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# Résultats de recherche

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## Entropic inequalities for unimodular networks (workshop ERC Nemo Processus ponctuels et graphes aléatoires unimodulaires

Voir la vidéole (55m6s)

## Comments and problems regarding large graphs. (workshop ERC Nemo Processus ponctuels et graphes aléatoires unimodulaires)

We will discuss a couple of results and questions regarding the structure of large graphs. These include vertex transitive graphs, expanders and random graphs. Voir la vidéole (50m57s)

## Point processes, cost and the growth of rank for locally compact groups (workshop ERC Nemo Processus ponctuels et graphes aléatoires unimodulaires)

The cost of a vertex transitive graph is the infimum of the expected degree of an invariant random wiring of the graph. Similarly, one can define the cost of a point process on a homogeneous space, as the infimum of the average degree of a factor wiring on its points. It turns out that the cost of a Poisson process is maximal among point processes of the same density, by proving that all free processes weakly contain the Poisson. The cost is related to the growth ... Voir la vidéole (43m18s)

## Spectral embedding for graph classification (workshop ERC Nemo Processus ponctuels et graphes aléatoires unimodulaires)

Learning on graphs requires a graph feature representation able to discriminate among different graphs while being amenable to fast computation. The graph isomorphism problem tells us that no fast representation of graphs is known if we require the representation to be both invariant to nodes permutation and able to discriminate two non-isomorphic graphs. Most graph representations explored so far require to be invariant. We explore new graph representations by relaxing this constraint. We present a generic embedding of graphs relying on spectral graph theory ... Voir la vidéole (39m53s)

## Emergence of extended states at zero in the spectrum of sparse random graphs (workshop ERC Nemo Processus ponctuels et graphes aléatoires unimodulaires)

We confirm the long-standing prediction that c=e≈2.718 is the threshold for the emergence of a non-vanishing absolutely continuous part (extended states) at zero in the limiting spectrum of the Erdős-Renyi random graph with average degree c. This is achieved by a detailed second-order analysis of the resolvent (A−z)−1 near the singular point z=0, where A is the adjacency operator of the Poisson-Galton-Watson tree with mean offspring c. More generally, our method applies to arbitrary unimodular Galton-Watson trees, yielding explicit criteria for the presence or ... Voir la vidéole (55m56s)

## A notion of entropy for limits of sparse marked graphs (workshop ERC Nemo Processus ponctuels et graphes aléatoires unimodulaires)

Bordenave and Caputo (2014) defined a notion of entropy for probability distributions on rooted graphs with finite expected degree at the root. When such a probability distribution \rho has finite BC entropy \Sigma(\rho), the growth in the number of vertices n of the number of graphs on n vertices whose associated rooted graph distribution is close to \rho is as d/2 n \log n + \Sigma(\rho) n + o(n), where d is expected degree of the root under \rho. We develop the parallel result for probability distributions on marked rooted graphs. Our graphs have vertex marks drawn from a finite ... Voir la vidéole (59m49s)

## Sampling cluster point processes: a review (workshop ERC Nemo Processus ponctuels et graphes aléatoires unimodulaires)

The theme of this talk is the sampling of cluster and iterated cluster point processes. It is partially a review, mainly of the Brix–Kendall exact sampling method for cluster point processes and its adaptation by Moller and Rasmussen to Hawkes branching point processes on the real line with light-tail fertility rate. A formal proof via Laplace transforms of the validity of the method in terms of general clusters that are not necessarily point processes fits this purpose and allows to include the exact sampling of Boolean ... Voir la vidéole (52m52s)

## Absence of percolation for Poisson outdegree-one graphs (workshop ERC Nemo Processus ponctuels et graphes aléatoires unimodulaires)

A Poisson outdegree-one graph is a directed graph based on a marked Poisson point process such that each vertex has only one outgoing edge. We state the absence of percolation for such graphs satisfying two assumptions. The Shield assumption roughly says that the graph is locally determined with possible random horizons. The Loop assumption ensures that any forward branch merges on a loop provided that the Poisson point process is augmented with a finite collection of well-chosen points. This result allows to solve a ... Voir la vidéole (53m51s)