Section: New Results

Random Graphs and Combinatorial Optimization

Participants : Hamed Amini, Emilie Coupechoux, Mathieu Leconte, Marc Lelarge, Justin Salez.

Rank of Large Random Graphs

In [16] , with Charles Bordenave [CNRS-Université de Toulouse], we investigated the rank of the adjacency matrix of large diluted random graphs: for a sequence of graphs converging locally to a Galton-Watson tree, we provided an explicit formula for the asymptotic multiplicity of the eigenvalue 0 in terms of the degree generating function. In the first part, we showed that the adjacency operator associated with a Galton-Watson tree is self-adjoint with probability one ; we analyzed the associated spectral measure at the root and characterize the distribution of its atomic mass at 0. In the second part, we established a sufficient condition for the expectation of this atomic mass to be precisely the normalized limit of the dimension of the kernel of the adjacency matrices of the sequence of graphs. Our proofs borrow ideas from analysis of algorithms, functional analysis, random matrix theory, and statistical physics.

Matchings in infinite graphs

In [43] , we proved that for any sequence of (deterministic or random) graphs converging locally, the corresponding sequence of normalized matching numbers converges, and this limit depends only on the limit of the graph sequence. In the particular case where this limit is a unimodular Galton Watson tree, we were able to compute explicitly the value for the limit of the sequence of (normalized) matching numbers. This leads to an explicit formula that considerably extends the well-known one by Karp and Sipser for Erdős-Rényi random graphs.

We considered a natural family of Gibbs distributions over matchings on a finite graph, parameterized by a single positive number called the temperature. The correlation decay technique can be applied for the analysis of matchings at positive temperature and allowed us to establish the weak convergence of the Gibbs marginal as the underlying graph converges locally. However for the zero temperature problem (i.e. maximum matchings), we showed that there is no correlation decay even in very simple cases. By using a complex temperature and a half-plane property due to Heilmann and Lieb, we were able to let the temperature tend to zero and obtained a limit theorem for the asymptotic size of a maximum matching in the graph sequence.

Counting spanning subgraphs subject to local constraints

In [53] , we use negative association and local weak convergence to establish the validity of the cavity method for counting spanning subgraphs subject to local constraints. Specifically, the normalized logarithm of the associated generating polynomial (or partition function) is shown to converge along any sequence of graphs whose random weak limit is a tree, and the limit is directly expressed in terms of the unique solution to a limiting cavity equation. On a Galton-Watson tree, the latter simplifies into a recursive distributional equation which can be solved explicitly. As an illustration, we provide an asymptotic formula for the maximal size of a spanning subgraph with maximal degree $b$ in the Erdős-Rényi model with fixed average degree and diverging size, for any $b\in N$.

Bipartite graph structures for efficient balancing of heterogeneous loads

With Laurent Massoulié [Technicolor], we extend the results obtained previously on the asymptotic size of maximum matchings in random graphs converging locally to Galton-Watson trees to so-called b-matchings (with non-unitary capacity at vertices as well as constraints on individual edges). Compared to the matching case, this involves studying the convergence of a message passing algorithms which transmits vectors instead of single real numbers. We also look further into an application of these results to large scale distributed content service platforms, such as peer-to-peer video-on-demand systems. In this context, the density of maximum b-matchings corresponds to the maximum fraction of simultaneously satisfiable requests, when the service resources are limited and each server can only handle requests for a predetermined subset of the contents which it has stored in memory. An important design aspect of such systems is the content placement strategy onto the servers depending on the estimated content popularities; the results obtained allow to characterize the efficiency of such placement strategies and the optimal strategies in the limit of large storage capacity at servers are determined.

Flooding in Weighted Random Graphs

In a joint work [24] with Moez Draief [Imperial College London], we studied the impact of the edge weights on distances in diluted random graphs. We interpret these weights as delays, and take them as i.i.d exponential random variables. We analyzed the edge flooding time defined as the minimum time needed to reach all nodes from one uniformly chosen node, and the edge diameter corresponding to the worst case edge flooding time. Under some regularity conditions on the degree sequence of the random graph, we showed that these quantities grow as the logarithm of $n$, when the size of the graph $n$ tends to infinity. We also derived the exact value for the prefactors.

These allowed us to analyze an asynchronous randomized broadcast algorithm for random regular graphs. Our results show that the asynchronous version of the algorithm performs better than its synchronized version: in the large size limit of the graph, it will reach the whole network faster even if the local dynamics are similar on average.