Influential Simplices Mining via Simplicial Convolutional Network

TL;DR

This paper introduces ISMnet, a novel higher-order graph neural network model that effectively identifies influential simplices in simplicial complexes, outperforming existing methods in ranking simplices of various orders.

Contribution

The paper proposes a new higher-order graph learning model, ISMnet, utilizing hierarchical bipartite graphs and HoH Laplacians to identify influential simplices of arbitrary order.

Findings

ISMnet outperforms existing methods in ranking 0- and 2-simplices

The model effectively captures interactions among simplices using learnable graph convolutional operators

It can identify influential simplices of any order by adjusting the hub set

Abstract

Simplicial complexes have recently been in the limelight of higher-order network analysis, where a minority of simplices play crucial roles in structures and functions due to network heterogeneity. We find a significant inconsistency between identifying influential nodes and simplices. Therefore, it remains elusive how to characterize simplices' influence and identify influential simplices, despite the relative maturity of research on influential nodes (0-simplices) identification. Meanwhile, graph neural networks (GNNs) are potent tools that can exploit network topology and node features simultaneously, but they struggle to tackle higher-order tasks. In this paper, we propose a higher-order graph learning model, named influential simplices mining neural network (ISMnet), to identify vital h-simplices in simplicial complexes. It can tackle higher-order tasks by leveraging novel higher-order presentations: hierarchical bipartite graphs and higher-order hierarchical (HoH) Laplacians, where targeted simplices are grouped into a hub set and can interact with other simplices. Furthermore, ISMnet employs learnable graph convolutional operators in each HoH Laplacian domain to capture interactions among simplices, and it can identify influential simplices of arbitrary order by changing the hub set. Empirical results demonstrate that ISMnet significantly outperforms existing methods in ranking 0-simplices (nodes) and 2-simplices. In general, this novel framework excels in identifying influential simplices and promises to serve as a potent tool in higher-order network analysis.