The All-Paths and Cycles Graph Kernel

Abstract

With the recent rise in the amount of structured data available, there has been considerable interest in methods for machine learning with graphs. Many of these approaches have been kernel methods, which focus on measuring the similarity between graphs. These generally involving measuring the similarity of structural elements such as walks or paths. Borgwardt and Kriegel [1] proposed the all-paths kernel but emphasized that it is NP-hard to compute and infeasible in practice, favouring instead the shortest-path kernel. In this paper, we introduce a new algorithm for computing the all-paths kernel which is very efficient and enrich it further by including the simple cycles as well. We demonstrate how it is feasible even on large datasets to compute all the paths and simple cycles up to a moderate length. We show how to count labelled paths/simple cycles between vertices of a graph and evaluate a labelled path and simple cycles kernel. Extensive evaluations on a variety of graph datasets demonstrate that the all-paths and cycles kernel has superior performance to the shortest-path kernel and state-of-the-art performance overall.

Graf Çekirdek ve Graf Sinir Ağı Yöntemlerini Kullanarak RNA Moleküllerini Sınıflandırılmak İçin 3D RNA Graf Temsili Yöntemleri

“…Indeed, a graph kernel is a similarity technique that assesses pairwise graph similarities. Various newer techniques have emerged, such as the All Paths and Cycles (APC) Embedding (Giscard & Wilson, 2017), which explores the similarity in paths and cycles between graph pairs. The Weisfeiler Lehman Optimal Assignment Kernel (WL-OA) (Kriege et al, 2016) is another advanced method for comparing labelled pairwise graphs.…”

“…While graph kernels have produced excellent results in graph classification problems, one drawback is that they are typically restricted to limited/few numbers of discrete labels on the nodes and edges (Giscard & Wilson, 2017;Kriege et al, 2016). To overcome this limitation, we use LVQ for clustering node labels and obtain 1-dimensional discrete labels for use in kernel methods.…”

“…Graph kernels compare the similarity between two graphs and have been widely used for structural pattern recognition and RNA classification. Examples include Shortest Path (Borgwardt & Kriegel, 2005), random walk (Kang et al, 2012), Weisfeiler-Lehmann (Shervashidze et al, 2011), and All Paths and Cycles kernels (Giscard & Wilson, 2017). These methods typically have some limitations in the application of graph representations, such as restrictions on negative edge labels and a constrained set of node labels.…”

Enumerating Simple Paths from Connected Induced Subgraphs

“…Furthermore, given that Theorem 1 involves the labelled adjacency matrix W, the formulas of the Theorem permit the enumeration of the simple cycles and paths. By coding vertex labels using numerical values, this property was exploited to efficiently compare all label sequences corresponding to simple cycles in pairs of graphs, thereby reducing an important hurdle in automatic graph classification tasks [14].…”

“…A remarkable consequence is an expression that links the Hamiltonian paths of a graph to its dominating connected sets. While the problem of counting simple cycles and paths parametrised by length remains #W [1]-complete, the formulas we obtain form the base of a novel algorithm [15] that has proven to be efficient enough to effectively tackle the problem, up to length 20, on real-world networks [13,14].…”