BScNets: Block Simplicial Complex Neural Networks

Simplicial neural networks (SNN) have recently emerged as the newest direction in graph learning which expands the idea of convolutional architectures from node space to simplicial complexes on graphs. Instead of pre-dominantly assessing pairwise relations among nodes as in the current practice, simplicial complexes allow us to describe higher-order interactions and multi-node graph structures. By building upon connection between the convolution operation and the new block Hodge-Laplacian, we propose the first SNN for link prediction. Our new Block Simplicial Complex Neural Networks (BScNets) model generalizes the existing graph convolutional network (GCN) frameworks by systematically incorporating salient interactions among multiple higher-order graph structures of different dimensions. We discuss theoretical foundations behind BScNets and illustrate its utility for link prediction on eight real-world and synthetic datasets. Our experiments indicate that BScNets outperforms the state-ofthe-art models by a significant margin while maintaining low computation costs. Finally, we show utility of BScNets as the new promising alternative for tracking spread of infectious diseases such as COVID-19 and measuring the effectiveness of the healthcare risk mitigation strategies.

[1]  Glenn Fung,et al.  Simplicial 2-Complex Convolutional Neural Networks , 2020 .

[2]  Guido Mont'ufar,et al.  Weisfeiler and Lehman Go Topological: Message Passing Simplicial Networks , 2021, ICML.

[3]  Thomas Hofmann,et al.  Hyperbolic Neural Networks , 2018, NeurIPS.

[4]  Max Welling,et al.  Semi-Supervised Classification with Graph Convolutional Networks , 2016, ICLR.

[5]  Lek-Heng Lim,et al.  Hodge Laplacians on graphs , 2015, SIAM Rev..

[6]  Austin R. Benson,et al.  Random Walks on Simplicial Complexes and the normalized Hodge Laplacian , 2018, SIAM Rev..

[7]  M. Hajij,et al.  Cell Complex Neural Networks , 2020, ArXiv.

[8]  Dan A. Simovici.  Linear Algebra Tools for Data Mining , 2012 .

[9]  Jure Leskovec,et al.  Predicting multicellular function through multi-layer tissue networks , 2017, Bioinform..

[10]  U. Feige,et al.  Spectral Graph Theory , 2015 .

[11]  Sibo Wang,et al.  Inductive Link Prediction for Nodes Having Only Attribute Information , 2020, IJCAI.

[12]  Douwe Kiela,et al.  Poincaré Embeddings for Learning Hierarchical Representations , 2017, NIPS.

[13]  Yu Wang,et al.  Intrinsic and extrinsic operators for shape analysis , 2019, Handbook of Numerical Analysis.

[14]  Seth Lloyd,et al.  Quantum algorithms for topological and geometric analysis of data , 2016, Nature Communications.

[15]  Liangcai Gao,et al.  Link Prediction with Persistent Homology: An Interactive View , 2021, ICML.

[16]  Paul A. Bekker,et al.  THE POSITIVE SEMIDEFINITENESS OF PARTITIONED MATRICES , 1988 .

[17]  Santiago Segarra,et al.  Principled Simplicial Neural Networks for Trajectory Prediction , 2021, ICML.

[18]  Jon M. Kleinberg,et al.  Simplicial closure and higher-order link prediction , 2018, Proceedings of the National Academy of Sciences.

[19]  Kilian Q. Weinberger,et al.  Simplifying Graph Convolutional Networks , 2019, ICML.

[20]  Kathleen M. Carley,et al.  The effects of evolutionary adaptations on spreading processes in complex networks , 2018, Proceedings of the National Academy of Sciences.

[21]  Juan G. Restrepo,et al.  Topological synchronization: explosive transition and rhythmic phase , 2021, ArXiv.

[22]  Rafael Prieto Curiel,et al.  Vaccination strategies against COVID-19 and the diffusion of anti-vaccination views , 2020, Scientific Reports.

[23]  Maria A. Kazandjieva,et al.  A high-resolution human contact network for infectious disease transmission , 2010, Proceedings of the National Academy of Sciences.

[24]  S. Boccaletti,et al.  Contagion in simplicial complexes , 2021, 2107.03411.

[25]  Jure Leskovec,et al.  Hyperbolic Graph Convolutional Neural Networks , 2019, NeurIPS.

[26]  Chao Chen,et al.  Persistence Enhanced Graph Neural Network , 2020, AISTATS.

[27]  Martin Grohe,et al.  Weisfeiler and Leman Go Neural: Higher-order Graph Neural Networks , 2018, AAAI.

[28]  Gard Spreemann,et al.  Simplicial Neural Networks , 2020, ArXiv.

[29]  P. Kaye.  Infectious diseases of humans: Dynamics and control , 1993 .

[31]  James L. Johnson,et al.  Discrete Hodge Theory on Graphs: A Tutorial , 2013, Computing in Science & Engineering.

[32]  Jure Leskovec,et al.  node2vec: Scalable Feature Learning for Networks , 2016, KDD.

[33]  Serge J. Belongie,et al.  Higher order learning with graphs , 2006, ICML.

[34]  Vito Latora,et al.  Simplicial models of social contagion , 2018, Nature Communications.

[35]  A. Albert.  Conditions for Positive and Nonnegative Definiteness in Terms of Pseudoinverses , 1969 .

[36]  J. Leskovec,et al.  Open Graph Benchmark: Datasets for Machine Learning on Graphs , 2020, NeurIPS.

[37]  Tingyang Xu,et al.  DropEdge: Towards Deep Graph Convolutional Networks on Node Classification , 2020, ICLR.

[38]  Lise Getoor,et al.  Collective Classification in Network Data , 2008, AI Mag..

[39]  Yixin Chen,et al.  Link Prediction Based on Graph Neural Networks , 2018, NeurIPS.

[40]  Jure Leskovec,et al.  Inductive Representation Learning on Large Graphs , 2017, NIPS.

[41]  Pietro Liò,et al.  Graph Attention Networks , 2017, ICLR.

[42]  Yue Deng,et al.  Graph embedding-based novel protein interaction prediction via higher-order graph convolutional network , 2020, PloS one.

[43]  W. O. Kermack,et al.  A contribution to the mathematical theory of epidemics , 1927 .

[44]  Anand Louis,et al.  NHP: Neural Hypergraph Link Prediction , 2020, CIKM.

[45]  A. Marathe,et al.  Prioritizing allocation of COVID-19 vaccines based on social contacts increases vaccination effectiveness , 2021, medRxiv.

[46]  Howard Whitley Eves,et al.  Elementary Matrix Theory , 1980 .