Eigenvalues of Euclidean Random Matrices

We study the spectral measure of large Euclidean random matrices. The entries of these matrices are determined by the relative position of $n$ random points in a compact set $\Omega_n$ of $\R^d$. Under various assumptions we establish the almost sure convergence of the limiting spectral measure as the number of points goes to infinity. The moments of the limiting distribution are computed, and we prove that the limit of this limiting distribution as the density of points goes to infinity has a nice expression. We apply our results to the adjacency matrix of the geometric graph.Comment: 16 pages, 1 figur

A contact process with mutations on a tree

Consider the following stochastic model for immune response. Each pathogen gives birth to a new pathogen at rate $\lambda$. When a new pathogen is born, it has the same type as its parent with probability $1 - r$. With probability $r$, a mutation occurs, and the new pathogen has a different type from all previously observed pathogens. When a new type appears in the population, it survives for an exponential amount of time with mean 1, independently of all the other types. All pathogens of that type are killed simultaneously. Schinazi and Schweinsberg (2006) have shown that this model on $\Z^d$ behaves rather differently from its non-spatial version. In this paper, we show that this model on a homogeneous tree captures features from both the non-spatial version and the $\Z^d$ version. We also obtain comparison results between this model and the basic contact process on general graphs

SIRS Epidemics on Complex Networks: Concurrence of Exact Markov Chain and Approximated Models

We study the SIRS (Susceptible-Infected-Recovered-Susceptible) spreading processes over complex networks, by considering its exact $3^n$-state Markov chain model. The Markov chain model exhibits an interesting connection with its $2n$-state nonlinear "mean-field" approximation and the latter's corresponding linear approximation. We show that under the specific threshold where the disease-free state is a globally stable fixed point of both the linear and nonlinear models, the exact underlying Markov chain has an $O(\log n)$ mixing time, which means the epidemic dies out quickly. In fact, the epidemic eradication condition coincides for all the three models. Furthermore, when the threshold condition is violated, which indicates that the linear model is not stable, we show that there exists a unique second fixed point for the nonlinear model, which corresponds to the endemic state. We also investigate the effect of adding immunization to the SIRS epidemics by introducing two different models, depending on the efficacy of the vaccine. Our results indicate that immunization improves the threshold of epidemic eradication. Furthermore, the common threshold for fast-mixing of the Markov chain and global stability of the disease-free fixed point improves by the same factor for the vaccination-dominant model.Comment: A short version of this paper has been submitted to CDC 201

Inondation dans les réseaux dynamiques

International audienceCette note résume nos travaux sur l'inondation dans les réseaux dynamiques. Ces derniers sont définis à partir d'un processus Markovien de paramètres $p$ et $q$ générant des séquences de graphes $(G_0,G_1,G_2,\ldots)$ sur un même ensemble $[n]$ de sommets, et tels que $G_t$ est obtenu à partir de $G_{t-1}$ comme suit~: si $e\notin E(G_{t-1})$ alors $e\in E(G_{t})$ avec probabilité $p$, et si $e\in E(G_{t-1})$ alors $e\notin E(G_{t})$ avec probabilité $q$. Clementi et al. (PODC 2008) ont analysé différent processus de diffusion de l'information dans de tels réseaux, et ont en particulier établi un ensemble de bornes sur les performances de l'inondation. L'inondation consiste en un protocole élémentaire où chaque n{\oe}ud apprenant une information à un temps $t$ la retransmet à tous ses voisins à toutes les étapes suivantes. Evidemment, en dépit de ses avantages en terme de simplicité et de robustesse, le protocole d'inondation souffre d'une utilisation abusive des ressources en bande passante. Dans cette note, nous montrons que l'inondation dans les réseaux dynamiques peut être mis en {\oe}uvre de façon à limiter le nombre de retransmissions d'une même information, tout en préservant les performances en termes du temps mis par une information pour atteindre tous les n{\oe}uds du réseau. La principale difficulté de notre étude réside dans les dépendances temporelles entre les connexions du réseaux à différentes étapes de temps