## Section: Research Program

### The Cavity Method for Network Algorithms

The cavity method combined with geometric networks concepts has recently led to spectacular progresses in digital communications through error-correcting codes. More than fifty years after Shannon's theorems, some coding schemes like turbo codes and low-density parity-check codes (LDPC) now approach the limits predicted by information theory. One of the main ingredients of these schemes is message-passing decoding strategies originally conceived by Gallager, which can be seen as direct applications of the cavity method on a random bipartite graph (with two types of nodes representing information symbols and parity check symbols, see [57]).

Modern coding theory is only one example of application of the cavity method. The concepts and techniques developed for its understanding have applications in theoretical computer science and a rich class of *complex systems*, in the field of networking, economics and social sciences.
The cavity method can be used both for the analysis of randomized
algorithms and for the study of random ensembles of computational
problems representative real-world situations. In order to analyze the
performance of algorithms, one generally defines a family of instances
and endows it with a probability measure, in the same way as one
defines a family of samples in the case of spin glasses or LDPC
codes. The discovery that the hardest-to-solve instances, with all
existing algorithms, lie close to a *phase transition* boundary has spurred
a lot of interest. Theoretical physicists suggest that the reason is a structural one, namely a change in the geometry of the set of solutions related to the *replica symmetry breaking* in the cavity method.
Phase transitions, which lie at the core of statistical physics, also play a key role in computer
science [60], signal processing [44] and social sciences [49].
Their analysis is a major challenge, that may have a strong impact on the design of related algorithms.

We develop mathematical tools in the theory of discrete probabilities and theoretical computer science in order to contribute to a rigorous formalization of the cavity method, with applications to network algorithms, statistical inference, and at the interface between computer science and economics (EconCS).