Subgraph Matching Kernels for Attributed Graphs

We propose graph kernels based on subgraph matchings, i.e. structure-preserving bijections between subgraphs. While recently proposed kernels based on common subgraphs (Wale et al., 2008; Shervashidze et al., 2009) in general can not be applied to attributed graphs, our approach allows to rate mappings of subgraphs by a flexible scoring scheme comparing vertex and edge attributes by kernels. We show that subgraph matching kernels generalize several known kernels. To compute the kernel we propose a graph-theoretical algorithm inspired by a classical relation between common subgraphs of two graphs and cliques in their product graph observed by Levi (1973). Encouraging experimental results on a classification task of real-world graphs are presented.Comment: Appears in Proceedings of the 29th International Conference on Machine Learning (ICML 2012

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

Faster Algorithms for the Maximum Common Subtree Isomorphism Problem

The maximum common subtree isomorphism problem asks for the largest possible isomorphism between subtrees of two given input trees. This problem is a natural restriction of the maximum common subgraph problem, which is ${\sf NP}$-hard in general graphs. Confining to trees renders polynomial time algorithms possible and is of fundamental importance for approaches on more general graph classes. Various variants of this problem in trees have been intensively studied. We consider the general case, where trees are neither rooted nor ordered and the isomorphism is maximum w.r.t. a weight function on the mapped vertices and edges. For trees of order $n$ and maximum degree $\Delta$ our algorithm achieves a running time of $\mathcal{O}(n^2\Delta)$ by exploiting the structure of the matching instances arising as subproblems. Thus our algorithm outperforms the best previously known approaches. No faster algorithm is possible for trees of bounded degree and for trees of unbounded degree we show that a further reduction of the running time would directly improve the best known approach to the assignment problem. Combining a polynomial-delay algorithm for the enumeration of all maximum common subtree isomorphisms with central ideas of our new algorithm leads to an improvement of its running time from $\mathcal{O}(n^6+Tn^2)$ to $\mathcal{O}(n^3+Tn\Delta)$, where $n$ is the order of the larger tree, $T$ is the number of different solutions, and $\Delta$ is the minimum of the maximum degrees of the input trees. Our theoretical results are supplemented by an experimental evaluation on synthetic and real-world instances

Freshwater Mussels of the Greenup Navigational Pool, Ohio River, with a Comparison to Fish Host Communities

The Ohio River was historically a free-flowing system with diverse fish and freshwater mussel communities. Heavy industrialization, erosion from deforestation, and wide scale damming during the early-mid 20th century decimated riverine life. While mussel declines are well documented in the United States, in big river systems, freshwater mussel populations are poorly understudied. This thesis project mapped the mussel communities and site-specific sediments of the Greenup pool in the Ohio River for comparison to 2016 nighttime electrofishing data, provided by ORSANCO. Qualitative SCUBA surveys were performed at 18 randomly selected sites and two fixed sites between July and September. Each site consisted of six, 100 meter survey transects. Sediment was recorded in ten meter sections along each transect. I hypothesized that high fish-host richness and abundances will correlate with strong mussel communities. A secondary goal of my project was to identify areas which may warrent special protection due to the presence of federally endangered species. A total of 3,747 live mussels were collected from 23 species, including nine federally endangered Sheepnose (Plethobasus cyphyus). Using negative binomial regressions, fish host richness and abundances were not reliable predictors of freshwater mussel communities. The only exception was Aplodinotus grunniens, which acts as an inverse predictor of Ellipsaria lineolata populations. While there are few explanations to the broad spatial distribution of fish communities, freshwater mussel populations may be concentrated in the upper section of the pool due to heavy historical pollution and disturbances in the middle and lower Greenup pool