Dynamic graph embedding has emerged as an important technique for modeling complex time-evolving networks across diverse domains. While transformer-based models have shown promise in capturing long-range dependencies in temporal graph data, they face scalability challenges due to quadratic computational complexity. This study presents a comparative analysis of dynamic graph embedding approaches using transformers and the recently proposed Mamba architecture, a state-space model with linear complexity. We introduce three novel models: TransformerG2G augment with graph convolutional networks, \mathcal{DG}-Mamba, and \mathcal{GDG}-Mamba with graph isomorphism network edge convolutions. Our experiments on multiple benchmark datasets demonstrate that Mamba-based models achieve comparable or superior performance to transformer-based approaches in link prediction tasks while offering significant computational efficiency gains on longer sequences. Notably, \mathcal{DG}-Mamba variants consistently outperform transformer-based models on datasets with high temporal variability, such as UCI, Bitcoin, and Reality Mining, while maintaining competitive performance on more stable graphs like SBM. We provide insights into the learned temporal dependencies through analysis of attention weights and state matrices, revealing the models' ability to capture complex temporal patterns. By effectively combining state-space models with graph neural networks, our work addresses key limitations of previous approaches and contributes to the growing body of research on efficient temporal graph representation learning. These findings offer promising directions for scaling dynamic graph embedding to larger, more complex real-world networks, potentially enabling new applications in areas such as social network analysis, financial modeling, and biological system dynamics.
The influence maximization (IM) problem involves identifying a set of key individuals in a social network who can maximize the spread of influence through their network connections. With the advent of geometric deep learning on graphs, great progress has been made towards better solutions for the IM problem. In this paper, we focus on the dynamic non-progressive IM problem, which considers the dynamic nature of real-world social networks and the special case where the influence diffusion is non-progressive, i.e., nodes can be activated multiple times. We first extend an existing diffusion model to capture the non-progressive influence propagation in dynamic social networks. We then propose the method, DNIMRL, which employs deep reinforcement learning and dynamic graph embedding to solve the dynamic non-progressive IM problem. In particular, we propose a novel algorithm that effectively leverages graph embedding to capture the temporal changes of dynamic networks and seamlessly integrates with deep reinforcement learning. The experiments, on different types of real-world social network datasets, demonstrate that our method outperforms state-of-the-art baselines.
The notion of local intrinsic dimensionality (LID) has important theoretical implications and practical applications in the fields of data mining and machine learning. Recent research efforts indicate that LID measures defined for graphs can improve graph representational learning methods based on random walks. In this paper, we discuss how NC-LID, a LID measure designed for static graphs, can be adapted for dynamic networks. Focusing on dynnode2vec as the most representative dynamic graph embedding method based on random walks, we examine correlations between NC-LID and the intrinsic quality of 10 real-world dynamic network embeddings. The obtained results show that NC-LID can be used as a good indicator of nodes whose embedding vectors do not tend to preserve temporal graph structure well. Thus, our empirical findings constitute the first step towards LID-aware dynamic graph embedding methods.
We developed DyGETViz, a novel framework for effectively visualizing dynamic graphs (DGs) that are ubiquitous across diverse real-world systems. This framework leverages recent advancements in discrete-time dynamic graph (DTDG) models to adeptly handle the temporal dynamics inherent in dynamic graphs. DyGETViz effectively captures both micro- and macro-level structural shifts within these graphs, offering a robust method for representing complex and massive dynamic graphs. The application of DyGETViz extends to a diverse array of domains, including ethology, epidemiology, finance, genetics, linguistics, communication studies, social studies, and international relations. Through its implementation, DyGETViz has revealed or confirmed various critical insights. These include the diversity of content sharing patterns and the degree of specialization within online communities, the chronological evolution of lexicons across decades, and the distinct trajectories exhibited by aging-related and non-related genes. Importantly, DyGETViz enhances the accessibility of scientific findings to non-domain experts by simplifying the complexities of dynamic graphs. Our framework is released as an open-source Python package for use across diverse disciplines. Our work not only addresses the ongoing challenges in visualizing and analyzing DTDG models but also establishes a foundational framework for future investigations into dynamic graph representation and analysis across various disciplines.
