Abstract : Motivated by the discovery of hard-to-find social networks in epidemiology, we consider the question of exploring the topology of random structures (such as a random graph G) by random walks. The usual random walk jumps from a vertex of G to a neighboring vertex, with providing information on the connected components of the graph G. The number of these connected components is the Betti number $beta_{0}$. To gather further information on the higher Betti numbers that describe the topology of the graph, we can consider the simplicial complex C associated to the graph G: a k-simplex (edge for k = 1, triangle for k = 2, tetrahedron for k = 3 etc.) belongs to C if all the lower (k-1)-simplices that constitute it also belong to C. For example, a triangle belongs to C if its three edges are in the graph G. Several random walks have already been proposed recently to explore these structures. We introduce a new random walk, whose generator is related to a Laplacian of higher order of the graph and to the Betti number betak. A rescaling of the walk for k = 2 (cycle-valued random walk), and on regular triangulation of the torus, is also detailed. We embed the space of chains into spaces of currents to establish the limiting theorem.
Joint work with T. Bonis, L. Decreusefond and Z. Zhang.
https://perso.math.u-pem.fr/tran.viet-chi/

Keywords : random walks; simplicial complex; random graph; stochastic geometry