SAT Encoding of Partial Ordering Models for Graph Coloring Problems

In this paper, we suggest new SAT encodings of the partial-ordering based ILP model for the graph coloring problem (GCP) and the bandwidth coloring problem (BCP). The GCP asks for the minimum number of colors that can be assigned to the vertices of a given graph such that each two adjacent vertices get different colors. The BCP is a generalization, where each edge has a weight that enforces a minimal "distance" between the assigned colors, and the goal is to minimize the "largest" color used. For the widely studied GCP, we experimentally compare our new SAT encoding to the state-of-the-art approaches on the DIMACS benchmark set. Our evaluation confirms that this SAT encoding is effective for sparse graphs and even outperforms the state-of-the-art on some DIMACS instances. For the BCP, our theoretical analysis shows that the partial-ordering based SAT and ILP formulations have an asymptotically smaller size than that of the classical assignment-based model. Our practical evaluation confirms not only a dominance compared to the assignment-based encodings but also to the state-of-the-art approaches on a set of benchmark instances. Up to our knowledge, we have solved several open instances of the BCP from the literature for the first time.

A Temporal Graphlet Kernel for Classifying Dissemination in Evolving Networks

We introduce the \emph{temporal graphlet kernel} for classifying dissemination processes in labeled temporal graphs. Such dissemination processes can be spreading (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., infectious 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. We show that our kernels are faster to compute and provide better accuracy than state-of-the-art methods.

Temporal Walk Centrality: Ranking Nodes in Evolving Networks

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.

TUDataset: A collection of benchmark datasets for learning with graphs

Recently, there has been an increasing interest in (supervised) learning with graph data, especially using graph neural networks. However, the development of meaningful benchmark datasets and standardized evaluation procedures is lagging, consequently hindering advancements in this area. To address this, we introduce the TUDataset for graph classification and regression. The collection consists of over 120 datasets of varying sizes from a wide range of applications. We provide Python-based data loaders, kernel and graph neural network baseline implementations, and evaluation tools. Here, we give an overview of the datasets, standardized evaluation procedures, and provide baseline experiments. All datasets are available at www.graphlearning.io. The experiments are fully reproducible from the code available at www.github.com/chrsmrrs/tudataset.

Temporal Graph Kernels for Classifying Dissemination Processes

Many real-world graphs or networks are temporal, e.g., in a social network persons only interact at specific points in time. This information directs dissemination processes on the network, 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. Specifically, 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 a wide range of 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. Hence, we confirm that taking temporal information into account is crucial for the successful classification of dissemination processes.

Towards a practical $k$-dimensional Weisfeiler-Leman algorithm

The $k$-dimensional Weisfeiler-Leman algorithm is a well-known heuristic for the graph isomorphism problem. Moreover, it recently emerged as a powerful tool for supervised graph classification. The algorithm iteratively partitions the set of $k$-tuples, defined over the set of vertices of a graph, by considering neighboring $k$-tuples. Here, we propose a \new{local} variant which considers a subset of the original neighborhood in each iteration step. The cardinality of this local neighborhood, unlike the original one, only depends on the sparsity of the graph. Surprisingly, we show that the local variant has at least the same power as the original algorithm in terms of distinguishing non-isomorphic graphs. In order to demonstrate the practical utility of our local variant, we apply it to supervised graph classification. Our experimental study shows that our local algorithm leads to improved running times and classification accuracies on established benchmark datasets.

Recognizing Cuneiform Signs Using Graph Based Methods

The cuneiform script constitutes one of the earliest systems of writing and is realized by wedge-shaped marks on clay tablets. A tremendous number of cuneiform tablets have already been discovered and are incrementally digitalized and made available to automated processing. As reading cuneiform script is still a manual task, we address the real-world application of recognizing cuneiform signs by two graph based methods with complementary runtime characteristics. We present a graph model for cuneiform signs together with a tailored distance measure based on the concept of the graph edit distance. We propose efficient heuristics for its computation and demonstrate its effectiveness in classification tasks experimentally. To this end, the distance measure is used to implement a nearest neighbor classifier leading to a high computational cost for the prediction phase with increasing training set size. In order to overcome this issue, we propose to use CNNs adapted to graphs as an alternative approach shifting the computational cost to the training phase. We demonstrate the practicability of both approaches in an extensive experimental comparison regarding runtime and prediction accuracy. Although currently available annotated real-world data is still limited, we obtain a high accuracy using CNNs, in particular, when the training set is enriched by augmented examples.

Global Weisfeiler-Lehman Graph Kernels

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 propertiesinto account may not scale well to large graph databases. Here we propose to start exploring the space between local and global 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.

A Unifying View of Explicit and Implicit Feature Maps for Structured Data: Systematic Studies of Graph Kernels

Non-linear kernel methods can be approximated by fast linear ones using suitable explicit feature maps allowing their application to large scale problems. To this end, explicit feature maps of kernels for vectorial data have been extensively studied. As many real-world data is structured, various kernels for complex data like graphs have been proposed. Indeed, many of them directly compute feature maps. However, the kernel trick is employed when the number of features is very large or the individual vertices of graphs are annotated by real-valued attributes. Can we still compute explicit feature maps efficiently under these circumstances? Triggered by this question, we investigate how general convolution kernels are composed from base kernels and construct corresponding feature maps. We apply our results to widely used graph kernels and analyze for which kernels and graph properties computation by explicit feature maps is feasible and actually more efficient. In particular, we derive feature maps for random walk and subgraph matching kernels and apply them to real-world graphs with discrete labels. Thereby, 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. Moreover, 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.

Faster Kernels for Graphs with Continuous Attributes via Hashing

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.