying and sparse, capturing key dynamic node interactions and representing signal diffusion to both near and distant neighbors over time. The Dynamic Multi-hop Estimation algorithm is further proposed, accurately representing the interaction dynamics among node signals while enabling adaptive estimation of time-varying multivariate signals spatially and temporally. The Dynamic Multi-hop Estimation is evaluated under two real-world datasets of brain network and stock market for the online estimation of partially observed time-varying signals corrupted by noise.

Binarized Simplicial Convolutional Neural Networks

Graph Neural Networks have a limitation of solely processing features on graph nodes, neglecting data on high-dimensional structures such as edges and triangles. Simplicial Convolutional Neural Networks (SCNN) represent higher-order structures using simplicial complexes to break this limitation albeit still lacking time efficiency. In this paper, we propose a novel neural network architecture on simplicial complexes named Binarized Simplicial Convolutional Neural Networks (Bi-SCNN) based on the combination of simplicial convolution with a binary-sign forward propagation strategy. The usage of the Hodge Laplacian on a binary-sign forward propagation enables Bi-SCNN to efficiently and effectively represent simplicial features that have higher-order structures than traditional graph node representations. Compared to the previous Simplicial Convolutional Neural Networks, the reduced model complexity of Bi-SCNN shortens the execution time without sacrificing the prediction performance and is less prone to the over-smoothing effect. Experimenting with real-world citation and ocean-drifter data confirmed that our proposed Bi-SCNN is efficient and accurate.

Adaptive Least Mean pth Power Graph Neural Networks

In the presence of impulsive noise, and missing observations, accurate online prediction of time-varying graph signals poses a crucial challenge in numerous application domains. We propose the Adaptive Least Mean $p^{th}$ Power Graph Neural Networks (LMP-GNN), a universal framework combining adaptive filter and graph neural network for online graph signal estimation. LMP-GNN retains the advantage of adaptive filtering in handling noise and missing observations as well as the online update capability. The incorporated graph neural network within the LMP-GNN can train and update filter parameters online instead of predefined filter parameters in previous methods, outputting more accurate prediction results. The adaptive update scheme of the LMP-GNN follows the solution of a $l_p$-norm optimization, rooting to the minimum dispersion criterion, and yields robust estimation results for time-varying graph signals under impulsive noise. A special case of LMP-GNN named the Sign-GNN is also provided and analyzed, Experiment results on two real-world datasets of temperature graph and traffic graph under four different noise distributions prove the effectiveness and robustness of our proposed LMP-GNN.

Adaptive Graph Normalized Sign Algorithm

Efficient and robust prediction of graph signals is challenging when the signals are under impulsive noise and have missing data. Exploiting graph signal processing (GSP) and leveraging the simplicity of the classical adaptive sign algorithm, we propose an adaptive algorithm on graphs named the Graph Normalized Sign (GNS). GNS approximated a normalization term into the update, therefore achieving faster convergence and lower error compared to previous adaptive GSP algorithms. In the task of the online prediction of multivariate temperature data under impulsive noise, GNS outputs fast and robust predictions.

Adaptive Least Mean Squares Graph Neural Networks and Online Graph Signal Estimation

The online prediction of multivariate signals, existing simultaneously in space and time, from noisy partial observations is a fundamental task in numerous applications. We propose an efficient Neural Network architecture for the online estimation of time-varying graph signals named the Adaptive Least Mean Squares Graph Neural Networks (LMS-GNN). LMS-GNN aims to capture the time variation and bridge the cross-space-time interactions under the condition that signals are corrupted by noise and missing values. The LMS-GNN is a combination of adaptive graph filters and Graph Neural Networks (GNN). At each time step, the forward propagation of LMS-GNN is similar to adaptive graph filters where the output is based on the error between the observation and the prediction similar to GNN. The filter coefficients are updated via backpropagation as in GNN. Experimenting on real-world temperature data reveals that our LMS-GNN achieves more accurate online predictions compared to graph-based methods like adaptive graph filters and graph convolutional neural networks.

Online Signal Estimation on the Graph Edges via Line Graph Transformation

We propose the Line Graph Normalized Least Mean Square (LGNLMS) algorithm for online time-varying graph edge signals prediction. LGNLMS utilizes the Line Graph to transform graph edge signals into the node of its edge-to-vertex dual. This enables edge signals to be processed using established GSP concepts without redefining them on graph edges.