Random Walks Across Dimensions: Exploring Simplicial Complexes

TL;DR

This paper introduces a new operator for random walks on simplicial complexes, enabling exploration across various dimensions and providing insights into the importance of higher order structures.

Contribution

It presents a novel framework for random walks on simplicial complexes, extending traditional graph-based methods to higher-dimensional structures.

Findings

Asymptotic distribution of walkers ranks higher order simplices.

Optimal search strategies with stochastic teleportation are developed.

Interplay of noise and higher order structures is analyzed.

Abstract

We introduce a novel operator to describe a random walk process on a simplicial complex. Walkers are allowed to wonder across simplices of various dimensions, bridging nodes to edges, and edges to triangles, via a nested organization that hierarchically extends to higher structures of arbitrary 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 presence of stochastic teleportation are addressed and the peculiar interplay of noise with higher order structures unraveled.