Influential simplices mining via simplicial convolutional networks

The identification of influential simplices is crucial for understanding higher-order network dynamics. Yet, despite relatively mature research on influential nodes (0-simplices) mining, characterizing simplices’ influence and identifying influential simplices remain challenging due to notable discrepancies in vital nodes and vital simplices mining. In this paper, we propose a higher-order graph learning model, named influential simplices mining neural networks (ISMnet), to identify vital simplices in simplicial complexes. ISMnet leverages novel higher-order representations: hierarchical bipartite graphs and higher-order hierarchical (HoH) Laplacians, where target simplices are grouped into a hub set and can interact with other simplices. It also employs learnable graph convolution operators in each HoH Laplacian domain to capture interactions among simplices and can identify influential simplices of arbitrary order by changing the hub set. Notably, ISMnet addresses the limitations inherent in traditional graph neural networks that struggle with higher-order tasks, while seamlessly retaining the capability to exploit network topology and node features concurrently. Numerical results on empirical and synthetic datasets demonstrate that ISMnet significantly outperforms existing methods by at least 12% and 4%, respectively, in ranking 2-simplices. In general, this novel framework promises to serve as a potent tool in higher-order network analysis.