Abstract Inferring tie strengths ( strong vs. weak ) is a core task in network analysis, often guided by the Strong Triadic Closure (STC) principle. In multilayer networks, such as social platforms or biological systems, applying STC independently to each layer can lead to inconsistent tie labels, undermining interpretations that rely on coherent relationship semantics across layers. We propose new formulations, multilayer STC and its extension STC+, which are axiomatically grounded and enforce cross-layer consistency. These problems are NP-hard; we present efficient 2- and 6-approximation algorithms alongside exact solutions. Experiments on real-world networks demonstrate that our methods produce consistent tie strength labelings with a transparent structural justification, significantly improving over the baselines.

A temporal network is a dynamic graph where every edge is assigned an integer time label that indicates at which discrete time step the edge is available. We consider the problem of hierarchically decomposing the network and introduce an edge-based decomposition framework that unifies the core and truss decompositions for temporal networks while allowing us to consider the network's temporal dimension. Based on our new framework, we introduce the (k,∆)-core and (k,∆)-truss decompositions, which are generalizations of the classic k-core and k-truss decompositions for multigraphs. Moreover, we show how (k,∆)-cores and (k,∆)-trusses can be efficiently further decomposed to obtain spatially and temporally connected components. We evaluate the characteristics of our new decompositions and the efficiency of our algorithms. Moreover, we demonstrate how our (k,∆)-decompositions can be applied to analyze malicious content in a Twitter network to obtain insights that state-of-the-art baselines cannot obtain.

Abstract Inferring tie strengths in social networks is an essential task in social network analysis. Common approaches classify the ties as weak and strong ties based on the strong triadic closure (STC). The STC states that if for three nodes, A, B, and C, there are strong ties between A and B, as well as A and C, there has to be a (weak or strong) tie between B and C. A variant of the STC called STC+ allows adding a few new weak edges to obtain improved solutions. So far, most works discuss the STC or STC+ in static networks. However, modern large-scale social networks are usually highly dynamic, providing user contacts and communications as streams of edge updates. Temporal networks capture these dynamics. To apply the STC to temporal networks, we first generalize the STC and introduce a weighted version such that empirical a priori knowledge given in the form of edge weights is respected by the STC. Similarly, we introduce a generalized weighted version of the STC+. The weighted STC is hard to compute, and our main contribution is an efficient 2-approximation (resp. 3-approximation) streaming algorithm for the weighted STC (resp. STC+) in temporal networks. As a technical contribution, we introduce a fully dynamic k-approximation for the minimum weighted vertex cover problem in hypergraphs with edges of size k, which is a crucial component of our streaming algorithms. An empirical evaluation shows that the weighted STC leads to solutions that better capture the a priori knowledge given by the edge weights than the non-weighted STC. Moreover, we show that our streaming algorithm efficiently approximates the weighted STC in real-world large-scale social networks.

We consider a variant of the densest subgraph problem in networks with single or multiple edge attributes. For example, in a social network, the edge attributes may describe the type of relationship between users, such as friends, family, or acquaintances, or different types of communication. For conceptual simplicity, we view the attributes as edge colors. The new problem we address is to find a diverse densest subgraph that fulfills given requirements on the numbers of edges of specific colors. When searching for a dense social network community, our problem will enforce the requirement that the community is diverse according to criteria specified by the edge attributes. We show that the decision versions for finding exactly, at most, and at least h colored edges densest subgraph, where h is a vector of color requirements, are NP-complete, for already two colors. For the problem of finding a densest subgraph with at least h colored edges, we provide a linear-time constant-factor approximation algorithm when the input graph is sparse. On the way, we introduce the related at least h (non-colored) edges densest subgraph problem, show its hardness, and also provide a linear-time constant-factor approximation. In our experiments, we demonstrate the efficacy and efficiency of our new algorithms.

The H-index of a node in a static network is the maximum value h such that at least h of its neighbors have a degree of at least h. Recently, a generalized version, the n-th order H-index, was introduced, allowing to relate degree centrality, H-index, and the k-core of a node. We extend the n-th order H-index to temporal networks and define corresponding temporal centrality measures and temporal core decompositions. Our n-th order temporal H-index respects the reachability in temporal networks leading to node rankings, which reflect the importance of nodes in spreading processes. We derive natural decompositions of temporal networks into subgraphs with strong temporal coherence. We analyze a recursive computation scheme and develop a highly scalable streaming algorithm. Our experimental evaluation demonstrates the efficiency of our algorithms and the conceptional validity of our approach. Specifically, we show that the n-th order temporal H-index is a strong heuristic for identifying possible super-spreaders in evolving social networks and detects temporally well-connected components.

