The graph edit distance (GED) is among the most widely used graph similarity measures in practice. It asks for a minimum cost edit path between two given labeled graphs G and H , where the edit path is defined as a sequence of operations (e.g., node and edge insertions, deletions or substitutions) that successively transform the graph G into H. In this work, we suggest a new ILP formulation (FORI) based on orienting the corresponding edge variables. Moreover, we suggest enhancing two state-of-the-art ILP formulations by incorporating additional inequalities. We theoretically compare the strength of the formulations with respect to their Linear Programming relaxations. The result is a hierarchy with (FORI) at the top. Our extensive evaluation on widely used benchmark sets shows that our improved formulations run significantly faster than the previous ones. These allow to solve to proven optimality all the reference instances from common databases, such as the IAM Graph Database, many of which were prohibitive with state-of-the-art methods. Moreover, we are able to compute the GED of a small pattern and a large graph such as CORA and PUBMED, having up to 19,717 nodes and 44,327 edges.

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.

Abstract Non-linear kernel methods can be approximated by fast linear ones using suitable explicit feature maps allowing their application to large scale problems. We investigate how convolution kernels for structured data are composed from base kernels and construct corresponding feature maps. On this basis we propose exact and approximative feature maps for widely used graph kernels based on the kernel trick. We analyze for which kernels and graph properties computation by explicit feature maps is feasible and actually more efficient. In particular, we derive approximative, explicit feature maps for state-of-the-art kernels supporting real-valued attributes including the GraphHopper and graph invariant kernels. In extensive experiments we show that our approaches often achieve a classification accuracy close to the exact methods based on the kernel trick, but require only a fraction of their running time. Moreover, we propose and analyze algorithms for computing random walk, shortest-path and subgraph matching kernels by explicit and implicit feature maps. Our theoretical results are confirmed experimentally by observing a phase transition when comparing running time with respect to label diversity, walk lengths and subgraph size, respectively.

Most state-of-the-art graph kernels only take local graph properties into account, i.e., the kernel is computed with regard to properties of the neighborhood of vertices or other small substructures. On the other hand, kernels that do take global graph properties into account may not scale well to large graph databases. Here we propose to start exploring the space between local and global graph kernels, so called glocalized graph kernels, striking the balance between both worlds. Specifically, we introduce a novel graph kernel based on the k-dimensional Weisfeiler-Lehman algorithm. Unfortunately, the k-dimensional Weisfeiler-Lehman algorithm scales exponentially in k. Consequently, we devise a stochastic version of the kernel with provable approximation guarantees using conditional Rademacher averages. On bounded-degree graphs, it can even be computed in constant time. We support our theoretical results with experiments on several graph classification benchmarks, showing that our kernels often outperform the state-of-the-art in terms of classification accuracies.

While state-of-the-art kernels for graphs with discrete labels scale well to graphs with thousands of nodes, the few existing kernels for graphs with continuous attributes, unfortunately, do not scale well. To overcome this limitation, we present hash graph kernels, a general framework to derive kernels for graphs with continuous attributes from discrete ones. The idea is to iteratively turn continuous attributes into discrete labels using randomized hash functions. We illustrate hash graph kernels for the Weisfeiler-Lehman subtree kernel and for the shortest-path kernel. The resulting novel graph kernels are shown to be, both, able to handle graphs with continuous attributes and scalable to large graphs and data sets. This is supported by our theoretical analysis and demonstrated by an extensive experimental evaluation.

As many real-world data can elegantly be represented as graphs, various graph kernels and methods for computing them have been proposed. Surprisingly, many of the recent graph kernels do not employ the kernel trick anymore but rather compute an explicit feature map and report higher efficiency. So, is there really no benefit of the kernel trick when it comes to graphs? Triggered by this question, we investigate under which conditions it is possible to compute a graph kernel explicitly and for which graph properties this computation is actually more efficient. We give a sufficient condition for R-convolution kernels that enables kernel computation by explicit mapping. We theoretically and experimentally analyze efficiency and flexibility of implicit kernel functions and dot products of explicitly computed feature maps for widely used graph kernels such as random walk kernels, sub graph matching kernels, and shortest-path kernels. For walk kernels we observe a phase transition when comparing runtime with respect to label diversity and walk lengths leading to the conclusion that explicit computations are only favourable for smaller label sets and walk lengths whereas implicit computation is superior for longer walk lengths and data sets with larger label diversity.

Efficient subgraph queries in large databases are a time-critical task in many application areas as e.g. biology or chemistry, where biological networks or chemical compounds are modeled as graphs. The NP-completeness of the underlying subgraph isomorphism problem renders an exact subgraph test for each database graph infeasible. Therefore efficient methods have to be found that avoid most of these tests but still allow to identify all graphs containing the query pattern. We propose a new approach based on the filter-verification paradigm, using a new hash-key fingerprint technique with a combination of tree and cycle features for filtering and a new subgraph isomorphism test for verification. Our approach is able to cope with edge and vertex labels and also allows to use wild card patterns for the search. We present an experimental comparison of our approach with state-of-the-art methods using a benchmark set of both real world and generated graph instances that shows its practicability. Our approach is implemented as part of the Scaffold Hunter software, a tool for the visual analysis of chemical compound databases.