Recent studies successfully learned static graph embeddings that are structurally fair by preventing the effectiveness disparity of high- and low-degree vertex groups in downstream graph mining tasks. However, achieving structure fairness in dynamic graph embedding remains an open problem. Neglecting degree changes in dynamic graphs will significantly impair embedding effectiveness without notably improving structure fairness. This is because the embedding performance of high-degree and low-to-high-degree vertices will significantly drop close to the generally poorer embedding performance of most slightly changed vertices in the long-tail part of the power-law distribution. We first identify biased structural evolutions in a dynamic graph based on the evolving trend of vertex degree and then propose FairDGE, the first structurally Fair Dynamic Graph Embedding algorithm. FairDGE learns biased structural evolutions by jointly embedding the connection changes among vertices and the long-short-term evolutionary trend of vertex degrees. Furthermore, a novel dual debiasing approach is devised to encode fair embeddings contrastively, customizing debiasing strategies for different biased structural evolutions. This innovative debiasing strategy breaks the effectiveness bottleneck of embeddings without notable fairness loss. Extensive experiments demonstrate that FairDGE achieves simultaneous improvement in the effectiveness and fairness of embeddings.
Dynamic text-attributed graphs (DyTAGs) are prevalent in various real-world scenarios, where each node and edge are associated with text descriptions, and both the graph structure and text descriptions evolve over time. Despite their broad applicability, there is a notable scarcity of benchmark datasets tailored to DyTAGs, which hinders the potential advancement in many research fields. To address this gap, we introduce Dynamic Text-attributed Graph Benchmark (DTGB), a collection of large-scale, time-evolving graphs from diverse domains, with nodes and edges enriched by dynamically changing text attributes and categories. To facilitate the use of DTGB, we design standardized evaluation procedures based on four real-world use cases: future link prediction, destination node retrieval, edge classification, and textual relation generation. These tasks require models to understand both dynamic graph structures and natural language, highlighting the unique challenges posed by DyTAGs. Moreover, we conduct extensive benchmark experiments on DTGB, evaluating 7 popular dynamic graph learning algorithms and their variants of adapting to text attributes with LLM embeddings, along with 6 powerful large language models (LLMs). Our results show the limitations of existing models in handling DyTAGs. Our analysis also demonstrates the utility of DTGB in investigating the incorporation of structural and textual dynamics. The proposed DTGB fosters research on DyTAGs and their broad applications. It offers a comprehensive benchmark for evaluating and advancing models to handle the interplay between dynamic graph structures and natural language. The dataset and source code are available at https://github.com/zjs123/DTGB.
Dynamic graphs provide a flexible data abstraction for modelling many sorts of real-world systems, such as transport, trade, and social networks. Graph neural networks (GNNs) are powerful tools allowing for different kinds of prediction and inference on these systems, but getting a handle on uncertainty, especially in dynamic settings, is a challenging problem. In this work we propose to use a dynamic graph representation known in the tensor literature as the unfolding, to achieve valid prediction sets via conformal prediction. This representation, a simple graph, can be input to any standard GNN and does not require any modification to existing GNN architectures or conformal prediction routines. One of our key contributions is a careful mathematical consideration of the different inference scenarios which can arise in a dynamic graph modelling context. For a range of practically relevant cases, we obtain valid prediction sets with almost no assumptions, even dispensing with exchangeability. In a more challenging scenario, which we call the semi-inductive regime, we achieve valid prediction under stronger assumptions, akin to stationarity. We provide real data examples demonstrating validity, showing improved accuracy over baselines, and sign-posting different failure modes which can occur when those assumptions are violated.
Influence estimation aims to predict the total influence spread in social networks and has received surged attention in recent years. Most current studies focus on estimating the total number of influenced users in a social network, and neglect susceptibility estimation that aims to predict the probability of each user being influenced from the individual perspective. As a more fine-grained estimation task, susceptibility estimation is full of attractiveness and practical value. Based on the significance of susceptibility estimation and dynamic properties of social networks, we propose a task, called susceptibility estimation in dynamic social networks, which is even more realistic and valuable in real-world applications. Susceptibility estimation in dynamic networks has yet to be explored so far and is computationally intractable to naively adopt Monte Carlo simulation to obtain the results. To this end, we propose a novel end-to-end framework DySuse based on dynamic graph embedding technology. Specifically, we leverage a structural feature module to independently capture the structural information of influence diffusion on each single graph snapshot. Besides, {we propose the progressive mechanism according to the property of influence diffusion,} to couple the structural and temporal information during diffusion tightly. Moreover, a self-attention block {is designed to} further capture temporal dependency by flexibly weighting historical timestamps. Experimental results show that our framework is superior to the existing dynamic graph embedding models and has satisfactory prediction performance in multiple influence diffusion models.
