Temporal Graph Classes: A View Through Temporal Separators

Dynamics of interactions play an increasingly important role in the analysis of complex networks. A modeling framework to capture this is temporal graphs which consist of a set of vertices (entities in the network) and a set of time-stamped binary interactions between the vertices. We focus on enumerating Δ-cliques, an extension of the concept of cliques to temporal graphs: for a given time period Δ, a Δ-clique in a temporal graph is a set of vertices and a time interval such that all vertices interact with each other at least after every Δ time steps within the time interval. Viard, Latapy, and Magnien (ASONAM 2015, TCS 2016) proposed a greedy algorithm for enumerating all maximal Δ-cliques in temporal graphs. In contrast to this approach, we adapt the Bron–Kerbosch algorithm—an efficient, recursive backtracking algorithm which enumerates all maximal cliques in static graphs—to the temporal setting. We obtain encouraging results both in theory (concerning worst-case running time analysis based on the parameter “Δ-slice degeneracy” of the underlying graph) as well as in practice with experiments on real-world data. The latter culminates in an improvement for most interesting Δ-values concerning running time in comparison with the algorithm of Viard, Latapy, and Magnien.

Maximum 0-1 timed matching on temporal graphs

SimRank is a significant metric to measure the similarity of nodes in graph data analysis. The problem of SimRank computation has been studied extensively, however there is no existing work that can provide one unified algorithm to support the SimRank computation both on static and temporal graphs. In this work, we first propose CrashSim, an index-free algorithm for single-source SimRank computation in static graphs. CrashSim can provide provable approximation guarantees for the computational results in an efficient way. In addition, as the reallife graphs are often represented as temporal graphs, CrashSim enables efficient computation of SimRank in temporal graphs. We formally define two typical SimRank queries in temporal graphs, and then solve them by developing an efficient algorithm based on CrashSim, called CrashSim-T. From the extensive experimental evaluation using five real-life and synthetic datasets, it can be seen that the CrashSim algorithm and CrashSim-T algorithm substantially improve the efficiency of the state-of-the-art SimRank algorithms by about 30%, while achieving the precision of the result set with about 97%.

Multiplex graphs have been recently proposed as a model to represent high-level complexity in real-world networks such as heterogeneous social networks where actors could be characterized by heterogeneous properties and could be linked with different types of social interactions. This has brought new challenges in community detection, which aims to identify pertinent groups of nodes in a complex graph. In this context, great efforts have been made to tackle the problem of community detection in multiplex graphs. However, most of the proposed methods until recently deal with static multiplex graph and ignore the temporal dimension, which is a key characteristic of real networks. Even more, the few methods that consider temporal graphs, they just propose to follow communities over time and none of them use the temporal aspect directly to detect stable communities, which are often more meaningful in reality. Thus, this paper proposes a new two-step method to detect stable communities in temporal multiplex graphs. The first step aims to find the best static graph partition at each instant by applying a new hybrid community detection algorithm, which considers both relations heterogeneities and nodes similarities. Then, the second step considers the temporal dimension in order to find final stable communities. Finally, experiments on synthetic graphs and a real social network show that this method is competitive and it is able to extract high-quality communities.

Being an influencer is hard: The complexity of influence maximization in temporal graphs with a fixed source

We consider the influence maximization problem over a temporal graph, where there is a single fixed source. We deviate from the standard model of influence maximization, where the goal is to choose the set of most influential vertices. Instead, in our model we are given a fixed vertex, or source, and the goal is to find the best time steps to transmit so that the influence of this vertex is maximized. We frame this problem as a spreading process that follows a variant of the susceptible-infected-susceptible (SIS) model and we focus on three objective functions. In the MaxSpread objective, the goal is to maximize the total number of vertices that get infected at least once. In the MaxViral objective, the goal is to maximize the number of vertices that are infected at the same time step. Finally, in MaxViralTstep, the goal is to maximize the number of vertices that are infected at a given time step. We perform a thorough complexity theoretic analysis for these three objectives over three different scenarios: (1) the unconstrained setting where the source can transmit whenever it wants; (2) the window-constrained setting where the source has to transmit at either a predetermined, or a shifting window; (3) the periodic setting where the temporal graph has a small period. We prove that all of these problems, with the exception of MaxSpread for periodic graphs, are intractable even for very simple underlying graphs.

