  Focusing on coupling between edges, we generalize the relationship between
the normalized graph Laplacian and random walks on graphs by devising an
appropriate normalization for the Hodge Laplacian -- the generalization of the
graph Laplacian for simplicial complexes -- and relate this to a random walk on
edges. Importantly, these random walks are intimately connected to the topology
of the simplicial complex, just as random walks on graphs are related to the
topology of the graph. This serves as a foundational step towards incorporating
Laplacian-based analytics for higher-order interactions. We demonstrate how to
use these dynamics for data analytics that extract information about the
edge-space of a simplicial complex that complements and extends graph-based
analysis. Specifically, we use our normalized Hodge Laplacian to derive
spectral embeddings for examining trajectory data of ocean drifters near
Madagascar and also develop a generalization of personalized PageRank for the
edge-space of simplicial complexes to analyze a book co-purchasing dataset.
