A Survey on Graph Kernels

Graph kernels have become an established and widely-used technique for solving classification tasks on graphs. This survey gives a comprehensive overview of techniques for kernel-based graph classification developed in the past 15 years. We describe and categorize graph kernels based on properties inherent to their design, such as the nature of their extracted graph features, their method of computation and their applicability to problems in practice. In an extensive experimental evaluation, we study the classification accuracy of a large suite of graph kernels on established benchmarks as well as new datasets. We compare the performance of popular kernels with several baseline methods and study the effect of applying a Gaussian RBF kernel to the metric induced by a graph kernel. In doing so, we find that simple baselines become competitive after this transformation on some datasets. Moreover, we study the extent to which existing graph kernels agree in their predictions (and prediction errors) and obtain a data-driven categorization of kernels as result. Finally, based on our experimental results, we derive a practitioner's guide to kernel-based graph classification

Analysis of Trajectories by Preserving Structural Information

The analysis of trajectories from traffic data is an established and yet fast growing area of research in the related fields of Geo-analytics and Geographic Information Systems (GIS). It has a broad range of applications that impact lives of millions of people, e.g., in urban planning, transportation and navigation systems and localized search methods. Most of these applications share some underlying basic tasks which are related to matching, clustering and classification of trajectories. And, these tasks in turn share some underlying problems, i.e., dealing with the noisy and variable length spatio-temporal sequences in the wild. In our view, these problems can be handled in a better manner by exploiting the spatio-temporal relationships (or structural information) in sampled trajectory points that remain considerably unharmed during the measurement process. Although, the usage of such structural information has allowed breakthroughs in other fields related to the analysis of complex data sets [18], surprisingly, there is no existing approach in trajectory analysis that looks at this structural information in a unified way across multiple tasks. In this thesis, we build upon these observations and give a unified treatment of structural information in order to improve trajectory analysis tasks. This treatment explores for the first time that sequences, graphs, and kernels are common to machine learning and geo-analytics. This common language allows to pool the corresponding methods and knowledge to help solving the challenges raised by the ever growing amount of movement data by developing new analysis models and methods. This is illustrated in several ways. For example, we introduce new problem settings, distance functions and a visualization scheme in the area of trajectory analysis. We also connect the broad fild of kernel methods to the analysis of trajectories, and, we strengthen and revisit the link between biological sequence methods and analysis of trajectories. Finally, the results of our experiments show that - by incorporating the structural information - our methods improve over state-of-the-art in the focused tasks, i.e., map matching, clustering and traffic event detection

EmbAssi: Embedding Assignment Costs for Similarity Search in Large Graph Databases

The graph edit distance is an intuitive measure to quantify the dissimilarity of graphs, but its computation is NP-hard and challenging in practice. We introduce methods for answering nearest neighbor and range queries regarding this distance efficiently for large databases with up to millions of graphs. We build on the filter-verification paradigm, where lower and upper bounds are used to reduce the number of exact computations of the graph edit distance. Highly effective bounds for this involve solving a linear assignment problem for each graph in the database, which is prohibitive in massive datasets. Index-based approaches typically provide only weak bounds leading to high computational costs verification. In this work, we derive novel lower bounds for efficient filtering from restricted assignment problems, where the cost function is a tree metric. This special case allows embedding the costs of optimal assignments isometrically into $\ell_1$ space, rendering efficient indexing possible. We propose several lower bounds of the graph edit distance obtained from tree metrics reflecting the edit costs, which are combined for effective filtering. Our method termed EmbAssi can be integrated into existing filter-verification pipelines as a fast and effective pre-filtering step. Empirically we show that for many real-world graphs our lower bounds are already close to the exact graph edit distance, while our index construction and search scales to very large databases