A Generic Framework for Engineering Graph Canonization Algorithms

The state-of-the-art tools for practical graph canonization are all based on the individualization-refinement paradigm, and their difference is primarily in the choice of heuristics they include and in the actual tool implementation. It is thus not possible to make a direct comparison of how individual algorithmic ideas affect the performance on different graph classes. We present an algorithmic software framework that facilitates implementation of heuristics as independent extensions to a common core algorithm. It therefore becomes easy to perform a detailed comparison of the performance and behaviour of different algorithmic ideas. Implementations are provided of a range of algorithms for tree traversal, target cell selection, and node invariant, including choices from the literature and new variations. The framework readily supports extraction and visualization of detailed data from separate algorithm executions for subsequent analysis and development of new heuristics. Using collections of different graph classes we investigate the effect of varying the selections of heuristics, often revealing exactly which individual algorithmic choice is responsible for particularly good or bad performance. On several benchmark collections, including a newly proposed class of difficult instances, we additionally find that our implementation performs better than the current state-of-the-art tools

edge2vec: Representation learning using edge semantics for biomedical knowledge discovery

Representation learning provides new and powerful graph analytical approaches and tools for the highly valued data science challenge of mining knowledge graphs. Since previous graph analytical methods have mostly focused on homogeneous graphs, an important current challenge is extending this methodology for richly heterogeneous graphs and knowledge domains. The biomedical sciences are such a domain, reflecting the complexity of biology, with entities such as genes, proteins, drugs, diseases, and phenotypes, and relationships such as gene co-expression, biochemical regulation, and biomolecular inhibition or activation. Therefore, the semantics of edges and nodes are critical for representation learning and knowledge discovery in real world biomedical problems. In this paper, we propose the edge2vec model, which represents graphs considering edge semantics. An edge-type transition matrix is trained by an Expectation-Maximization approach, and a stochastic gradient descent model is employed to learn node embedding on a heterogeneous graph via the trained transition matrix. edge2vec is validated on three biomedical domain tasks: biomedical entity classification, compound-gene bioactivity prediction, and biomedical information retrieval. Results show that by considering edge-types into node embedding learning in heterogeneous graphs, \textbf{edge2vec}\ significantly outperforms state-of-the-art models on all three tasks. We propose this method for its added value relative to existing graph analytical methodology, and in the real world context of biomedical knowledge discovery applicability.Comment: 10 page

Contact handles, duality, and sutured Floer homology

We give an explicit construction of the Honda--Kazez--Mati\'c gluing maps in terms of contact handles. We use this to prove a duality result for turning a sutured manifold cobordism around, and to compute the trace in the sutured Floer TQFT. We also show that the decorated link cobordism maps on the hat version of link Floer homology defined by the first author via sutured manifold cobordisms and by the second author via elementary cobordisms agree.Comment: 86 pages, 54 figures, to appear in Geometry and Topolog

Genus expansion for real Wishart matrices

We present an exact formula for moments and cumulants of several real compound Wishart matrices in terms of an Euler characteristic expansion, similar to the genus expansion for complex random matrices. We consider their asymptotic values in the large matrix limit: as in a genus expansion, the terms which survive in the large matrix limit are those with the greatest Euler characteristic, that is, either spheres or collections of spheres. This topological construction motivates an algebraic expression for the moments and cumulants in terms of the symmetric group. We examine the combinatorial properties distinguishing the leading order terms. By considering higher cumulants, we give a central limit-type theorem for the asymptotic distribution around the expected value

Joint Vertex Degrees in an Inhomogeneous Random Graph Model

In a random graph, counts for the number of vertices with given degrees will typically be dependent. We show via a multivariate normal and a Poisson process approximation that, for graphs which have independent edges, with a possibly inhomogeneous distribution, only when the degrees are large can we reasonably approximate the joint counts as independent. The proofs are based on Stein's method and the Stein-Chen method with a new size-biased coupling for such inhomogeneous random graphs, and hence bounds on distributional distance are obtained. Finally we illustrate that apparent (pseudo-) power-law type behaviour can arise in such inhomogeneous networks despite not actually following a power-law degree distribution.Comment: 30 pages, 9 figure

Initial Draft of a Possible Declarative Semantics for the Language

This article introduces a preliminary declarative semantics for a subset of the language Xcerpt (so-called grouping-stratifiable programs) in form of a classical (Tarski style) model theory, adapted to the specific requirements of Xcerpt’s constructs (e.g. the various aspects of incompleteness in query terms, grouping constructs in rule heads, etc.). Most importantly, the model theory uses term simulation as a replacement for term equality to handle incomplete term specifications, and an extended notion of substitutions in order to properly convey the semantics of grouping constructs. Based upon this model theory, a fixpoint semantics is also described, leading to a first notion of forward chaining evaluation of Xcerpt program