We study a family of reachability problems under waiting-time restrictions in temporal and vertex-colored temporal graphs. Given a temporal graph and a set of source vertices, we find the set of vertices that are reachable from a source via a time-respecting path, where the difference in timestamps between consecutive edges is at most a resting time. Given a vertex-colored temporal graph and a multiset query of colors, we find the set of vertices reachable from a source via a time-respecting path such that the vertex colors of the path agree with the multiset query and the difference in timestamps between consecutive edges is at most a resting time. These kinds of problems have applications in understanding the spread of a disease in a network, tracing contacts in epidemic outbreaks, finding signaling pathways in the brain network, and recommending tours for tourists, among others. We present an algebraic algorithmic framework based on constrained multilinear sieving for solving the restless reachability problems we propose. In particular, parameterized by the length k of a path sought, we show that the proposed problems can be solved in O(2^k k m Delta ) time and O(n Delta ) space, where n is the number of vertices, m the number of edges, and Delta the maximum resting time of an input temporal graph. The approach can be extended to extract paths and connected subgraphs in both static and temporal graphs, thus improving the work of Björklund et al. (in Proceedings of the European symposium on algorithms, 2014) and Thejaswi et al. (Big Data 8:335–362, 2020). In addition, we prove that our algorithms for the restless reachability problems in vertex-colored temporal graphs are optimal under plausible complexity-theoretic assumptions. Finally, with an open-source implementation, we demonstrate that our algorithm scales to large graphs with up to one billion temporal edges, despite the problems being NP-hard. Specifically, we present extensive experiments to evaluate our scalability claims both on synthetic and on real-world graphs. Our implementation is efficiently engineered and highly optimized. For instance, we can solve the restless reachability problem by restricting the path length to 9 in a real-world graph dataset with over 36 million directed edges in less than one hour on a commodity desktop with a 4-core Haswell CPU.

With very fast global instant messaging services, communication networks of today are changing rapidly. To accurately represent such dynamic networks, a special type of a graph, known as temporal graph is indispensable. Majority of the community detection algorithms are designed for static graphs. So, an efficient conversion of temporal graphs to static graphs, while retaining important temporal information, enables the use of any standard community detection algorithm easily. This paper proposes a novel community detection method by constructing static graphs from temporal networks. The proposed method is validated through experiments. Results show that our technique leads to considerable runtime reduction upon dealing with large graphs comprise multiple community structures.

In static graphs, the betweenness centrality of a graph vertex measures how many times this vertex is part of a shortest path between any two graph vertices. Betweenness centrality is efficiently computable and it is a fundamental tool in network science. Continuing and extending previous work, we study the efficient computability of betweenness centrality in temporal graphs (graphs with fixed vertex set but time-varying arc sets). Unlike in the static case, there are numerous natural notions of being a “shortest” temporal path (walk). Depending on which notion is used, it was already observed that the problem is #P-hard in some cases while polynomial-time solvable in others. In this conceptual work, we contribute towards classifying what a “shortest path (walk) concept” has to fulfill in order to gain polynomial-time computability of temporal betweenness centrality.

The recent deep generative models for static graphs that are now being actively developed have achieved significant success in areas such as molecule design. However, many real-world problems involve temporal graphs whose topology and attribute values evolve dynamically over time, including important applications such as protein folding, human mobility networks, and social network growth. As yet, deep generative models for temporal graphs are not yet well understood and existing techniques for static graphs are not adequate for temporal graphs since they cannot 1) encode and decode continuously-varying graph topology chronologically, 2) enforce validity via temporal constraints, or 3) ensure efficiency for information-lossless temporal resolution. To address these challenges, we propose a new model, called ``Temporal Graph Generative Adversarial Network'' (TG-GAN) for continuous-time temporal graph generation, by modeling the deep generative process for truncated temporal random walks and their compositions. Specifically, we first propose a novel temporal graph generator that jointly model truncated edge sequences, time budgets, and node attributes, with novel activation functions that enforce temporal validity constraints under recurrent architecture. In addition, a new temporal graph discriminator is proposed, which combines time and node encoding operations over a recurrent architecture to distinguish the generated sequences from the real ones sampled by a newly-developed truncated temporal random walk sampler. Extensive experiments on both synthetic and real-world datasets demonstrate TG-GAN significantly outperforms the comparison methods in efficiency and effectiveness.

A variety of complex systems, including social and communication networks, financial markets, biology, and neuroscience are modeled using temporal graphs that contain a set of nodes and directed timestamped edges. Temporal motifs in temporal graphs are generalized from subgraph patterns in static graphs in that they also account for edge ordering and time duration, in addition to the graph structure. Mining temporal motifs is a fundamental problem used in several application domains. However, existing software frameworks offer suboptimal performance due to high algorithmic complexity and irregular memory accesses of temporal motif mining.This paper presents $\mathsf{Mint}$—a novel accelerator architecture and a programming model for mining temporal motifs efficiently. We first divide this workload into three fundamental tasks: search, book-keeping, and backtracking. Based on this, we propose a task-centric programming model that enables decoupled, asynchronous execution. This model unlocks massive opportunities for parallelism, and allows storing task context information on-chip. To best utilize the proposed programming model, we design a domain-specific hardware accelerator using its data path and memory subsystem design to cater to the unique workload characteristics of temporal motif mining. To further improve performance, we propose a novel optimization called search index memoization that significantly reduces memory traffic. We comprehensively compare the performance of $\mathsf{Mint}$ with state-of-the-art temporal motif mining software frameworks (both approximate and exact) running on both CPU and GPU, and show $9\times-2576\times$ benefit in performance.

