Journées MAS de la SMAI
Modélisation et Statistiques des Réseaux
Rennes, 27-29 août 2008

Philippe ROBERT

Equilibrium and dynamics of large stochastic networks

The recent development of very large networks with self-organizing algorithms (large distributed systems) is a major challenge for the mathematical community. There is a large class of problems related to such networks, one can mention :

• Diffusion of information in very large peer to peer networks.
• Analysis of dynamic aspects of distributed algorithms.
• Allocation Bandwidth Algorithms in large mobile networks.

A general problem, classical in statistical physics, concern the qualitative properties of the network that can be inferred from the local specification of the algorithms. In general, the equilibrium distributions of stochastic networks do not have a Gibbsian representation. In fact equilibrium distributions are not known at all in general. And even when a Gibbsian representation holds, the dynamics of the networks are not Gibbsian. This has implications to analyze the stability properties of these networks. The purpose of the talk is to present and discuss several examples of stochastic networks.