Experimental Evaluation of Graph Classification with Hadamard Code Graph Kernels

  • Tetsuya Kataoka
  • Akihiro InokuchiEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10163)


When mining information from a database comprising graphs, kernel methods are used to efficiently classify graphs that have similar structures into the same classes. Instances represented by graphs usually have similar properties if their graph representations have high structural similarity. The neighborhood hash kernel (NHK) and Weisfeiler–Lehman subtree kernel (WLSK) have previously been proposed as kernels that compute more quickly than the random-walk kernel; however, they each have drawbacks. NHK can produce hash collision and WLSK must sort vertex labels. We propose a novel graph kernel equivalent to NHK in terms of time and space complexities, and comparable to WLSK in terms of expressiveness. The proposed kernel is based on the Hadamard code. Labels assigned by our graph kernel follow a binomial distribution with zero mean. The expected value of a label is zero; thus, such labels do not require large memory. This allows the compression of vertex labels in graphs, as well as fast computation. This paper presents the Hadamard code kernel (HCK) and shortened HCK (SHCK), a version of HCK that compresses vertex labels in graphs. The performance and practicality of the proposed method are demonstrated in experiments that compare the computation time, scalability and classification accuracy of HCK and SHCK with those of NHK and WLSK for both artificial and real-world datasets. The effect of assigning initial labels is also investigated.


Graph classification Support vector machine Graph kernel Hadamard code 


  1. 1.
    Borgwardt, K.M., Cheng, S.O., Schonauer, S., Vishwanathan, S.V.N., Smola, A.J., Kriegel, H.-P.: Protein function prediction via graph kernels. Bioinfomatics 21(suppl 1), 47–56 (2005)CrossRefGoogle Scholar
  2. 2.
    Chang, C.-C., Lin, C.-J.: LIBSVM: A Library for Support Vector Machines (2001).
  3. 3.
    Cristianini, N., Shawe-Taylor, J.: An Introduction to Support Vector Machines and Other Kernel-based Learning Methods. Cambridge University Press, Cambridge (2000)CrossRefzbMATHGoogle Scholar
  4. 4.
    Debnath, A.K., Lopez de Compadre, R.L., Debnath, G., Shusterman, A.J., Hansch, C.: Structure-activity relationship of mutagenic aromatic and heteroaromatic nitro compounds. Correlation with molecular orbital energies and hydrophobicity. J. Med. Chem. 34, 786–797 (1991)CrossRefGoogle Scholar
  5. 5.
    Dobson, P.D., Doig, A.J.: Distinguishing enzyme structures from non-enzymes without alignments. J. Mol. Biol. 330(4), 771–783 (2003)CrossRefGoogle Scholar
  6. 6.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman, Gordonsville (1979)zbMATHGoogle Scholar
  7. 7.
    Hido, S., Kashima, H.: A linear-time graph kernel. In: Proceedings of the IEEE International Conference on Data Mining (ICDM), pp. 179–188 (2009)Google Scholar
  8. 8.
    Kashima, H., Tsuda, K., Inokuchi, A.: Marginalized kernels between labeled graphs. In: Proceedings of the International Conference on Machine Learning (ICML), pp. 321–328 (2003)Google Scholar
  9. 9.
    Shervashidze, N., Schweitzer, P., van Leeuwen, E.J., Mehlhorn, K., Borgwardt, K.M.: Weisfeiler-Lehman graph kernels. J. Mach. Learn. Res. (JMLR) 12, 2539–2561 (2011)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Schölkopf, B., Smola, A.J.: Learning with Kernels. MIT Press, Cambridge (2002)zbMATHGoogle Scholar
  11. 11.
    Schölkopf, B., Tsuda, K., Vert, J.-P.: Kernel Methods in Computational Biology. MIT Press, Cambridge (2004)Google Scholar
  12. 12.
    Schomburg, I., Chang, A., Ebeling, C., Gremse, M., Heldt, C., Huhn, G., Schomburg, D.: BRENDA, the enzyme database: updates and major new developments. Nucleic Acids Res. 32D, 431–433 (2004)CrossRefGoogle Scholar
  13. 13.
    Kataoka, T., Inokuchi, A.: Hadamard code graph kernels for classifying graphs. In: Proceedings of the International Conference on Pattern Recognition Applications and Methods (ICPRAM), pp. 24–32 (2016)Google Scholar
  14. 14.
    Vinh, N.D., Inokuchi, A., Washio, T.: Graph classification based on optimizing graph spectra. In: Pfahringer, B., Holmes, G., Hoffmann, A. (eds.) DS 2010. LNCS (LNAI), vol. 6332, pp. 205–220. Springer, Heidelberg (2010). doi: 10.1007/978-3-642-16184-1_15 CrossRefGoogle Scholar
  15. 15.
    Wale, N., Karypis, G.: Comparison of descriptor spaces for chemical compound retrieval and classification. In: Proceedings of the IEEE International Conference on Data Mining (ICDM), Hong Kong, pp. 678–689 (2006)Google Scholar

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© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.School of Science and TechnologyKwansei Gakuin UniversitySandaJapan

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