Section: Scientific Foundations

Scientific Foundations

Here is the scientific content of each of our main research directions.

• Modeling and control of communication networks. Here we mean admission control, flow regulation and feedback control à la TCP . Our aim is a mathematical representation of the dynamics of the most commonly used control protocols, from which one could predict and optimize the resulting end user bandwidth sharing and quality of service. We currently use our understanding of the dynamics of these protocols on Split TCP, as used in wireless access networks and in peer-to-peer overlays, and on variants of TCP meant to reach higher throughputs such as scalable TCP.

• Modeling and performance analysis of wireless networks. The main focus is on the following three classes of wireless networks: cellular networks, mobile ad hoc networks (MANETs) and WiFi mesh networks.

  Concerning cellular networks, our mathematical representation of interferences based on shot-noise has led to a variety of results on coverage and capacity of large CDMA networks when taking into account intercell interferences and power control. Our general goals are twofold: 1) to propose a strategy for the densification and parameterization of UMTS and future OFDM networks that is optimized for both voice and data traffic; 2) to design new self organization and self optimization protocols for cellular networks e.g. for power control, sub-carrier selection, load balancing, etc.

  Using a similar approach, we currently investigate MAC layer scheduling algorithms and power control protocols for MANETs and their vehicular variants called VANETs. We concentrate on cross layer optimizations allowing one to maximize the transport capacity of multihop MANETs. A recent example within this class of problems is that of opportunistic routing for MANETs. Our main quantitative results concern one-hop analysis as well as scaling laws for end-to-end delays on long routes. This last question is treated studying an appropriate first passage percolation problem on a new class of random graphs called space-time SINR graphs .

• Theory of network dynamics. TREC is also pursuing the elaboration of a stochastic network calculus, that would allow the analysis of network dynamics by algebraic methods. The mathematical tools are those of discrete event dynamical systems: semi-rings, ergodic theory, perfect simulation, stochastic comparison, inverse problems, large deviations, etc.

• Economics of networks The premise this relatively new direction of research, developed jointly with Jean Bolot (SPRINT) is that economic incentives drive the development and deployment of technology. Such incentives exist if there is a market where suppliers and buyers can meet. In today's Internet, such a market is missing. We started by looking at the general problem of security on Internet from an economic perspective and derived a model showing that network externalities and misaligned incentives are responsible for a low investment in security measures. We then analyzed the possible impact of insurance.

• The development of mathematical tools based on stochastic geometry and random geometric graphs Classical Stochastic Geometry. Stochastic geometry is a rich branch of applied probability which allows one to quantify random phenomena on the plane or in higher dimension. It is intrinsically related to the theory of point processes and also to random geometric graphs. Our research is centered on the development of a methodology for the analysis, the synthesis, the optimization and the comparison of architectures and protocols to be used in wireless communication networks. The main strength of this method is its capacity for taking into account the specific properties of wireless links, as well as the fundamental question of scalability.

• Combinatorial optimization and analysis of algorithms. In this research direction started in 2007, we build upon our expertise on random trees/graphs and our collaboration with D. Aldous in Berkeley. Sparse graph structures have proved useful in a number of applications from information processing tasks to the modeling of social networks. We obtained new results for stochastic processes taking place on such graphs. Thereby, we were able to analyze an iterative message passing algorithm for the random assignement problem and to characterize its performance. Likewise, we made a sensitivity analysis of such processes and computed the corresponding scaling exponents (for a dynamic programming optimization problem). We also derived analytic formula for the spectrum of the adjacency matrix of diluted random graphs.