Invariants of distance k-graphs for graph embedding

Abstract

Graph comparison algorithms based on metric space embedding have been proven robust in graph clustering and classification. In this paper we propose graph embedding method exploiting ordered invariants of distance k-graphs, which encode structure of shortest-paths. We study degree histograms of those graphs and use them to construct permutation invariant graph representation called vertex B-matrix. In order to extract more information from structural patterns we also define edge distance k-graphs and associated edge B-matrix. Next, several new graph characteristics obtained by condensing information stored in B-matrices are introduced. We demonstrate that our approach provides stable embedding, which captures relevant graph features. Experiments on classification with satellite photo and mutagenicity benchmark datasets revealed, that new descriptors allow for distinguishing graphs with non trivial structural differences. Moreover, they appear to outperform descriptors based on heat kernel matrix, being at the same time more effective computationally. In the end we test feature selection on B-matrices showing that selecting right B-submatrix can improve classification rate on testing datasets.