arXiv subjects

Maria Deijfen

Publications and source records attributed to Maria Deijfen.

43 records · Page 3Linked to original sources

arXiv2011

Stable Poisson Graphs in One Dimension

Let each point of a homogeneous Poisson process on $\RR$ independently be equipped with a random number of stubs (half-edges) according to a given probability distribution $μ$ on the positive integers. We consider schemes based on Gale-Shapley stable marriage for perfectly matching the stubs to obtain a simple graph with degree distribution $μ$. We prove results on the existence of an infinite component and on the length of the edges, with focus on the case $μ(\{2\})=1$. In this case, for the random direction stable matching scheme introduced by Deijfen and Meester we prove that there is no infinite component, while for the stable matching of Deijfen, Häggström and Holroyd we prove that existence of an infinite component follows from a certain statement involving a {\em finite} interval, which is overwhelmingly supported by simulation evidence.

math.PR↗

arXiv2011

Scale-free percolation

We formulate and study a model for inhomogeneous long-range percolation on $\Zbold^d$. Each vertex $x\in\Zbold^d$ is assigned a non-negative weight $W_x$, where $(W_x)_{x\in\Zbold^d}$ are i.i.d.\ random variables. Conditionally on the weights, and given two parameters $α,λ>0$, the edges are independent and the probability that there is an edge between $x$ and $y$ is given by $p_{xy}=1-\exp\{-λW_xW_y/|x-y|^α\}$. The parameter $λ$ is the percolation parameter, while $α$ describes the long-range nature of the model. We focus on the degree distribution in the resulting graph, on whether there exists an infinite component and on graph distance between remote pairs of vertices. First, we show that the tail behavior of the degree distribution is related to the tail behavior of the weight distribution. When the tail of the distribution of $W_x$ is regularly varying with exponent $τ-1$, then the tail of the degree distribution is regularly varying with exponent $γ=α(τ-1)/d$. The parameter $γ$ turns out to be crucial for the behavior of the model. Conditions on the weight distribution and $γ$ are formulated for the existence of a critical value $λ_c\in(0,\infty)$ such that the graph contains an infinite component when $λ>λ_c$ and no infinite component when $λ<λ_c$. Furthermore, a phase transition is established for the graph distances between vertices in the infinite component at the point $γ=2$, that is, at the point where the degrees switch from having finite to infinite second moment. The model can be viewed as an interpolation between long-range percolation and models for inhomogeneous random graphs, and we show that the behavior shares the interesting features of both these models.

math.PR↗

arXiv2011

Epidemics and vaccination on weighted graphs

A Reed-Frost epidemic with inhomogeneous infection probabilities on a graph with prescribed degree distribution is studied. Each edge $(u,v)$ in the graph is equipped with two weights $W_{(u,v)}$ and $W_{(v,u)}$ that represent the (subjective) strength of the connection and determine the probability that $u$ infects $v$ in case $u$ is infected and vice versa. Expressions for the epidemic threshold are derived for i.i.d.\ weights and for weights that are functions of the degrees. For i.i.d.\ weights, a variation of the so called acquaintance vaccination strategy is analyzed where vertices are chosen randomly and neighbors of these vertices with large edge weights are vaccinated. This strategy is shown to outperform the strategy where the neighbors are chosen randomly in the sense that the basic reproduction number is smaller for a given vaccination coverage.

math.PR↗

arXiv2010

On the speed of biased random walk in translation invariant percolation

For biased random walk on the infinite cluster in supercritical i.i.d.\ percolation on $\Z^2$, where the bias of the walk is quantified by a parameter $β>1$, it has been conjectured (and partly proved) that there exists a critical value $β_c>1$ such that the walk has positive speed when $β<β_c$ and speed zero when $β>β_c$. In this paper, biased random walk on the infinite cluster of a certain translation invariant percolation process on $\Z^2$ is considered. The example is shown to exhibit the opposite behavior to what is expected for i.i.d.\ percolation, in the sense that it has a critical value $β_c$ such that, for $β<β_c$, the random walk has speed zero, while, for $β>β_c$, the speed is positive. Hence the monotonicity in $β$ that is part of the conjecture for i.i.d.\ percolation cannot be extended to general translation invariant percolation processes.

math.PR↗

arXiv2010

Percolation in invariant Poisson graphs with i.i.d. degrees

Let each point of a homogeneous Poisson process in R^d independently be equipped with a random number of stubs (half-edges) according to a given probability distribution mu on the positive integers. We consider translation-invariant schemes for perfectly matching the stubs to obtain a simple graph with degree distribution mu. Leaving aside degenerate cases, we prove that for any mu there exist schemes that give only finite components as well as schemes that give infinite components. For a particular matching scheme that is a natural extension of Gale-Shapley stable marriage, we give sufficient conditions on mu for the absence and presence of infinite components.

math.PR↗

arXiv2007

The two-type Richardson model with unbounded initial configurations

The two-type Richardson model describes the growth of two competing infections on $\mathbb{Z}^d$ and the main question is whether both infection types can simultaneously grow to occupy infinite parts of $\mathbb{Z}^d$. For bounded initial configurations, this has been thoroughly studied. In this paper, an unbounded initial configuration consisting of points $x=(x_1,...,x_d)$ in the hyperplane $\mathcal{H}=\{x\in\mathbb{Z}^d:x_1=0\}$ is considered. It is shown that, starting from a configuration where all points in $\mathcal{H} {\mathbf{0}\}$ are type 1 infected and the origin $\mathbf{0}$ is type 2 infected, there is a positive probability for the type 2 infection to grow unboundedly if and only if it has a strictly larger intensity than the type 1 infection. If, instead, the initial type 1 infection is restricted to the negative $x_1$-axis, it is shown that the type 2 infection at the origin can also grow unboundedly when the infection types have the same intensity.

math.PR↗

arXiv2007

Epidemics on random graphs with tunable clustering

In this paper, a branching process approximation for the spread of a Reed-Frost epidemic on a network with tunable clustering is derived. The approximation gives rise to expressions for the epidemic threshold and the probability of a large outbreak in the epidemic. It is investigated how these quantities varies with the clustering in the graph and it turns out for instance that, as the clustering increases, the epidemic threshold decreases. The network is modelled by a random intersection graph, in which individuals are independently members of a number of groups and two individuals are linked to each other if and only if they share at least one group.

math.PR↗