  Triggered by limitations of graph-based deep learning methods in terms of
computational expressivity and model flexibility, recent years have seen a
surge of interest in computational models that operate on higher-order
topological domains such as hypergraphs and simplicial complexes. While the
increased expressivity of these models can indeed lead to a better
classification performance and a more faithful representation of the underlying
system, the computational cost of these higher-order models can increase
dramatically. To this end, we here explore a simplicial complex neural network
learning architecture based on random walks and fast 1D convolutions (SCRaWl),
in which we can adjust the increase in computational cost by varying the length
and number of random walks considered while accounting for higher-order
relationships. Importantly, due to the random walk-based design, the
expressivity of the proposed architecture is provably incomparable to that of
existing message-passing simplicial neural networks. We empirically evaluate
SCRaWl on real-world datasets and show that it outperforms other simplicial
neural networks.
