Kernels for acyclic digraphs

This paper proposes two efficient kernels for comparing acyclic, directed graphs. The first kernel counts the number of common paths and allows for weighing according to path-length and/or according to the vertices contained in each particular path. The second kernel counts the number of paths in common minors of the graphs involved and allows for length- and vertex-weighting too. Both kernels have algorithmic complexity that is cubic in the size of the vertex-set. The performance of the algorithms is concisely demonstrated using synthetic and real data.

[1]  Jérôme Euzenat,et al.  A Survey of Schema-Based Matching Approaches , 2005, J. Data Semant..

[2]  Horst Bunke,et al.  A New Algorithm for Error-Tolerant Subgraph Isomorphism Detection , 1998, IEEE Trans. Pattern Anal. Mach. Intell..

[3]  Kaspar Riesen,et al.  Approximate graph edit distance computation by means of bipartite graph matching , 2009, Image Vis. Comput..

[4]  Thomas Gärtner,et al.  On Graph Kernels: Hardness Results and Efficient Alternatives , 2003, COLT.

[5]  Tatsuya Akutsu,et al.  Extensions of marginalized graph kernels , 2004, ICML.

[6]  Marcus Liwicki,et al.  Faster subgraph isomorphism detection by well-founded total order indexing , 2012, Pattern Recognit. Lett..

[7]  Zhaoqi Wang,et al.  Indexing and retrieval of human motion data by a hierarchical tree , 2009, VRST '09.

[8]  Sebastian Nowozin,et al.  On feature combination for multiclass object classification , 2009, 2009 IEEE 12th International Conference on Computer Vision.

[9]  Kurt Mehlhorn,et al.  Weisfeiler-Lehman Graph Kernels , 2011, J. Mach. Learn. Res..

[10]  David Haussler,et al.  Convolution kernels on discrete structures , 1999 .

[11]  Nello Cristianini,et al.  An Introduction to Support Vector Machines and Other Kernel-based Learning Methods , 2000 .

[12]  Ronald L. Rivest,et al.  Introduction to Algorithms , 1990 .

[13]  Nello Cristianini,et al.  Kernel Methods for Pattern Analysis , 2003, ICTAI.

[14]  John Skvoretz,et al.  8. Comparing Networks across Space and Time, Size and Species , 2002 .

[15]  Sven Rahmann,et al.  Algorithms for subsequence combinatorics , 2008, Theor. Comput. Sci..

[16]  Horst Bunke,et al.  On a relation between graph edit distance and maximum common subgraph , 1997, Pattern Recognit. Lett..

[17]  Abraham Kandel,et al.  Graph-Theoretic Techniques for Web Content Mining , 2005, Series in Machine Perception and Artificial Intelligence.

[18]  M. Kanehisa,et al.  Development of a chemical structure comparison method for integrated analysis of chemical and genomic information in the metabolic pathways. , 2003, Journal of the American Chemical Society.

[19]  Volker Sperschneider,et al.  Bioinformatics: Problem Solving Paradigms , 2008 .

[20]  Tommy R. Jensen,et al.  Graph Coloring Problems , 1994 .

[21]  Gabriel Valiente,et al.  A graph distance metric combining maximum common subgraph and minimum common supergraph , 2001, Pattern Recognit. Lett..

[22]  D. Baum.  Concordance trees, concordance factors, and the exploration of reticulate genealogy , 2007 .

[23]  Kaspar Riesen,et al.  IAM Graph Database Repository for Graph Based Pattern Recognition and Machine Learning , 2008, SSPR/SPR.

[24]  Zhiwei Lin,et al.  A Novel Algorithm for Counting All Common Subsequences , 2007 .

[25]  Lawrence B. Holder,et al.  Mining Graph Data , 2006 .

[26]  Abraham Kandel,et al.  Applied Graph Theory in Computer Vision and Pattern Recognition , 2007 .

[27]  Sally I. McClean,et al.  Measuring Tree Similarity for Natural Language Processing Based Information Retrieval , 2010, NLDB.

[28]  Jan Kleinnijenhuis,et al.  Reasoning in Economic Discourse: An Application of the Network Approach to the Dutch Press , 1997 .

[29]  Zhiwei Lin,et al.  Concordance and consensus , 2011, Inf. Sci..

[30]  Eric D. Widmer,et al.  Family Configurations: A Structural Approach to Family Diversity , 2010 .

[31]  Reinhard Diestel,et al.  Graph Theory , 1997 .

[32]  Julian R. Ullmann,et al.  An Algorithm for Subgraph Isomorphism , 1976, J. ACM.

[33]  William H. Press,et al.  Numerical recipes in C. The art of scientific computing , 1987 .

[34]  Hans-Peter Kriegel,et al.  Shortest-path kernels on graphs , 2005, Fifth IEEE International Conference on Data Mining (ICDM'05).

[35]  Nello Cristianini,et al.  Classification using String Kernels , 2000 .

[36]  Hasan M. Jamil.  Computing subgraph isomorphic queries using structural unification and minimum graph structures , 2011, SAC '11.

[37]  Kaspar Riesen,et al.  Recent advances in graph-based pattern recognition with applications in document analysis , 2011, Pattern Recognit..

[38]  Ronald L. Rivest,et al.  Introduction to Algorithms, third edition , 2009 .

[39]  Horst Bunke,et al.  A graph distance metric based on the maximal common subgraph , 1998, Pattern Recognit. Lett..

[40]  Karsten M. Borgwardt,et al.  Kernel Methods in Bioinformatics , 2011, Handbook of Statistical Bioinformatics.

[41]  R. Bapat.  Graphs and Matrices , 2014 .

[42]  S. V. N. Vishwanathan,et al.  Graph kernels , 2007 .

[43]  Hisashi Kashima,et al.  Marginalized Kernels Between Labeled Graphs , 2003, ICML.