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Graph sparsification with graph convolutional networks

Abstract

Graphs are ubiquitous across the globe and within science and engineering. Some powerful classifiers are proposed to classify nodes in graphs, such as Graph Convolutional Networks (GCNs). However, as graphs are growing in size, node classification on large graphs can be space and time consuming due to using whole graphs. Hence, some questions are raised, particularly, whether one can prune some of the edges of a graph while maintaining prediction performance for node classification, or train classifiers on specific subgraphs instead of a whole graph with limited performance loss in node classification. To address these questions, we propose Sparsified Graph Convolutional Network (SGCN), a neural network graph sparsifier that sparsifies a graph by pruning some edges. We formulate sparsification as an optimization problem and solve it by an Alternating Direction Method of Multipliers (ADMM). The experiment illustrates that SGCN can identify highly effective subgraphs for node classification in GCN compared to other sparsifiers such as Random Pruning, Spectral Sparsifier and DropEdge. We also show that sparsified graphs provided by SGCN can be inputs to GCN, which leads to better or comparable node classification performance with that of original graphs in GCN, DeepWalk, GraphSAGE, and GAT. We provide insights on why SGCN performs well by analyzing its performance from the view of a low-pass filter.

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Notes

  1. 1.

    We cannot consider NeuralSparse [37] as the code of NeuralSparse cannot be accessed.

  2. 2.

    http://linqs.soe.ucsc.edu/data.

References

  1. 1.

    Benczúr, A.A., Karger, D.R.: Approximating s-t minimum cuts in Õ(n\({}^{\text{2}}\)) time. In: Proceedings of the Twenty-Eighth Annual ACM Symposium on the Theory of Computing, Philadelphia, Pennsylvania, USA, May 22–24, 1996, pp. 47–55 (1996)

  2. 2.

    Bhagat, S., Cormode, G., Muthukrishnan, S.: Node classification in social networks. In: Aggarwal, C.C. (eds.) Social Network Data Analytics, pp. 115–148. Springer, Boston (2011). https://doi.org/10.1007/978-1-4419-8462-3_5

  3. 3.

    Boyd, S., Parikh, N., Chu, E., Peleato, B., Eckstein, J., et al.: Distributed optimization and statistical learning via the alternating direction method of multipliers. Found. Trends® Mach. Learn. 3(1), 1–122 (2011)

  4. 4.

    Carlson, A., Betteridge, J., Kisiel, B., Settles, B., Hruschka, E.R., Mitchell, T.M.: Toward an architecture for never-ending language learning. In: Proceedings of the Twenty-Fourth AAAI Conference on Artificial Intelligence, AAAI’10, pp. 1306–1313. AAAI Press (2010)

  5. 5.

    Chen, J., Ma, T., Xiao, C.: FastGCN: fast learning with graph convolutional networks via importance sampling. In: International Conference on Learning Representations (2018)

  6. 6.

    Chen, S., Varma, R., Singh, A., Kovacevic, J.: Signal representations on graphs: tools and applications. CoRR (2015). arXiv:1512.05406

  7. 7.

    Chiang, W.-L., Liu, X., Si, S., Li, Y., Bengio, S., Hsieh, C.-J.: Cluster-GCN: an efficient algorithm for training deep and large graph convolutional networks (2019)

  8. 8.

    Cho, K., Merrienboer, B.V., Gulcehre, C., Bougares, F., Schwenk, H., Bengio, Y.: Learning phrase representations using RNN encoder-decoder for statistical machine translation. Comput. Sci. CoRR (2014). arXiv:1406.1078

  9. 9.

    Defferrard, M., Bresson, X., Vandergheynst, P.: Convolutional neural networks on graphs with fast localized spectral filtering . In: Lee, D., Sugiyama, M., Luxburg, U., Guyon, I., Garnett, R. (eds.) Advances in Neural Information Processing Systems, vol. 29. Curran Associates, Inc. (2016). https://proceedings.neurips.cc/paper/2016/file/04df4d434d481c5bb723be1b6df1ee65-Paper.pdf

  10. 10.

