Speaker: Camille Poignard

Abstract: This work deals with the spectrum of Laplacian matrices over weighted graphs, for which Fiedler [70's] showed their topological descriptions rely on two objects of fundamental importance: the second eigenvalue of the spectrum ("spectral gap") and one of its associated eigenvectors, the so called "Fiedler eigenvector".
Since the seminal work of Fiedler, the use of Spectral graph theory in the study of dynamical networks has been really successful.
First, I will show that given a Laplacian matrix, it is possible to perturb slightly the weights of its existing links so that its spectrum be composed of only simple eigenvalues, and its Fiedler eigenvector be composed of only non zero entries. These genericity properties with the constraint of not adding links in the underlying network are stronger than the classical ones, for which any "topological" perturbation is allowed.
Then, I will show how this result can be useful for the synchronization of diffusively coupled networks, more precisely for the problem of identifying the links for which the perturbations of the weights modify this dynamics, i.e decreases or enhances the synchronization.
The talk does not require any background on graphs theory or on dynamical systems