Section: New Results

Random Graphs

Participant : Nicolas Broutin.

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) [3] 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 [2] .

In a stochastic context, the minimum spanning tree is tightly 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 in [2] permits to describe precisely the metric space scaling limit of a random minimum spanning tree [21] , which identifies a novel continuum random tree which is truly different from that of Aldous.

Analysis of recursive partitions

This is joint work with R. Neininger (Frankfurt) and H. Sulzbach (Frankfurt/McGill).

The quadtrees are essential data structures that permit to store and manipulate geometric data by building a recursive partition of the space. In order to evaluate their performance, Flajolet and his co-authors have estimated the average cost of reporting all the data matching certain random queries. When the query does not fully specify all the fields, one talk about a partial match query. Such queries are ubiquitous, but analyzing their behaviour turns out to be intricate, and no performance guarantee was available in the form of a bound on the probability that any query would take much more time that one expects. [14] provides such guarantees by analysing the behaviour of all the queries at the same time, as a process. This yields estimates for the cost of the worst possible query (not a uniformly random one), as well as asymptotics for the variance and higher moments.

This line of research has motivated the analysis of the related combinatorial model of recursive lamination of the disk. The model had been recently introduced, but no full analysis was available. The techniques developed in the context of quadtrees have inspired a proof that the dual tree of the recursive lamination does converge to a limit tree-like metric space which is identified [23] .

Navigation and point location in Poisson Delaunay triangulation

Nicolas Broutin has recently initiated a project with O. Devillers (Inria Sophia) and R. Hemsley (Inria Sophia) concerning the performance of local routing algorithms in plane subdivisions. Such algorithms also turn out to be important for the point location problem: for instance, finding the face of the subdivision which contains a query point is the first step towards inserting this point as a vertex. The aim is to prove that when the subdivision consists of the faces of a Delaunay triangulation, and when the points are random, any natural strategy which would take you closer to the aim performs well. Preliminary results about a specific routing algorithm, the cone walk, that we designed for its amenability to analysis appear in [22] .