    Feng, Z.: Spectral graph sparsification in nearly-linear time leveraging efficient spectral perturbation analysis. In: Proceedings of the 53rd Annual Design Automation Conference, DAC ’16, pp. 57:1–57:6. ACM, New York (2016)

  11. 11.

    Fung, W.S., Hariharan, R., Harvey, N.J., Panigrahi, D.: A general framework for graph sparsification. In: Proceedings of the Forty-Third Annual ACM Symposium on Theory of Computing, STOC ’11, pp. 71–80. ACM, New York (2011)

  12. 12.

    Geng, X., Zhang, H., Bian, J., Chua, T.: Learning image and user features for recommendation in social networks. In: 2015 IEEE International Conference on Computer Vision (ICCV), pp. 4274–4282 (2015)

  13. 13.

    Gilmer, J., Schoenholz, S.S., Riley, P.F., Vinyals, O., Dahl, G.E.: Neural message passing for quantum chemistry . In: Precup, D., Teh, Y.W. (eds.) Proceedings of the 34th International Conference on Machine Learning. Proceedings of Machine Learning Research, vol. 70, pp. 1263–1272. PMLR (2017). http://proceedings.mlr.press/v70/gilmer17a/gilmer17a.pdf

  14. 14.

    Hamilton, W.L., Ying, R., Leskovec, J.: Inductive representation learning on large graphs. In: NIPS (2017)

  15. 15.

    Hong, M., Luo, Z.Q., Razaviyayn, M.: Convergence analysis of alternating direction method of multipliers for a family of nonconvex problems. SIAM J. Optim. 26(1), 337–364 (2016)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Karger, D.R.: Random sampling in cut, flow, and network design problems. In: Proceedings of the Twenty-Sixth Annual ACM Symposium on Theory of Computing, 23–25 May 1994, Montréal, Québec, Canada, pp. 648–657 (1994)

  17. 17.

    Kipf, T.N., Welling, M.: Semi-supervised classification with graph convolutional networks . CoRR (2016). arXiv:1609.02907

  18. 18.

    Li, J., Zhang, T., Tian, H., Jin, S., Fardad, M., Zafarani, R.: SGCN: a graph sparsifier based on graph convolutional networks. In: Lauw, H.W., Wong, R.C.W., Ntoulas, A., Lim, E.P., Ng, S.K., Pan, S.J. (eds.) Advances in Knowledge Discovery and Data Mining, pp. 275–287. Springer, Cham (2020)

    Chapter  Google Scholar 

  19. 19.

    Lindner, G., Staudt, C.L., Hamann, M., Meyerhenke, H., Wagner, D.: Structure-preserving sparsification of social networks. In: Proceedings of the 2015 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining 2015, ASONAM ’15, pp. 448–454. ACM, New York (2015)

  20. 20.

    Lü, L., Zhou, T.: Link prediction in complex networks: a survey. Physica A Stat. Mech. Appl. 390(6), 1150–1170 (2011)

    Article  Google Scholar 

  21. 21.

    Niepert, M., Ahmed, M., Kutzkov, K.: Learning convolutional neural networks for graphs . In: Balcan, M.F., Weinberger, K.Q. (eds.) Proceedings of the 33rd International Conference on Machine Learning. Proceedings of Machine Learning Research, vol. 48, pp. 2014–2023. PMLR, New York (2016). http://proceedings.mlr.press/v48/niepert16.pdf

  22. 22.

    Perozzi, B., Al-Rfou, R., Skiena, S.: Deepwalk: online learning of social representations. In: Proceedings of the 20th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD ’14, pp. 701–710. ACM, New York (2014)

  23. 23.

    Rong, Y., Huang, W., Xu, T., Huang, J.: Dropedge: towards deep graph convolutional networks on node classification. In: International Conference on Learning Representations (2020)

  24. 24.

    Satuluri, V., Parthasarathy, S., Ruan, Y.: Local graph sparsification for scalable clustering. In: Proceedings of the 2011 ACM SIGMOD International Conference on Management of Data, SIGMOD ’11, pp. 721–732. ACM, New York (2011)

  25. 25.

    Scarselli, F., Gori, M., Tsoi, A.C., Hagenbuchner, M., Monfardini, G.: The graph neural network model. Trans. Neural Netw. 20(1), 61–80 (2009)

    Article  Google Scholar 

  26. 26.

