Graph Metrics for Temporal Networks

Temporal networks, i.e., networks in which the interactions among a set of elementary units change over time, can be modelled in terms of time-varying graphs, which are time-ordered sequences of graphs over a set of nodes. In such graphs, the concepts of node adjacency and reachability crucially depend on the exact temporal ordering of the links. Consequently, all the concepts and metrics proposed and used for the characterisation of static complex networks have to be redefined or appropriately extended to time-varying graphs, in order to take into account the effects of time ordering on causality. In this chapter we discuss how to represent temporal networks and we review the definitions of walks, paths, connectedness and connected components valid for graphs in which the links fluctuate over time. We then focus on temporal node-node distance, and we discuss how to characterise link persistence and the temporal small-world behaviour in this class of networks. Finally, we discuss the extension of classic centrality measures, including closeness, betweenness and spectral centrality, to the case of time-varying graphs, and we review the work on temporal motifs analysis and the definition of modularity for temporal graphs.Comment: 26 pages, 5 figures, Chapter in Temporal Networks (Petter Holme and Jari Saram\"aki editors). Springer. Berlin, Heidelberg 201

DYMOND: DYnamic MOtif-NoDes Network Generative Model

Motifs, which have been established as building blocks for network structure, move beyond pair-wise connections to capture longer-range correlations in connections and activity. In spite of this, there are few generative graph models that consider higher-order network structures and even fewer that focus on using motifs in models of dynamic graphs. Most existing generative models for temporal graphs strictly grow the networks via edge addition, and the models are evaluated using static graph structure metrics -- which do not adequately capture the temporal behavior of the network. To address these issues, in this work we propose DYnamic MOtif-NoDes (DYMOND) -- a generative model that considers (i) the dynamic changes in overall graph structure using temporal motif activity and (ii) the roles nodes play in motifs (e.g., one node plays the hub role in a wedge, while the remaining two act as spokes). We compare DYMOND to three dynamic graph generative model baselines on real-world networks and show that DYMOND performs better at generating graph structure and node behavior similar to the observed network. We also propose a new methodology to adapt graph structure metrics to better evaluate the temporal aspect of the network. These metrics take into account the changes in overall graph structure and the individual nodes' behavior over time.Comment: In Proceedings of the Web Conference 2021 (WWW '21

Modulus on temporal graphs

Doctor of PhilosophyDepartment of MathematicsNathan Albinp-Modulus was adopted by Graph Theory from Complex Analysis and became a powerful and versatile tool for exploring the structure of graphs and networks. Clustering and community detection, the construction of a large class of graph metrics, measures of centrality, hierarchical graph decomposition, and the solution to game-theoretic models of secure network broadcast are successful applications. The flexibility of p-modulus allows it to be applied to a wide variety of graph types; the essential definitions are easily adapted for directed or undirected graphs, for weighted or unweighted graphs, for simple graphs or multigraphs, or even for hypergraphs. In this work, we study application of p-modulus to temporal networks --- that is, networks whose structure can change over time. We introduce a definition of temporal p-modulus for a temporal graph with discrete time availability moments assigned to edges. We provide theorems allowing to adjust static p-modulus algorithms to compute temporal p-modulus. We present the dual problem. For the p=2 case, it has a probabilistic interpretation. Also, we give a mass-transportation interpretation of temporal Modulus for various values of p between 1 and infinity including the endpoints. We consider different types of penalty function and study properties of temporal p-modulus with it. We show examples on several standard graphs modified to temporal ones. Time-respecting paths are given special attention. In order to obtain an efficient numerical algorithm to compute p-modulus we adjust the well-know Dijkstra algorithm to find optimal paths by two criteria --- that is, time of arrival and static length. We compare this algorithm with another way: building a directed static graph equivalent to a given temporal one and running classic Dijkstra algorithm. Computational complexity estimates for both algorithms are the same

Explainable Spatio-Temporal Graph Neural Networks

Spatio-temporal graph neural networks (STGNNs) have gained popularity as a powerful tool for effectively modeling spatio-temporal dependencies in diverse real-world urban applications, including intelligent transportation and public safety. However, the black-box nature of STGNNs limits their interpretability, hindering their application in scenarios related to urban resource allocation and policy formulation. To bridge this gap, we propose an Explainable Spatio-Temporal Graph Neural Networks (STExplainer) framework that enhances STGNNs with inherent explainability, enabling them to provide accurate predictions and faithful explanations simultaneously. Our framework integrates a unified spatio-temporal graph attention network with a positional information fusion layer as the STG encoder and decoder, respectively. Furthermore, we propose a structure distillation approach based on the Graph Information Bottleneck (GIB) principle with an explainable objective, which is instantiated by the STG encoder and decoder. Through extensive experiments, we demonstrate that our STExplainer outperforms state-of-the-art baselines in terms of predictive accuracy and explainability metrics (i.e., sparsity and fidelity) on traffic and crime prediction tasks. Furthermore, our model exhibits superior representation ability in alleviating data missing and sparsity issues. The implementation code is available at: https://github.com/HKUDS/STExplainer.Comment: 32nd ACM International Conference on Information and Knowledge Management (CIKM' 23

Intrinsically Dynamic Network Communities

Community finding algorithms for networks have recently been extended to dynamic data. Most of these recent methods aim at exhibiting community partitions from successive graph snapshots and thereafter connecting or smoothing these partitions using clever time-dependent features and sampling techniques. These approaches are nonetheless achieving longitudinal rather than dynamic community detection. We assume that communities are fundamentally defined by the repetition of interactions among a set of nodes over time. According to this definition, analyzing the data by considering successive snapshots induces a significant loss of information: we suggest that it blurs essentially dynamic phenomena - such as communities based on repeated inter-temporal interactions, nodes switching from a community to another across time, or the possibility that a community survives while its members are being integrally replaced over a longer time period. We propose a formalism which aims at tackling this issue in the context of time-directed datasets (such as citation networks), and present several illustrations on both empirical and synthetic dynamic networks. We eventually introduce intrinsically dynamic metrics to qualify temporal community structure and emphasize their possible role as an estimator of the quality of the community detection - taking into account the fact that various empirical contexts may call for distinct `community' definitions and detection criteria.Comment: 27 pages, 11 figure

Brain Connectivity in Late-Life Depression and Aging Revealed by Network Analysis

Objective To use novel methods to examine age associations across an integrated brain network in healthy older adults (HOA) and individuals with late-life depression (LLD). Graph theory metrics describe the organizational configuration of both the global network and specified brain regions. Methods Cross-sectional data were acquired. Graph theory was used to explore diffusion tensor imaging–derived white matter networks. Forty-eight HOA and 28 adults with LLD were recruited from the community. Global and local metrics in prefrontal, cingulate, and temporal regions were calculated. Group differences and associations with age were explored. Results Group differences were noted in local metrics of the right prefrontal and temporal regions, but no significant differences were observed on global metrics. Local (not global) metrics were associated with age differently across groups. For HOA, local metrics across all regions correlated with age, whereas in adults with LLD, correlations were only observed within temporal regions. In keeping with hypothesized regions impacted by LLD, stronger hubs in right temporal regions were observed among HOA, whereas LLD individuals were characterized by robust hubs in frontal regions. Conclusion We demonstrate widespread age-related changes in local network properties among HOA with different and more restricted local changes in LLD. Although a preliminary analysis, different patterns of correlations in local networks coupled with equivalent global metrics may reflect altered local structural brain networks in patients with LLD