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Section: New Results

Random Graphs

Participants : Nicolas Broutin, Henning Sulzbach.

Universality of scaling limits of random graphs

Random graphs are one of the most studied models of networks, and they turn out to be related to crucial questions in physics about the behaviour of matter at the phase transition, or in combinatorial optimization about the hardness of computation. In recent years, we have constructed the scaling limit of the classical Erdos-Renyi random graph model, and conjectured that this limit also happened to be universal.

The funding of the Associated Team RNA has permitted to invite Shankar Bhamidi. During his visit, we have worked and found a new way to construct the scaling limit of random graph processes in the critical window. This method is especially important since it is robust enough to prove universality of the limit, that is that many models have the same limit. The method relies on the dynamics of the coalescence of clusters as the edges are added, and allows us to hope for proofs that would be able to treat the more complex geometric models.

Cutting down random tree and the genealogy of fragmentations

The study of the internal structure of random combinatorial object such as graphs and trees led to question about whether such objects exhibit invariance by certain complex surgical operations (disconnect some pieces, and re-attach them somewhere else). In the context of graphs, this is related to the so-called self-organized criticality: certain distributions that yield fractal objects should naturally appear in nature because they are the fixed points of some recombination procedures. In the context of trees, it turns out that certain fragmentations arising when chopping off a random tree have a genealogy that has the same distribution as the original tree. We have investigated this with Minmin Wang, and obtained results about p-trees and the genealogy of the fragmentation on Aldous' celebrated continuum random tree. These may also be interpreted in terms of complex path transformations for Brownian excursions and other random processes with exchangeable increments, and hence relate to very classical questions in probability theory.

New encodings for combinatorial coalescent processes

In 2013, we had constructed the scaling limit of the minimum spanning tree of a complete graph using crucial information about the scaling limit of random graphs, and especially about the way the cluster merge as the edges are added in the graph. With J.-F. Marckert (LaBRI, Bordeaux) we have found a novel construction of the important multiplicative coalescent that describes how the connected components of a random graph coalesce as the edges are added. This unveils yet more interesting links between the minimum spanning tree and the random graph, since Prim's celebrated algorithm is used to construct a consistent ordering of the vertices that ensures that the connected components are intervals.

Navigation in random Delaunay triangulations

Navigation or routing algorithms are fundamental routines: in order to solve many problems, one of the first steps consists in locating a node in a data structure. Unfortunately, the current algorithms are based on heuristics and very few rigorous results about the performance of such algorithms are known when the model for data is more realistic than the worst-case.

With O. Devillers and R. Hemsley, we have initiated a program that aims at finding rigourous estimates for the performance of routing algorithms in geometric structures such as Delaunay tesselations. So far we have managed to develop some tools that permitted us to analyse a simple algorithm. Although this algorithm has been designed for most of the analysis to work, this work paves the way towards the rigorous analysis of other more natural and widely used algorithms.

Connectivity and sparsification of sparse wireless networks

Many models of wireless networks happen to be connected only when the average degree is tending to infinity with the size of the network, more precisely when it is about the logarithm of the number of nodes. This raises questions about the potential issues in scaling such models. With L. Devroye (McGill) and G. Lugosi (ICREA and Pompeu Fabra), we have worked at analysing models in which we try to construct connected or almost connected networks in a distributed way (that is that no global optimization is allowed in designing the network, and every device should proceed in the same way to choose its neighbors). We have managed to analyse an algorithm for constructing such a network, and to obtain tight results about the number of links that a typical device should have in order for the global network to be connected. We further proved that this is asymptotically optimal when one only requires that most nodes should be in the same connected component.