Optimal Navigation on Simplicial Complexes

Abstract

The navigation time and optimal search strategies deriving from random dynamical processes on binary graphs have been extensively explored and analyzed, being of prominent interest in the network science field. In this work, we study an extension of these topological measures for simplicial complexes: a specific type of geometric and algebraic structures that encapsulates higher-order interactions. Here, the explorability analysis of simplicial complexes has been conducted in terms of the mean first passage times between nodes, i.e. the 0th-order simplices, with the inclusion of a long-range stochastic teleportation term modulated with respect to the local random walk hopping across the various dimensions. We also provide a perturbative approximation scheme recovering the modulation parameter between pure random walk and teleportation mode (for higher-order setting) acting as the expansion parameter.