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Fast depth-based subgraph kernels for unattributed graphs. (English) Zbl 1394.68272
Summary: In this paper, we investigate two fast subgraph kernels based on a depth-based representation of graph-structure. Both methods gauge depth information through a family of \(K\)-layer expansion subgraphs rooted at a vertex [F. Escolano et al., “Heat diffusion: thermodynamic depth complexity of networks”, Phys. Rev. E 85, No. 3, 036206, 15 p. (2012; doi:10.1103/physreve.85.036206)]. The first method commences by computing a centroid-based complexity trace for each graph, using a depth-based representation rooted at the centroid vertex that has minimum shortest path length variance to the remaining vertices [the authors, Pattern Recognition 47, No. 3, 1172–1186 (2014; Zbl 1326.68241)]. This subgraph kernel is computed by measuring the Jensen-Shannon divergence between centroid-based complexity entropy traces. The second method, on the other hand, computes a depth-based representation around each vertex in turn. The corresponding subgraph kernel is computed using isomorphisms tests to compare the depth-based representation rooted at each vertex in turn. For graphs with \(n\) vertices, the time complexities for the two new kernels are \(O(n^2)\) and \(O(n^3)\), in contrast to \(O(n^6)\) for the classic Gärtner graph kernel [T. Gärtner et al., Lect. Notes Comput. Sci. 2777, 129–143 (2003; Zbl 1274.68312)]. Key to achieving this efficiency is that we compute the required Shannon entropy of the random walk for our kernels with \(O(n^2)\) operations. This computational strategy enables our subgraph kernels to easily scale up to graphs of reasonably large sizes and thus overcome the size limits arising in state-of-the-art graph kernels. Experiments on standard bioinformatics and computer vision graph datasets demonstrate the effectiveness and efficiency of our new subgraph kernels.

MSC:
68T05 Learning and adaptive systems in artificial intelligence
68R10 Graph theory (including graph drawing) in computer science
68T45 Machine vision and scene understanding
94A17 Measures of information, entropy
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