Alejandro Pérez Riascos (Université Mariana, San Juan de Pasto, Colombie). Random Walks on Networks: Lévy Flights and Fractional Transport.

We study fractional random walks on networks defined from the equivalent of the fractional diffusion equation in graphs and their relation with Lévy walk navigation strategies on networks. We explore these processes for different structures; in particular, rings, circulant networks, n-dimensional lattices, complete graphs and random networks. From the spectra and the eigenvectors of the Laplacian matrix, we deduce explicit results for different quantities that characterize this dynamical process and their capacity to explore networks. We obtain analytical expressions for the fractional transition matrix, the fractional degree and the average probability of return of the random walker. Also, we discuss the Kemeny constant that gives the average number of steps necessary to reach any site of the network. Through all this work, we analyze the mechanisms behind the fractional transport on networks and how this long-range process induces dynamically the small-world property in different structures.