Facilitating Graph Neural Networks with Random Walk on Simplicial
  Complexes
Cai Zhou Xiyuan Wang Muhan Zhang
Node-level random walk has been widely used to improve Graph Neural Networks.
However, there is limited attention to random walk on edge and, more generally,
on $k$-simplices. This paper systematically analyzes how random walk on
different orders of simplicial complexes (SC) facilitates GNNs in their
theoretical expressivity. First, on $0$-simplices or node level, we establish a
connection between existing positional encoding (PE) and structure encoding
(SE) methods through the bridge of random walk. Second, on $1$-simplices or
edge level, we bridge edge-level random walk and Hodge $1$-Laplacians and
design corresponding edge PE respectively. In the spatial domain, we directly
make use of edge level random walk to construct EdgeRWSE. Based on the spectral
analysis of Hodge $1$-Laplcians, we propose Hodge1Lap, a permutation
equivariant and expressive edge-level positional encoding. Third, we generalize
our theory to random walk on higher-order simplices and propose the general
principle to design PE on simplices based on random walk and Hodge Laplacians.
Inter-level random walk is also introduced to unify a wide range of simplicial
networks. Extensive experiments verify the effectiveness of our random
walk-based methods.