Évènements

This talk will give an overview of how random graphs can be used to model complex networks as well as processes that run on them. The main motivating examples are the analysis of network centrality and community structure on directed networks, such as the web, and the evolution of opinions on a social network. The main focus of the talk will be on the various types of random graph models that are commonly used, including both static and dynamic ones. I will discuss the notion of “sparsity” and explain how it impacts both the analysis and the type of results that one can obtain. For sparse graphs I will describe the notion of “local weak convergence” and explain how it can be used to obtain tractable approximations for stochastic processes on random graphs that depend only on the network’s statistical properties.

The DeGroot and the Johnsen-Friedkin model are popular models for opinion formation on directed networks. In both of them, individuals update their opinions at each time-step by taking a weighted average of their neighbors’ opinions, either synchronously or asynchronously. Given a graph, the synchronous DeGroot model can be seen as the power iterations of a stochastic matrix, and provided the weight matrix is irreducible and aperiodic, it is known to converge to consensus, i.e., a common value that all individuals agree on. The Johnsen-Friedkin model allows an additional parameter which can be used to represent random external influences, giving rise to an interactive particle system that converges geometrically fast to a stationary distribution. This talk explains how to analyze the limiting behavior of both models when the underlying social network is assumed to be a locally tree-like random graph (e.g., a configuration model, an inhomogeneous random digraph, or a stochastic block model). This analysis allows us to understand the time to stationarity, as well as characterize the limit, in the DeGroot model, or the stationary distribution, in the Friedkin-Johnsen model, in terms of the statistical properties of the underlying random graph.

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We study the large-time $\ell^p$ behavior of transition densities of an isotropic random walk in homogeneous trees, which are infinite, connected, acyclic graphs in which every vertex has the same degree, and can be thought as discrete counterparts of hyperbolic space. Caloric functions of interest are then convolutions of these transition densities with a finitely supported initial condition, and we are interested in their large time behavior in $\ell^p$ norm.

For each $p \in [1, \infty]$, we introduce a notion of a $p$-mass function and prove that caloric functions with compactly supported initial data, asymptotically decouple as the product of this mass function the transition density. Using tools of Fourier analysis available on such graphs, we show that this function even boils down to a constant, still depending on $p$, if the initial condition is radial, that is, depends only on the distance to the origin. Determining the spatial concentration of the densities in $p$-norm plays an important role, in turn clarifying the interplay between the exponential volume growth of the graph and heat diffusion. The results extend to affine buildings, even exotic ones beyond the Bruhat–Tits framework.

Joint work with B. Trojan.

En mécanique quantique, le principe d’incertitude d’Heisenberg stipule qu’on ne peut connaître simultanément avec précision la position et la vitesse d’une particule. Cette célèbre inégalité relie en réalité une fonction et sa transformée de Fourier.
En 1989, motivés par des applications en traitement du signal, Donoho et Stark donnent un nouveau principe d’incertitude, non plus pour des fonctions définies sur $\mathbb{R}$ mais sur un groupe abélien fini. Ce dernier a ensuite été significativement amélioré : en 2006, Tao prouve ce qu’on appelle un principe d’incertitude fort pour des fonctions définies sur $\mathbb{Z}/p\mathbb{Z}$, où $p$ est premier. Plus récemment, en 2021, Garcia, Karaali et Katz généralisent ce principe aux corps finis, pour des fonctions vérifiant une certaine condition de symétrie qu’on détaillera.
Dans cet exposé, on présentera une généralisation du principe d’incertitude fort pour des groupes abéliens finis quelconques. Nous verrons à quel point ce dernier est restrictif, et nous décrirons les cas pour lesquels il est vérifié. Enfin, nous terminerons avec une application en combinatoire additive, plus précisément un théorème de type Cauchy-Davenport sur les corps finis.
Cet exposé est issu d’un travail en collaboration avec Angelot Behajaina.

The discrepancy of a distribution of $N$ points in the torus $T^d$ with respect to a given family of test sets measures how far the points are from being uniformly distributed over that family. When the family consists of all translates of a fixed set, one can consider the $L^2$-average of the discrepancy over translations and use Fourier analytical methods to understand its size. Sharp lower bounds for such $L^2$ discrepancy in terms of $N$ are known for wide classes of sets in $T^2$, but much less is known in higher dimensions. In this talk, I will report on recent progress in this direction, focusing on a family of test
sets with “cylindrical” symmetry that can be defined in any dimension. In three dimensions, these sets have the shape of a barrel. They are particularly
interesting because they exhibit geometric features known to play a key role in discrepancy theory: flat regions, curved regions, and corners. Joint work with
Luca Brandolini and Alessandro Monguzzi.