Dynamic graph embedding has emerged as a very effective technique for addressing diverse temporal graph analytic tasks (i.e., link prediction, node classification, recommender systems, anomaly detection, and graph generation) in various applications. Such temporal graphs exhibit heterogeneous transient dynamics, varying time intervals, and highly evolving node features throughout their evolution. Hence, incorporating long-range dependencies from the historical graph context plays a crucial role in accurately learning their temporal dynamics. In this paper, we develop a graph embedding model with uncertainty quantification, TransformerG2G, by exploiting the advanced transformer encoder to first learn intermediate node representations from its current state ($t$) and previous context (over timestamps [$t-1, t-l$], $l$ is the length of context). Moreover, we employ two projection layers to generate lower-dimensional multivariate Gaussian distributions as each node's latent embedding at timestamp $t$. We consider diverse benchmarks with varying levels of ``novelty" as measured by the TEA (Temporal Edge Appearance) plots. Our experiments demonstrate that the proposed TransformerG2G model outperforms conventional multi-step methods and our prior work (DynG2G) in terms of both link prediction accuracy and computational efficiency, especially for high degree of novelty. Furthermore, the learned time-dependent attention weights across multiple graph snapshots reveal the development of an automatic adaptive time stepping enabled by the transformer. Importantly, by examining the attention weights, we can uncover temporal dependencies, identify influential elements, and gain insights into the complex interactions within the graph structure. For example, we identified a strong correlation between attention weights and node degree at the various stages of the graph topology evolution.
This paper presents ATEM, a novel framework for studying topic evolution in scientific archives. ATEM is based on dynamic topic modeling and dynamic graph embedding techniques that explore the dynamics of content and citations of documents within a scientific corpus. ATEM explores a new notion of contextual emergence for the discovery of emerging interdisciplinary research topics based on the dynamics of citation links in topic clusters. Our experiments show that ATEM can efficiently detect emerging cross-disciplinary topics within the DBLP archive of over five million computer science articles.
Dynamic interaction graphs have been widely adopted to model the evolution of user-item interactions over time. There are two crucial factors when modelling user preferences for link prediction in dynamic interaction graphs: 1) collaborative relationship among users and 2) user personalized interaction patterns. Existing methods often implicitly consider these two factors together, which may lead to noisy user modelling when the two factors diverge. In addition, they usually require time-consuming parameter learning with back-propagation, which is prohibitive for real-time user preference modelling. To this end, this paper proposes FreeGEM, a parameter-free dynamic graph embedding method for link prediction. Firstly, to take advantage of the collaborative relationships, we propose an incremental graph embedding engine to obtain user/item embeddings, which is an Online-Monitor-Offline architecture consisting of an Online module to approximately embed users/items over time, a Monitor module to estimate the approximation error in real time and an Offline module to calibrate the user/item embeddings when the online approximation errors exceed a threshold. Meanwhile, we integrate attribute information into the model, which enables FreeGEM to better model users belonging to some under represented groups. Secondly, we design a personalized dynamic interaction pattern modeller, which combines dynamic time decay with attention mechanism to model user short-term interests. Experimental results on two link prediction tasks show that FreeGEM can outperform the state-of-the-art methods in accuracy while achieving over 36X improvement in efficiency. All code and datasets can be found in https://github.com/FudanCISL/FreeGEM.
Dynamic graphs refer to graphs whose structure dynamically changes over time. Despite the benefits of learning vertex representations (i.e., embeddings) for dynamic graphs, existing works merely view a dynamic graph as a sequence of changes within the vertex connections, neglecting the crucial asynchronous nature of such dynamics where the evolution of each local structure starts at different times and lasts for various durations. To maintain asynchronous structural evolutions within the graph, we innovatively formulate dynamic graphs as temporal edge sequences associated with joining time of vertices (ToV) and timespan of edges (ToE). Then, a time-aware Transformer is proposed to embed vertices' dynamic connections and ToEs into the learned vertex representations. Meanwhile, we treat each edge sequence as a whole and embed its ToV of the first vertex to further encode the time-sensitive information. Extensive evaluations on several datasets show that our approach outperforms the state-of-the-art in a wide range of graph mining tasks. At the same time, it is very efficient and scalable for embedding large-scale dynamic graphs.