Community detection has gained significant research interest within the data mining field. It involves identifying subsets of nodes with dense internal connections and sparse external connections. Most studies on community detection focus solely on identifying non-overlapping communities in a static graph. However, in practice, communities often overlap, and the structure of the graphs is dynamically evolving. This dynamic nature leads to community changes and poses a significant challenge in detecting overlapping communities on temporal graphs (T-OCD). While graph neural networks have shown great performance in generating node representations for community detection, learning representations that capture temporal graph structures and support overlapping community detection remain an open question. To address these challenges, we present T-OCDIB , a novel approach for T emporal O verlapping C ommunity D etection guided by I nformation B ottleneck. Specifically, we first propose an overlapping community detection approach for static graphs, under the guidance of a community-oriented information bottleneck. This approach allows us to learn discriminative node representations specific to each community, facilitating the detection of overlapping communities. Following this, we extend this method to temporal graphs by presenting a temporal convolution module. This module uses adaptive weight matrices based on evolving graph structures to capture temporal dependencies for community detection. Additionally, to promote smooth transitions between consecutive communities, we introduce a temporal smoothing module to further constrain changes in community structure. We evaluate the proposed approach on both real-world and synthetic temporal networks. Experimental results illustrate the superiority of T-OCDIB over other community detection methods.

Many real-world networks evolve over time, that is, new contacts appear and old contacts may disappear. They can be modeled as temporal graphs where interactions between vertices (which represent people in the case of social networks) are represented by time-stamped edges. One of the most fundamental problems in (social) network analysis is community detection, and one of the most basic primitives to model a community is a clique. Addressing the problem of finding communities in temporal networks, Viard et al. [TCS 2016] introduced $Δ$-cliques as a natural temporal version of cliques. Himmel et al. [SNAM 2017] showed how to adapt the well-known Bron-Kerbosch algorithm to enumerate $Δ$-cliques. We continue this work and improve and extend the algorithm of Himmel et al. to enumerate temporal $k$-plexes (notably, cliques are the special case $k=1$). We define a $Δ$-$k$-plex as a set of vertices and a time interval, where during this time interval each vertex has in each consecutive $Δ+ 1$ time steps at least one edge to all but at most $k-1$ vertices in the chosen set of vertices. We develop a recursive algorithm for enumerating all maximal $Δ$-$k$-plexs and perform experiments on real-world social networks that demonstrate the practical feasibility of our approach. In particular, for the special case of $Δ$-1-plexes (that is, $Δ$-cliques), we observe that our algorithm is on average significantly faster than the previous algorithms by Himmel et al. [SNAM 2017] and Viard et al. [IPL 2018] for enumerating $Δ$-cliques.

Hypothesis Generation (HG) aims to expedite biomedical researches by generating novel hypotheses from existing scientific literature. Most existing studies focused on modeling static snapshots of the corpus, neglecting the temporal evolution of scientific terms. Despite recent efforts to learn term evolution from Knowledge Bases (KBs) for HG, the temporal information from multi-source KBs is still overlooked, which contains important, up-to-date knowledge. In this paper, an innovative Temporal Contrastive Learning (TCL) framework is introduced to uncover latent associations between entities by jointly modeling their co-evolution across multi-source temporal KBs. Specifically, we first construct a temporal relation graph based on PubMed papers and a biomedical relation database (such as Comparative Toxicogenomics Database (CTD)). Then the constructed temporal relation graph and a temporal concept graph (such as Medical Subject Headings (MeSH)) are used to train two GCN-based recurrent networks for learning the entity temporal evolutional embeddings, respectively. Finally, a cross-view temporal prediction task is designed for learning knowledge enriched temporal embeddings by contrasting the temporal embeddings learned from the two Temporal Knowledge Graphs (TKGs). Findings from experiments conducted on three real-world biomedical term relationship datasets demonstrate that the proposed approach is clearly superior to approaches based on single TKG, achieving the state-of-the-art performance.

Contemporary data-systems empowering the daily human activity are routinely represented with graphs. During the last decade, the volume growth of such systems has been unprece-dented. This hinders the timely analysis of the formed networks due to existing physical memory limitations and significant I/O overheads. Graph compression techniques have managed to reduce memory requirements and allow for representing such networks using a few bits-per-edge. Respective approaches offer succinct mappings for social, biological, and information networks while allowing for the efficient access of sought graph elements. Despite their success, such methods mostly focus on static graphs, and predominantly offer access to either a snapshot or an aggregated view of a network. In reality however, networks change over time and, in many instances, we are interested in capturing and studying this evolution. In this paper we propose a framework for compressing emerging temporal graphs based on a dual-representation which articulates both network structure and corresponding temporal information. We empirically establish properties exhibited by community-networks regarding their time aspect(s) and harness these features in our proposed repre-sentation. Our experimental evaluation demonstrates that our approach for compressing temporal graphs readily outperforms competing techniques, attaining compression ratios that are on average around 60% of the space required by state-of-the-art techniques. Moreover, our memory-efficient representation yields more than 70 % faster graph compression and orders of magnitude quicker retrieval of graphs' elements, especially when it comes to large-scale networks. Finally, our framework is the first effort we are aware of, that considers actual time instead of time steps. This helps us attain better control for the size of our representation and reap further memory savings.