Abstract
We propose a novel method to study properties of graph-structured data by means of a geometric construction called Dowker complex. We study this simplicial complex through the use of persistent homology, which has shown to be a prominent tool to uncover relevant geometric and topological information in data. A positively weighted graph induces a distance in its sets of vertices. A classical approach in persistent homology is to construct a filtered Vietoris-Rips complex with vertices on the vertices of the graph. However, when the size of the set of vertices of the graph is large, the obtained simplicial complex may be computationally hard to handle. A solution The Dowker complex is constructed on a sample in the set of vertices of the graph called landmarks. A way to guaranty sparsity and proximity of the set of landmarks to all the vertices of the graph is by considering \(\epsilon \)-nets. We provide theoretical proofs of the stability of the Dowker construction and comparison with the Vietorips-Rips construction. We perform experiments showing that the Dowker complex based neural networks model performs good with respect to baseline methods in tasks such as link prediction and resilience to attacks.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Agarwal, S., Branson, K., Belongie, S.: Higher order learning with graphs. In: ICML, pp. 17–24 (2006)
Aksoy, S.G., et al.: Seven open problems in applied combinatorics. arXiv preprint arXiv:2303.11464 (2023)
Arafat, N.A., Basu, D., Bressan, S.: \(\epsilon \)-net induced lazy witness complexes on graphs (2020)
Benson, A.R., Abebe, R., Schaub, M.T., Jadbabaie, A., Kleinberg, J.: Simplicial closure and higher-order link prediction. PNAS 115(48), E11221–E11230 (2018)
Bodnar, C., et al.: Weisfeiler and Lehman go topological: message passing simplicial networks. In: ICML, pp. 1026–1037 (2021)
Boissonnat, J.D., Guibas, L., Oudot, S.: Manifold reconstruction in arbitrary dimensions using witness complexes. In: SoCG, pp. 194–203 (2007)
Chami, I., Ying, R., Ré, C., Leskovec, J.: Hyperbolic graph convolutional neural networks (2019)
Chazal, F., De Silva, V., Oudot, S.: Persistence stability for geometric complexes. Geom. Dedicata. 173(1), 193–214 (2014)
Chazal, F., Michel, B.: An introduction to topological data analysis: fundamental and practical aspects for data scientists. Front. Artif. Intell 4 (2021)
Chen, Y., Gel, Y.R., Poor, H.V.: BScNets: block simplicial complex neural networks. Proc. AAAI Conf. Artif. Intell. 36(6), 6333–6341 (2022)
Chen, Y., Gel, Y.R., Marathe, M.V., Poor, H.V.: A simplicial epidemic model for COVID-19 spread analysis. Proc. Natl. Acad. Sci. 121(1), e2313171120 (2024)
Chen, Y., Jacob, R.A., Gel, Y.R., Zhang, J., Poor, H.V.: Learning power grid outages with higher-order topological neural networks. IEEE Trans. Power Syst. 39(1), 720–732 (2024)
Chen, Y., Jiang, T., Gel, Y.R.: H\(^2\)-nets: hyper-Hdge convolutional neural networks for time-series forecasting. In: Koutra, D., Plant, C., Gomez Rodriguez, M., Baralis, E., Bonchi, F. (eds.) ECML PKDD 2023. LNCS, vol. 14713, pp. 271–289. Springer, Cham (2023). https://doi.org/10.1007/978-3-031-43424-2_17
Cohen-Steiner, D., Edelsbrunner, H., Harer, J.: Stability of persistence diagrams. In: SoCG, pp. 263–271 (2005)
De Silva, V., Carlsson, G.: Topological estimation using witness complexes. In: PBG, pp. 157–166 (2004)
Dey, T.K., Fan, F., Wang, Y.: Graph induced complex on point data. In: SoCG, pp. 107–116 (2013)
Ebli, S., Defferrard, M., Spreemann, G.: Simplicial neural networks. In: NeurIPS 2020 Workshop on TDA and Beyond (2020)
Hensel, F., Moor, M., Rieck, B.: A survey of topological machine learning methods. Front. Artif. Intell. 4, 52 (2021)
Johnson, J.L., Goldring, T.: Discrete Hodge theory on graphs: a tutorial. Comput. Sci. Eng. 15(5), 42–55 (2013)
Kipf, T.N., Welling, M.: Variational graph auto-encoders (2016)
Kipf, T.N., Welling, M.: Semi-supervised classification with gcns. In: ICLR (2017)
Liu, X., Feng, H., Wu, J., Xia, K.: Dowker complex based machine learning (DCML) models for protein-ligand binding affinity prediction. PLoS Comp. Biol. 18(4), e1009943 (2022)
Mavromatis, C., Karypis, G.: Graph InfoClust: maximizing coarse-grain mutual information in graphs. In: Karlapalem, K., et al. (eds.) Advances in Knowledge Discovery and Data Mining. PAKDD 2021, Part I. LNCS, vol. 12712, pp. 541–553. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-75762-5_43
Pei, H., Wei, B., Chang, K.C.C., Lei, Y., Yang, B.: Geom-GCN: geometric graph convolutional networks. In: ICLR (2019)
Roddenberry, T.M., Glaze, N., Segarra, S.: Principled simplicial neural networks for trajectory prediction. In: ICML, pp. 9020–9029 (2021)
Schaub, M.T., Benson, A.R., Horn, P., Lippner, G., Jadbabaie, A.: Random walks on simplicial complexes and the normalized Hodge 1-laplacian. SIAM Rev. 62(2), 353–391 (2020)
Sen, P., Namata, G., Bilgic, M., Getoor, L., Galligher, B., Eliassi-Rad, T.: Collective classification in network data. AI Mag. 29(3), 93–93 (2008)
Veličković, P., Cucurull, G., Casanova, A., Romero, A., Liò, P., Bengio, Y.: Graph attention networks. In: ICLR (2018)
Yan, Z., Ma, T., Gao, L., Tang, Z., Chen, C.: Link prediction with persistent homology: an interactive view (2021)
Zhang, M., Chen, Y.: Link prediction based on graph neural networks (2018)
Acknowledgements
This project has been supported in part by NASA AIST 21-AIST21_2-0059, NSF ECCS 2039701, TIP-2333703, and ONR N00014-21-1-2530 grants. The paper is based upon work supported by (while Y.R.G. was serving at) the NSF. The views expressed in the article do not necessarily represent the views of NSF, ONR or NASA.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2024 The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd.
About this paper
Cite this paper
Choi, J.W., Chen, Y., Frías, J., Castillo, J., Gel, Y. (2024). Revisiting Link Prediction with the Dowker Complex. In: Yang, DN., Xie, X., Tseng, V.S., Pei, J., Huang, JW., Lin, J.CW. (eds) Advances in Knowledge Discovery and Data Mining. PAKDD 2024. Lecture Notes in Computer Science(), vol 14646. Springer, Singapore. https://doi.org/10.1007/978-981-97-2253-2_33
Download citation
DOI: https://doi.org/10.1007/978-981-97-2253-2_33
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-97-2252-5
Online ISBN: 978-981-97-2253-2
eBook Packages: Computer ScienceComputer Science (R0)