Speaker
Dimitri Volchenkov

Description Most of the networks and databases humans have deal with contain large albeit finite number of units. Their structure maintaining functional consistency of the components is essentially not random and calls for a precise quantitative description of relations between nodes or data units and all network parts, as having important implications for the network robustness. A network can be seen as a discrete time dynamical system possessing a finite number of states. The behavior of such a dynamical system can be studied by means of transfer operators which describe the time evolution of distributions in phase space. The transfer operator can be represented by a stochastic matrix determining a discrete time random walk on the graph in which a walker picks at each node between the various available edges with equal probability. The Laplace operator associated to random walks possesses a group generalized inverse that can be used in order to define a probabilistic Riemannian manifold with a random metric on any finite connected undirected graph, or a database. In contrast to classical graph theory paying attention to the shortest paths of least cost, in the developed probabilistic approach all possible paths between a pair of vertices in a connected graph or a pair of units in a database are taken into account, although some paths shall be more probable than others. In such a formulation of graph theory, the distance is nothing else as a "path integral".The probabilistic geometrization of data enables us to attack applied problems which could not even be started otherwise. In particular, we report on the applications of the probabilistic approach for the analysis of urban structures, evolution of languages and musical compositions.

Speaker
Philip Maini

Speaker
Jordi Garcia-Ojalvo

Speaker
Aurora Hernandez Machado