Viet-Chi TRAN -- Random walks on simplicial complexes,
schedule le mardi 07 janvier 2020 de 14h00 à 15h00
Organisé par : LPSM
Intervenant : Viet-Chi TRAN (UPEM)
Lieu : Jussieu, tours 16-26, 2ème étage, salle 209.
Sujet : Viet-Chi TRAN -- Random walks on simplicial complexes,
A natural and well-known way to discover the topology of random structures (such as a random graph $G$), is to have them explored by random walks. The usual random walk jumps from a vertex of $G$ to a neighboring vertex, 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 $G$. More generally, a simplicial complex $C$ is made of $k$-simplices (edge for $k=1$, triangle for $k=2$, tetrahedron for $k=3$ etc.) with the constraint that a $k$ simplex can belong to $C$ only if all the lower $k-1$-simplices that constitute it also belong to $C$. For example, if a triangle belongs to $C$, this implies that its three edges are in $C$. Several random walks have already been proposed recently to explore these structures, mostly in Informatics Theory. We propose new random walks, whose generators are related to the combinatorial Laplacians of the simplicial complex, and to the Betti numbers $\beta_k$. A rescaling of the walk is also proposed.