We introduce the temporal graphlet kernel for classifying dissemination processes in labeled temporal graphs. Such processes can be the spreading of (fake) news, infectious diseases, or computer viruses in dynamic networks. The networks are modeled as labeled temporal graphs, in which the edges exist at specific points in time, and node labels change over time. The classification problem asks to discriminate dissemination processes of different origins or parameters, e.g., diseases with different infection probabilities. Our new kernel represents labeled temporal graphs in the feature space of temporal graphlets, i.e., small subgraphs distinguished by their structure, time-dependent node labels, and chronological order of edges. We introduce variants of our kernel based on classes of graphlets that are efficiently countable. For the case of temporal wedges, we propose a highly efficient approximative kernel with low error in expectation. Our experimental evaluation shows that our kernels are computed faster than state-of-the-art methods and provide higher accuracy in many cases.

Temporal closeness is a generalization of the classical closeness centrality measure for analyzing evolving networks. The temporal closeness of a vertex v is defined as the sum of the reciprocals of the temporal distances to the other vertices. Ranking all vertices of a network according to the temporal closeness is computationally expensive as it leads to a single-source-all-destination (SSAD) temporal distance query starting from each vertex of the graph. To reduce the running time of temporal closeness computations, we introduce an index to speed up SSAD temporal distance queries called Substream index. We show that deciding if a Substream index of a given size exists is NP-complete and provide an efficient greedy approximation. Moreover, we improve the running time of the approximation using min- hashing and parallelization. Our evaluation with real-world temporal networks shows a running time improvement of up to one order of magnitude compared to the state-of-the-art temporal closeness ranking algorithms.

We propose the Temporal Walk Centrality, which quantifies the importance of a node by measuring its ability to obtain and distribute information in a temporal network. In contrast to the widely-used betweenness centrality, we assume that information does not necessarily spread on shortest paths but on temporal random walks that satisfy the time constraints of the network. We show that temporal walk centrality can identify nodes playing central roles in dissemination processes that might not be detected by related betweenness concepts and other common static and temporal centrality measures. We propose exact and approximation algorithms with different running times depending on the properties of the temporal network and parameters of our new centrality measure. A technical contribution is a general approach to lift existing algebraic methods for counting walks in static networks to temporal networks. Our experiments on real-world temporal networks show the efficiency and accuracy of our algorithms. Finally, we demonstrate that the rankings by temporal walk centrality often differ significantly from those of other state-of-the-art temporal centralities.

Abstract The closeness centrality of a vertex in a classical static graph is the reciprocal of the sum of the distances to all other vertices. However, networks are often dynamic and change over time. Temporal distances take these dynamics into account. In this work, we consider the harmonic temporal closeness with respect to the shortest duration distance. We introduce an efficient algorithm for computing the exact top-ktemporal closeness values and the corresponding vertices. The algorithm can be generalized to the task of computing all closeness values. Furthermore, we derive heuristic modifications that perform well on real-world data sets and drastically reduce the running times. For the case that edge traversal takes an equal amount of time for all edges, we lift two approximation algorithms to the temporal domain. The algorithms approximate the transitive closure of a temporal graph (which is an essential ingredient for the top-kalgorithm) and the temporal closeness for all vertices, respectively, with high probability. We experimentally evaluate all our new approaches on real-world data sets and show that they lead to drastically reduced running times while keeping high quality in many cases. Moreover, we demonstrate that the top-ktemporal and static closeness vertex sets differ quite largely in the considered temporal networks.

We consider the problem of efficiently computing the top- k temporal closeness values and the corresponding vertex sets in a given temporal network. The closeness centrality of a vertex in a classical static graph is the reciprocal of the sum of the distances to all other vertices. Temporal distances in networks that change over time take the dynamics into account. In this work, we consider the harmonic temporal closeness with respect to the shortest duration distance. We introduce an efficient algorithm for computing the exact top- k temporal closeness values and the corresponding vertices. The algorithm can be generalized to the task of computing all closeness values. Furthermore, we derive a modification leading to a heuristic that often performs well on the majority of real-world data sets and drastically reduces the running times. For the case that edge traversal takes an equal amount of time for all edges, we lift two approximation algorithms to the temporal domain. The algorithms approximate the transitive closure of a temporal graph (which is an essential ingredient for the top- k algorithm) and the temporal closeness for all vertices, respectively, with high probability. We experimentally evaluate all our new approaches on real-world data sets and show that they lead to drastically reduced running times while keeping high quality in many cases. Moreover, we demonstrate that the top- k temporal and static closeness vertex sets differ quite largely in real-world networks.

Many real-world graphs are temporal, e.g., in a social network persons only interact at specific points in time. This temporality directs possible dissemination processes on the graph, such as the spread of rumors, fake news, or diseases. However, the current state-of-the-art methods for supervised graph classification are designed mainly for static graphs and may not be able to capture temporal information. Hence, they are not powerful enough to distinguish between graphs modeling different dissemination processes. To address this, we introduce a framework to lift standard graph kernels to the temporal domain. We explore three different approaches and investigate the trade-offs between loss of temporal information and efficiency. Moreover, to handle large-scale graphs, we propose stochastic variants of our kernels with provable approximation guarantees. We evaluate our methods on various real-world social networks. Our methods beat static kernels by a large margin in terms of accuracy while still being scalable to large graphs and data sets. This confirms that taking temporal information into account is crucial for the successful classification of temporal graphs under consideration of dissemination processes.