    Schaeffer, S.E.: Survey: graph clustering. Comput. Sci. Rev. 1(1), 27–64 (2007)

    Article  Google Scholar 

  27. 27.

    Sen, P., Namata, G.M., Bilgic, M., Getoor, L., Gallagher, B., Eliassi-Rad, T.: Collective classification in network data. AI Mag. 29(3), 93–106 (2008)

    Google Scholar 

  28. 28.

    Serrano, M.Á., Boguñá, M., Vespignani, A.: Extracting the multiscale backbone of complex weighted networks. Proc. Natl. Acad. Sci. U. S. A. 106(16), 6483–8 (2009)

    Article  Google Scholar 

  29. 29.

    Shuman, D.I., Narang, S.K., Frossard, P., Ortega, A., Vandergheynst, P.: Signal processing on graphs: extending high-dimensional data analysis to networks and other irregular data domains . CoRR (2012). arXiv:1211.0053

  30. 30.

    Spielman, D.A., Srivastava, N.: Graph sparsification by effective resistances. In: Proceedings of the Fortieth Annual ACM Symposium on Theory of Computing. STOC’08, Victoria, British Columbia, Canada, pp. 563–568. Association for Computing Machinery, New York (2008). https://doi.org/10.1145/1374376.1374456

  31. 31.

    Spielman, D.A., Teng, S.: Nearly linear time algorithms for preconditioning and solving symmetric, diagonally dominant linear systems. SIAM J.Matrix Anal. Appl. 35(3), 835–885 (2014). https://doi.org/10.1137/090771430

  32. 32.

    Takapoui, R., Moehle, N., Boyd, S., Bemporad, A.: A simple effective heuristic for embedded mixed-integer quadratic programming. In: 2016 American Control Conference (ACC), pp. 5619–5625 (2016). https://doi.org/10.1109/ACC.2016.7526551

  33. 33.

    Veličković, P., Cucurull, G., Casanova, A., Romero, A., Liò, P., Bengio, Y.: Graph attention networks. In: International Conference on Learning Representations (2018)

  34. 34.

    Wu, F., Souza, A., Zhang, T., Fifty, C., Yu, T., Weinberger, K.: Simplifying graph convolutional networks. In: Chaudhuri, K., Salakhutdinov, R. (eds.) Proceedings of the 36th International Conference on Machine Learning, Proceedings of Machine Learning Research, vol. 97, pp. 6861–6871. PMLR, Long Beach (2019)

  35. 35.

    Wu, Z., Pan, S., Chen, F., Long, G., Zhang, C., Yu, P.S.: A comprehensive survey on graph neural networks. IEEE Trans. Neural Netw. Learn. Syst. 32(1), 4–24 (2019). https://doi.org/10.1109/TNNlS.2020.2978386

  36. 36.

    Yang, Z., Cohen, W.W., Salakhutdinov, R.: Revisiting semi-supervised learning with graph embeddings. In: Proceedings of the 33rd International Conference on International Conference on Machine Learning, vol. 48, ICML’16, pp. 40–48. JMLR.org (2016)

  37. 37.

    Zheng, C., Zong, B., Cheng, W., Song, D., Ni, J., Yu, W., Chen, H., Wang, W.: Robust graph representation learning via neural sparsification. In: III, H.D., Singh, A. (eds.) Proceedings of the 37th International Conference on Machine Learning, Proceedings of Machine Learning Research, vol. 119, pp. 11458–11468. PMLR (2020)

  38. 38.

    Zhu, X.: Semi-supervised learning with graphs. Ph.D. Thesis, Pittsburgh, PA, USA (2005). AAI3179046

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Acknowledgements

This research was supported in part by the National Science Foundation under awards CAREER IIS-1942929, CAREER CMMI-1750531, and ECCS-1609916.

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Correspondence to Jiayu Li.

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Li, J., Zhang, T., Tian , H. et al. Graph sparsification with graph convolutional networks. Int J Data Sci Anal (2021). https://doi.org/10.1007/s41060-021-00288-8

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Keywords

  • Graph sparsification
  • Node classification
  • Graph convolutional network