The phase transition in the configuration model

Let $G=G(d)$ be a random graph with a given degree sequence $d$, such as a random $r$-regular graph where $r\ge 3$ is fixed and $n=|G|\to\infty$. We study the percolation phase transition on such graphs $G$, i.e., the emergence as $p$ increases of a unique giant component in the random subgraph $G[p]$ obtained by keeping edges independently with probability $p$. More generally, we study the emergence of a giant component in $G(d)$ itself as $d$ varies. We show that a single method can be used to prove very precise results below, inside and above the `scaling window' of the phase transition, matching many of the known results for the much simpler model $G(n,p)$. This method is a natural extension of that used by Bollobas and the author to study $G(n,p)$, itself based on work of Aldous and of Nachmias and Peres; the calculations are significantly more involved in the present setting.Comment: 37 page

Connected Domination Critical Graphs

This thesis investigates the structure of connected domination critical graphs. The characterizations developed provide an important theoretical framework for addressing a number of difficult computational problems in the areas of operations research (for example, facility locations, industrial production systems), security, communication and wireless networks, transportation and logistics networks, land surveying and computational biology. In these application areas, the problems of interest are modelled by networks and graph parameters such as domination numbers reflect the efficiency and performance of the systems

Metastability for the contact process on the configuration model with infinite mean degree

We study the contact process on the configuration model with a power law degree distribution, when the exponent is smaller than or equal to two. We prove that the extinction time grows exponentially fast with the size of the graph and prove two metastability results. First the extinction time divided by its mean converges in distribution toward an exponential random variable with mean one, when the size of the graph tends to infinity. Moreover, the density of infected sites taken at exponential times converges in probability to a constant. This extends previous results in the case of an exponent larger than $2$ obtained in \cite{CD,MMVY,MVY}.Comment: Proposition 6.2 replaced by a weaker version (after a gap in its proof was mentioned to us by Daniel Valesin). Does not affect the two main theorems of the pape