  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.
