Random walks across dimensions: Exploring simplicial complexes

We introduce an operator to describe a random walk process on a simplicial complex. Walkers are allowed to wander across simplices of various dimensions, bridging nodes to edges, and edges to triangles, via a nested organization that hierarchically extends to higher structures of arbitrarily large, but finite, dimension. The asymptotic distribution of the walkers provides a natural ranking to gauge the relative importance of higher-order simplices. Optimal search strategies in the presence of stochastic teleportation are addressed, and the peculiar interplay of noise with higher-order structures is unraveled.