Graph simulation has recently received a surge of attention in graph processing and analytics. In real-life applications, e.g. social science, biology, and chemistry, many graphs are composed of a series of evolving graphs (i.e., temporal graphs). While most of the existing graph generators focus on static graphs, the temporal information of the graphs is ignored. In this paper, we focus on simulating temporal graphs, which aim to reproduce the structural and temporal properties of the observed real-life temporal graphs. In this paper, we first give an overview of the existing temporal graph generators, including recently emerged learning-based approaches. Most of these learning-based methods suffer from one of the limitations: low efficiency in training or slow generating, especially for temporal random walk-based methods. Therefore, we propose an efficient learning-based approach to generate graph snapshots, namely temporal graph autoencoder (TGAE). Specifically, we propose an attention-based graph encoder to encode temporal and structural characteristics on sampled ego-graphs. And we proposed an ego-graph decoder that can achieve a good trade-off between simulation quality and efficiency in temporal graph generation. Finally, the experimental evaluation is conducted among our proposed TGAE and representative temporal graph generators on real-life temporal graphs and synthesized graphs. It is reported that our proposed approach outperforms the state-of-the-art temporal graph generators by means of simulation quality and efficiency.

Time-evolving graphs, such as social and citation networks, often contain noise that distorts structural and temporal patterns, adversely affecting downstream tasks, such as node classification. Existing purification methods focus on static graphs, limiting their ability to account for critical temporal dependencies in dynamic graphs. In this work, we propose TiGer (Time-evolving Graph purifier), a self-supervised method explicitly designed for time-evolving graphs. TiGer assigns two different sub-scores to edges using (1) self-attention for capturing long-term contextual patterns shaped by both adjacent and distant past events of varying significance and (2) statistical distance measures for detecting inconsistency over a short-term period. These sub-scores are used to identify and filter out suspicious (i.e., noise-like) edges through an ensemble strategy, ensuring robustness without requiring noise labels. Our experiments on five real-world datasets show TiGer filters out noise with up to 10.2% higher accuracy and improves node classification performance by up to 5.3%, compared to state-of-the-art methods.

This perspective article is meant to be a self-contained introduction to and review of transfer operators such as the Koopman operator and the Perron-Frobenius operator as well as an overview of different applications. We will first introduce the required foundations and then show how transfer operators can not only be used to analyze highly nonlinear dynamical systems but also complex networks. In particular, we will focus on relationships between transfer operators for continuous-time stochastic processes defined on a continuous state space--whether they be reversible, non-reversible but time-homogeneous, or time-inhomogeneous--and their discrete counterparts associated with random walks on undirected, directed, and time-evolving graphs. Transfer operators play an important role in an increasing number of research fields. A few exemplary applications are illustrated in Figure 1.

While spectral clustering algorithms for undirected graphs are well established and have been successfully applied to unsupervised machine learning problems ranging from image segmentation and genome sequencing to signal processing and social network analysis, clustering directed graphs remains notoriously difficult. Two of the main challenges are that the eigenvalues and eigenvectors of graph Laplacians associated with directed graphs are in general complex-valued and that there is no universally accepted definition of clusters in directed graphs. We first exploit relationships between the graph Laplacian and transfer operators and in particular between clusters in undirected graphs and metastable sets in stochastic dynamical systems and then use a generalization of the notion of metastability to derive clustering algorithms for directed and time-evolving graphs. The resulting clusters can be interpreted as coherent sets, which play an important role in the analysis of transport and mixing processes in fluid flows.

Large-scale perturbations in the microbiome constitution are strongly correlated, whether as a driver or a consequence, with the health and functioning of human physiology. However, understanding the difference in the microbiome profiles of healthy and ill individuals can be complicated due to the large number of complex interactions among microbes. We propose to model these interactions as a time-evolving graph whose nodes are microbes and edges are interactions among them. Motivated by the need to analyse such complex interactions, we develop a method that learns a low-dimensional representation of the time-evolving graph and maintains the dynamics occurring in the high-dimensional space. Through our experiments, we show that we can extract graph features such as clusters of nodes or edges that have the highest impact on the model to learn the low-dimensional representation. This information can be crucial to identify microbes and interactions among them that are strongly correlated with clinical diseases. We conduct our experiments on both synthetic and real-world microbiome datasets.

The success of neural networks (NNs) in a wide range of applications has led to increased interest in understanding the underlying learning dynamics of these models. In this paper, we go beyond mere descriptions of the learning dynamics by taking a graph perspective and investigating the relationship between the graph structure of NNs and their performance. Specifically, we propose (1) representing the neural network learning process as a time-evolving graph (i.e., a series of static graph snapshots over epochs), (2) capturing the structural changes of the NN during the training phase in a simple temporal summary, and (3) leveraging the structural summary to predict the accuracy of the underlying NN in a classification or regression task. For the dynamic graph representation of NNs, we explore structural representations for fully-connected and convolutional layers, which are key components of powerful NN models. Our analysis shows that a simple summary of graph statistics, such as weighted degree and eigenvector centrality, over just a few epochs can be used to accurately predict the performance of NNs. For example, a weighted degree-based summary of the time-evolving graph that is constructed based on 5 training epochs of the LeNet architecture achieves classification accuracy of over 93%. Our findings are consistent for different NN architectures, including LeNet, VGG, AlexNet and ResNet.

More and more diseases have been found to be strongly correlated with disturbances in the microbiome constitution, e.g., obesity, diabetes, or some cancer types. Thanks to modern high-throughput omics technologies, it becomes possible to directly analyze human microbiome and its influence on the health status. Microbial communities are monitored over long periods of time and the associations between their members are explored. These relationships can be described by a time-evolving graph. In order to understand responses of the microbial community members to a distinct range of perturbations such as antibiotics exposure or diseases and general dynamical properties, the time-evolving graph of the human microbial communities has to be analyzed. This becomes especially challenging due to dozens of complex interactions among microbes and metastable dynamics. The key to solving this problem is the representation of the time-evolving graphs as fixed-length feature vectors preserving the original dynamics. We propose a method for learning the embedding of the time-evolving graph that is based on the spectral analysis of transfer operators and graph kernels. We demonstrate that our method can capture temporary changes in the time-evolving graph on both created synthetic data and real-world data. Our experiments demonstrate the efficacy of the method. Furthermore, we show that our method can be applied to human microbiome data to study dynamic processes.

In this paper we use a time-evolving graph which consists of a sequence of graph snapshots over time to model many real-world networks. We study the path classification problem in a time-evolving graph, which has many applications in real-world scenarios, for example, predicting path failure in a telecommunication network and predicting path congestion in a traffic network in the near future. In order to capture the temporal dependency and graph structure dynamics, we design a novel deep neural network named Long Short-Term Memory R-GCN (LRGCN). LRGCN considers temporal dependency between time-adjacent graph snapshots as a special relation with memory, and uses relational GCN to jointly process both intra-time and inter-time relations. We also propose a new path representation method named self-attentive path embedding (SAPE), to embed paths of arbitrary length into fixed-length vectors. Through experiments on a real-world telecommunication network and a traffic network in California, we demonstrate the superiority of LRGCN to other competing methods in path failure prediction, and prove the effectiveness of SAPE on path representation.