Optimal Navigation on Simplicial Complexes

Incidence-based random walks on simplicial complexes

We introduce an incidence-based random walk on the edges of a random two-dimensional simplicial complex with a complete $1$-skeleton and independently retained triangular faces. The dynamics combine two transport channels, one mediated by vertices and the other by triangular faces, through an effective transition opera...

Random Recursive Simplicial Complexes

We investigate random recursive simplicial complexes growing by adding, at each step, a vertex together with a simplex formed by joining the new vertex with a randomly chosen existing simplex. We also add all faces of the new simplex to ensure that the resulting object remains a simplicial complex. If the choice of an...

Stochastic Processes and their Applications

Vertex reinforced branching random walks and generalized time-dependent Polya urns

We consider a class of infinite critical tree-indexed random walks on $\mathbb Z$, where the motion of particles is subject to vertex reinforcement. We mainly focus on the strong reinforcement regime, where we expect the process to localize almost surely on two sites. Part of our analysis includes the study of a time-d...

On the maximal dimension of random simplicial complexes

The dimension of random simplicial complexes (defined as the maximal dimension among all faces) is a natural extreme value associated with the complex, and is closely related to other functionals defined by a maximum, such as the clique number of geometric graphs or scan statistics. We extend existing results in the...

Network parameters via equilibrium measures in Schrödinger random walks

This work demonstrates how equilibrium measures within the framework of Schr¨odinger random walks on networks can be leveraged to compute key network parameters such as the Mean First Passage Time (MFPT) and Kemeny's constant by expressing these parameters in terms of generalized inverses of the associated M-matrix.