Increased attention has been paid over the last four years to dynamic network embedding. Existing dynamic embedding methods, however, consider the problem as limited to the evolution of a topology over a sequence of global, discrete states. We propose a novel embedding algorithm, WalkingTime, based on a fundamentally different handling of time, allowing for the local consideration of continuously occurring phenomena; while others consider global time-steps to be first-order citizens of the dynamic environment, we hold flows comprised of temporally and topologically local interactions as our primitives, without any discretization or alignment of time-related attributes being necessary. Keywords: dynamic networks , representation learning , dynamic graph embedding , time-respecting paths , temporal-topological flows , temporal random walks , temporal networks , real-attributed knowledge graphs , streaming graphs , online networks , asynchronous graphs , asynchronous networks , graph algorithms , deep learning , network analysis , datamining , network science
Dynamic graph embedding has gained great attention recently due to its capability of learning low dimensional graph representations for complex temporal graphs with high accuracy. However, recent advances mostly focus on learning node embeddings as deterministic "vectors" for static graphs yet disregarding the key graph temporal dynamics and the evolving uncertainties associated with node embedding in the latent space. In this work, we propose an efficient stochastic dynamic graph embedding method (DynG2G) that applies an inductive feed-forward encoder trained with node triplet-based contrastive loss. Every node per timestamp is encoded as a time-dependent probabilistic multivariate Gaussian distribution in the latent space, hence we can quantify the node embedding uncertainty on-the-fly. We adopted eight different benchmarks that represent diversity in size (from 96 nodes to 87,626 and from 13,398 edges to 4,870,863) and diversity in dynamics. We demonstrate via extensive experiments on these eight dynamic graph benchmarks that DynG2G achieves new state-of-the-art performance in capturing the underlying temporal node embeddings. We also demonstrate that DynG2G can predict the evolving node embedding uncertainty, which plays a crucial role in quantifying the intrinsic dimensionality of the dynamical system over time. We obtain a universal relation of the optimal embedding dimension, $L_o$, versus the effective dimensionality of uncertainty, $D_u$, and we infer that $L_o=D_u$ for all cases. This implies that the uncertainty quantification approach we employ in the DynG2G correctly captures the intrinsic dimensionality of the dynamics of such evolving graphs despite the diverse nature and composition of the graphs at each timestamp. Moreover, this $L_0 - D_u$ correlation provides a clear path to select adaptively the optimum embedding size at each timestamp by setting $L \ge D_u$.
A smart Ponzi scheme is a new form of economic crime that uses Ethereum smart contract account and cryptocurrency to implement Ponzi scheme. The smart Ponzi scheme has harmed the interests of many investors, but researches on smart Ponzi scheme detection is still very limited. The existing smart Ponzi scheme detection methods have the problems of requiring many human resources in feature engineering and poor model portability. To solve these problems, we propose a data-driven smart Ponzi scheme detection system in this paper. The system uses dynamic graph embedding technology to automatically learn the representation of an account based on multi-source and multi-modal data related to account transactions. Compared with traditional methods, the proposed system requires very limited human-computer interaction. To the best of our knowledge, this is the first work to implement smart Ponzi scheme detection through dynamic graph embedding. Experimental results show that this method is significantly better than the existing smart Ponzi scheme detection methods.
Graph embedding, aiming to learn low-dimensional representations (aka. embeddings) of nodes, has received significant attention recently. Recent years have witnessed a surge of efforts made on static graphs, among which Graph Convolutional Network (GCN) has emerged as an effective class of models. However, these methods mainly focus on the static graph embedding. In this work, we propose an efficient dynamic graph embedding approach, Dynamic Graph Convolutional Network (DyGCN), which is an extension of GCN-based methods. We naturally generalizes the embedding propagation scheme of GCN to dynamic setting in an efficient manner, which is to propagate the change along the graph to update node embeddings. The most affected nodes are first updated, and then their changes are propagated to the further nodes and leads to their update. Extensive experiments conducted on various dynamic graphs demonstrate that our model can update the node embeddings in a time-saving and performance-preserving way.
