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.

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. 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 spatial domain, we directly make use of edge level random walk to construct EdgeRWSE.