A Mixed Weisfeiler-Lehman Graph Kernel

  • Lixiang XuEmail author
  • Jin Xie
  • Xiaofeng Wang
  • Bin Luo
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9069)


Using concepts from the Weisfeiler-Lehman (WL) test of isomorphism, we propose a mixed WL graph kernel (MWLGK) framework based on a family of efficient WL graph kernels for constructing mixed graph kernel. This family of kernels can be defined based on the WL sequence of graphs. We apply the MWLGK framework on WL graph sequence taking into account the structural information which was overlooked. Our MWLGK is competitive with or outperforms the corresponding single WL graph kernel on several classification benchmark data sets.


graph kernel Graph classification Weisfeiler-Lehman algorithm Mixed graph kernel 


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  1. 1.
    Babai, L., Kucera, L.: Canonical labelling of graphs in linear average time. In: 20th Annual Symposium on Foundations of Computer Science, pp. 39–46. IEEE (1979)Google Scholar
  2. 2.
    Bai, L., Hancock, E.R.: Graph clustering using the jensen-shannon kernel. In: Real, P., Diaz-Pernil, D., Molina-Abril, H., Berciano, A., Kropatsch, W. (eds.) CAIP 2011, Part I. LNCS, vol. 6854, pp. 394–401. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  3. 3.
    Bai, L., Hancock, E.R., Torsello, A., Rossi, L.: A quantum jensen-shannon graph kernel using the continuous-time quantum walk. In: Kropatsch, W.G., Artner, N.M., Haxhimusa, Y., Jiang, X. (eds.) GbRPR 2013. LNCS, vol. 7877, pp. 121–131. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  4. 4.
    Borgwardt, K.M., Kriegel, H.-P.: Shortest-path kernels on graphs. In: Fifth IEEE International Conference on Data Mining, 8 p. IEEE (2005)Google Scholar
  5. 5.
    Borgwardt, K.M., Ong, C.S., Schönauer, S., Vishwanathan, S.V.N., Smola, A.J., Kriegel, H.-P.: Protein function prediction via graph kernels. Bioinformatics 21(suppl. 1), i47–i56 (2005)Google Scholar
  6. 6.
    Borgwardt, K.M., Schraudolph, N.N., Vishwanathan, S.V.N.: Fast computation of graph kernels. In: Advances in Neural Information Processing Systems, pp. 1449–1456 (2006)Google Scholar
  7. 7.
    Chang, C.-C., Lin, C.-J.: Libsvm: a library for support vector machines. ACM Transactions on Intelligent Systems and Technology (TIST) 2(3), 27 (2011)Google Scholar
  8. 8.
    Fisher, M., Savva, M., Hanrahan, P.: Characterizing structural relationships in scenes using graph kernels. ACM Transactions on Graphics (TOG) 30, 34 (2011)CrossRefGoogle Scholar
  9. 9.
    Gärtner, T., Flach, P., Wrobel, S.: On Graph Kernels: Hardness Results and Efficient Alternatives. In: Schölkopf, B., Warmuth, M.K. (eds.) COLT/Kernel 2003. LNCS (LNAI), vol. 2777, pp. 129–143. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  10. 10.
    Kashima, H., Tsuda, K., Inokuchi, A.: Marginalized kernels between labeled graphs. In: ICML, vol. 3, pp. 321–328 (2003)Google Scholar
  11. 11.
    Kashima, H., Tsuda, K., Inokuchi, A.: Kernels for graphs. Kernel Methods in Computational Biology 39(1), 101–113 (2004)Google Scholar
  12. 12.
    Mahé, P., Ueda, N., Akutsu, T., Perret, J.-L., Vert, J.-P.: Extensions of marginalized graph kernels. In: Proceedings of the Twenty-First International Conference on Machine Learning, p. 70. ACM (2004)Google Scholar
  13. 13.
    Ramon, J., Gärtner, T.: Expressivity versus efficiency of graph kernels. In: First International Workshop on Mining Graphs, Trees and Sequences, pp. 65–74. Citeseer (2003)Google Scholar
  14. 14.
    Schölkopf, B., Smola, A.J.: Learning with kernels: support vector machines, regularization, optimization, and beyond. MIT Press (2002)Google Scholar
  15. 15.
    Shervashidze, N., Petri, T., Mehlhorn, K., Borgwardt, K.M., Vishwanathan, S.V.N.: Efficient graphlet kernels for large graph comparison. In: International Conference on Artificial Intelligence and Statistics, pp. 488–495 (2009)Google Scholar
  16. 16.
    Shervashidze, N., Schweitzer, P., Leeuwen, E.J.V., Mehlhorn, K., Borgwardt, K.M.: Weisfeiler-lehman graph kernels. The Journal of Machine Learning Research 12, 2539–2561 (2011)zbMATHGoogle Scholar
  17. 17.
    Vichy, S., Vishwanathan, N., Schraudolph, N.N., Kondor, R., Borgwardt, K.M.: Graph kernels. The Journal of Machine Learning Research 11, 1201–1242 (2010)Google Scholar
  18. 18.
    Ju Weisfeiler, B., Leman, A.A.: Reduction of a graph to a canonical form and an algebra which appears in the process. NTI, Ser. 2(9), 12–16 (1968)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Lixiang Xu
    • 1
    • 2
    • 4
    Email author
  • Jin Xie
    • 2
    • 4
  • Xiaofeng Wang
    • 3
  • Bin Luo
    • 1
  1. 1.School of Computer Science and TechnologyAnhui UniversityHefeiPeople’s Republic of China
  2. 2.Department of Mathmatics & PhysicsHefei UniversityHefeiPeople’s Republic of China
  3. 3.Department of Computer Science and TechnologyHefei UniversityHefeiPeople’s Republic of China
  4. 4.Institute of Scientific ComputingHefei UniversityHefeiPeople’s Republic of China

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