Dynamic graphs arise in a plethora of practical scenarios such as social networks, communication networks, and financial transaction networks. Given a dynamic graph, it is fundamental and essential to learn a graph representation that is expected not only to preserve structural proximity but also jointly capture the time-evolving patterns. Recently, graph convolutional network (GCN) has been widely explored and used in non-Euclidean application domains. The main success of GCN, especially in handling dependencies and passing messages within nodes, lies in its approximation to Laplacian smoothing. As a matter of fact, this smoothing technique can not only encourage must-link node pairs to get closer but also push cannot-link pairs to shrink together, which potentially cause serious feature shrink or oversmoothing problem, especially when stacking graph convolution in multiple layers or steps. For learning time-evolving patterns, a natural solution is to preserve historical state and combine it with the current interactions to obtain the most recent representation. Then the serious feature shrink or oversmoothing problem could happen when stacking graph convolution explicitly or implicitly according to current prevalent methods, which would make nodes too similar to distinguish each other. To solve this problem in dynamic graph embedding, we analyze the shrinking properties in the node embedding space at first, and then design a simple yet versatile method, which exploits L2 feature normalization constraint to rescale all nodes to hypersphere of a unit ball so that nodes would not shrink together, and yet similar nodes can still get closer. Extensive experiments on four real-world dynamic graph datasets compared with competitive baseline models demonstrate the effectiveness of the proposed method.
Embedding static graphs in low-dimensional vector spaces plays a key role in network analytics and inference, supporting applications like node classification, link prediction, and graph visualization. However, many real-world networks present dynamic behavior, including topological evolution, feature evolution, and diffusion. Therefore, several methods for embedding dynamic graphs have been proposed to learn network representations over time, facing novel challenges, such as time-domain modeling, temporal features to be captured, and the temporal granularity to be embedded. In this survey, we overview dynamic graph embedding, discussing its fundamentals and the recent advances developed so far. We introduce the formal definition of dynamic graph embedding, focusing on the problem setting and introducing a novel taxonomy for dynamic graph embedding input and output. We further explore different dynamic behaviors that may be encompassed by embeddings, classifying by topological evolution, feature evolution, and processes on networks. Afterward, we describe existing techniques and propose a taxonomy for dynamic graph embedding techniques based on algorithmic approaches, from matrix and tensor factorization to deep learning, random walks, and temporal point processes. We also elucidate main applications, including dynamic link prediction, anomaly detection, and diffusion prediction, and we further state some promising research directions in the area.
Graph analytics can lead to better quantitative understanding and control of complex networks, but traditional methods suffer from high computational cost and excessive memory requirements associated with the high-dimensionality and heterogeneous characteristics of industrial size networks. Graph embedding techniques can be effective in converting high-dimensional sparse graphs into low-dimensional, dense and continuous vector spaces, preserving maximally the graph structure properties. Another type of emerging graph embedding employs Gaussian distribution-based graph embedding with important uncertainty estimation. The main goal of graph embedding methods is to pack every node's properties into a vector with a smaller dimension, hence, node similarity in the original complex irregular spaces can be easily quantified in the embedded vector spaces using standard metrics. The generated nonlinear and highly informative graph embeddings in the latent space can be conveniently used to address different downstream graph analytics tasks (e.g., node classification, link prediction, community detection, visualization, etc.). In this Review, we present some fundamental concepts in graph analytics and graph embedding methods, focusing in particular on random walk-based and neural network-based methods. We also discuss the emerging deep learning-based dynamic graph embedding methods. We highlight the distinct advantages of graph embedding methods in four diverse applications, and present implementation details and references to open-source software as well as available databases in the Appendix for the interested readers to start their exploration into graph analytics.
Graph representation learning is a fundamental task in various applications that strives to learn low-dimensional embeddings for nodes that can preserve graph topology information. However, many existing methods focus on static graphs while ignoring evolving graph patterns. Inspired by the success of graph convolutional networks(GCNs) in static graph embedding, we propose a novel k-core based temporal graph convolutional network, the CTGCN, to learn node representations for dynamic graphs. In contrast to previous dynamic graph embedding methods, CTGCN can preserve both local connective proximity and global structural similarity while simultaneously capturing graph dynamics. In the proposed framework, the traditional graph convolution is generalized into two phases, feature transformation and feature aggregation, which gives the CTGCN more flexibility and enables the CTGCN to learn connective and structural information under the same framework. Experimental results on 7 real-world graphs demonstrate that the CTGCN outperforms existing state-of-the-art graph embedding methods in several tasks, including link prediction and structural role classification. The source code of this work can be obtained from \url{https://github.com/jhljx/CTGCN}.