Modularity Optimization as a Training Criterion for Graph Neural Networks

  • Tsuyoshi Murata
  • Naveed Afzal
Conference paper
Part of the Springer Proceedings in Complexity book series (SPCOM)


Graph convolution is a recent scalable method for performing deep feature learning on attributed graphs by aggregating local node information over multiple layers. Such layers only consider attribute information of node neighbors in the forward model and do not incorporate knowledge of global network structure in the learning task. In particular, the modularity function provides a convenient source of information about the community structure of networks. In this work, we investigate the effect on the quality of learned representations by the incorporation of community structure preservation objectives of networks in the graph convolutional model. We incorporate the objectives in two ways, through an explicit regularization term in the cost function in the output layer and as an additional loss term computed via an auxiliary layer. We report the effect of community-structure-preserving terms in the graph convolutional architectures. Experimental evaluation on two attributed bibliographic networks showed that the incorporation of the community-preserving objective improves semi-supervised node classification accuracy in the sparse label regime.



This work was supported by Tokyo Tech - Fuji Xerox Cooperative Research (Project Code KY260195), JSPS Grant-in-Aid for Scientific Research(B) (Grant Number 17H01785), and JST CREST (Grant Number JPMJCR1687).


  1. 1.
    Bruna, J., Zaremba, W., Szlam, A., LeCun, Y.: Spectral networks and locally connected networks on graphs (2013). arXiv:1312.6203
  2. 2.
    Defferrard, M., Bresson, X., Vandergheynst, P.: Convolutional neural networks on graphs with fast localized spectral filtering. In: Advances in Neural Information Processing Systems, pp. 3844–3852 (2016)Google Scholar
  3. 3.
    Hammond, D.K., Vandergheynst, P., Gribonval, R.: Wavelets on graphs via spectral graph theory. Appl. Comput. Harmon. Anal. 30(2), 129–150 (2011)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Hinton, G., Deng, L., Dong, Y., Dahl, G.E., Mohamed, A., Jaitly, N., Senior, A., Vanhoucke, V., Nguyen, P., Sainath, T.N., et al.: Deep neural networks for acoustic modeling in speech recognition: the shared views of four research groups. IEEE Signal Process. Maga. 29(6), 82–97 (2012)ADSCrossRefGoogle Scholar
  5. 5.
    Kipf, T.N., Welling, M.: Variational graph auto-encoders (2016). arXiv:1611.07308
  6. 6.
    Kipf, T.N., Welling, M.: Semi-supervised classification with graph convolutional networks (2017)Google Scholar
  7. 7.
    Krizhevsky, A., Sutskever, I., Hinton, G.E.: Imagenet classification with deep convolutional neural networks. In Advances in Neural Information Processing Systems, pp. 1097–1105 (2012)Google Scholar
  8. 8.
    LeCun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proc. IEEE 86(11), 2278–2324 (1998)CrossRefGoogle Scholar
  9. 9.
    Lin, F., Cohen, W.W.: Semi-supervised classification of network data using very few labels. In: 2010 International Conference on Advances in Social Networks Analysis and Mining (ASONAM), pp. 192–199. IEEE (2010)Google Scholar
  10. 10.
    Neville, J., Jensen, D.: Iterative classification in relational data. In: Proceedings of AAAI-2000 Workshop on Learning Statistical Models from Relational Data, pp. 13–20 (2000)Google Scholar
  11. 11.
    Newman, M.E.J.: The structure and function of complex networks. SIAM Rev. 45(2), 167–256 (2003)ADSMathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Newman, M.E.J.: Modularity and community structure in networks. Proc. Natl. Acad. Sci. 103(23), 8577–8582 (2006)ADSCrossRefGoogle Scholar
  13. 13.
    Sen, P., Namata, G., Bilgic, M., Getoor, L., Galligher, B., Eliassi-Rad, T.: Collective classification in network data. AI Mag. 29(3), 93 (2008)CrossRefGoogle Scholar
  14. 14.
    Tu, C., Wang, H., Zeng, X., Liu, Z., Sun, M.: Community-enhanced network representation learning for network analysis (2016). arXiv:1611.06645
  15. 15.
    Wang, X., Cui, P., Wang, J., Pei, J., Zhu, W., Yang, S.: Community preserving network embedding. In: AAAI, pp. 203–209 (2017)Google Scholar
  16. 16.
    Weston, J., Ratle, F., Mobahi, H., Collobert, R.: Deep learning via semi-supervised embedding. In: Neural Networks: Tricks of the Trade, pp. 639–655. Springer, Berlin (2012)Google Scholar
  17. 17.
    Yang, L., Cao, X., He, D., Wang, C., Wang, X., Zhang, W.: Modularity based community detection with deep learning. In: IJCAI, pp. 2252–2258 (2016)Google Scholar
  18. 18.
    Yang, Z., Cohen, W.W., Salakhutdinov, R.: Revisiting semi-supervised learning with graph embeddings (2016). arXiv:1603.08861

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Department of Computer Science, School of Computing Tokyo Institute of TechnologyMeguro, TokyoJapan

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