Section: New Results

Random Graphs

Participants : Nicolas Broutin, Henning Sulzbach.

Connectivity in models of wireless networks

This is joint work with S. Boucheron (Paris 7), L. Devroye (McGill), N. Fraiman (McGill), and G. Lugosi (Pompeu Fabra).

The traditional models for wireless networks rely on geometric random graphs. However, if one wants to ensure that the graph be fully connected the radius of influence (hence the power necessary, and number of links) is too large to be fully scalable. Recently some models have been proposed that skim the neighbours and only retain a random subset for each node, hence creating a sparser overlay that would hopefully be more scalable. The first results on the size of the subsets which guarantee connectivity of overlay (the irrigation graph) confirm that the average number of links per node is much smaller, but it remains large. These results motivate further investigations on the size of the largest connected component when one enforces a constant average degree which are in the process of being written.

Random graphs and minimum spanning trees

This is a long term collaboration with L. Addario-Berry (McGill), C. Goldschmidt (Oxford) and G. Miermont (ENS Lyon).

The random graph of Erdős and Rényi is one of the most studied models of random networks. Among the different ranges of density of edges, the “critical window” is the most interesting, both for its applications to the physics of phase transitions and its applications to combinatorial optimization (minimum spanning tree, constraint satisfaction problems). One of the major questions consists in determining the distribution of distances between the nodes. A limit object (a scaling limit) has been identified, that allows to describe precisely the first order asymptotics of pairwise distances between the nodes. This limit object is a random metric space whose definition allows to exhibit a strong connection between random graphs and the continuum random tree of Aldous. A variety of questions like the diameter, the size of cycles, etc, may be answered immediately by reading them on the limit metric space.

In a stochastic context, the minimum spanning tree is tigthly connected to random graphs via Kruskal's algorithm. Random minimum spanning trees have attracted much research because of their importance in combinatorial optimization and statistical physics; however, until now, only parameters that can be grasped by local arguments had been studied. The scaling limit of the random graphs obtained permits to describe precisely the metric space scaling limit of a random minimum spanning tree, which identifies a novel continuum random tree which is truely different from that of Aldous.

Analysis of recursive partitions

This is joint work with R. Neininger (Frankfurt)

The techniques that we developped in order to estimate the cost of partial match queries in random quad trees have been used to solve an open question about the recursive lamination of the disk. We have proved that the planar dual of the lamination, which is a tree, converges almost surely when suitably rescaled to a compact random tree encoded by a continuous function. We also pinned down the fractal dimension of the limit object.