arXiv subjects

H. Don

Publications and source records attributed to H. Don.

4 recordsLinked to original sources

arXiv2024

The SIS process on Erd\"os-R\'enyi graphs: determining the infected fraction

There are many methods to estimate the quasi-stationary infected fraction of the SIS process on (random) graphs. A challenge is to adequately incorporate correlations, which is especially important in sparse graphs. Methods typically are either significantly biased in sparse graphs, or computationally very demanding already for small network sizes. The former applies to Heterogeneous Mean Field and to the N-intertwined Mean Field Approximation, the latter to most higher order approximations. In this paper we present a new method to determine the infected fraction in sparse graphs, which we test on Erd\H{o}s-R\'enyi graphs. Our method is based on degree-pairs, does take into account correlations and gives accurate estimates. At the same time, computations are very feasible and can easily be done even for large networks.

cond-mat.stat-mech↗

arXiv2023

Random walks on $\mathbb{Z}$ with metastable Gaussian distribution caused by linear drift with application to the contact process on the complete graph

We study random walks on $\mathbb{Z}$ which have a linear (or almost linear) drift towards 0 in a range around 0. This drift leads to a metastable Gaussian distribution centered at zero. We give specific, fast growing, time windows where we can explicitely bound the distance of the distribution of the walk to an appropriate Gaussian. In this way we give a solid theoretical foundation to the notion of metastability. We show that the supercritical contact process on the complete graph has a drift towards its equilibrium point which is locally linear and that our results for random walks apply. This leads to the conclusion that the infected fraction of the population in metastability (when properly scaled) converges in distribution to a Gaussian, uniformly for all times in a fast growing interval.

math.PR↗

arXiv2019

A lower bound for point-to-point connection probabilities in critical percolation

Consider critical site percolation on $\mathbb{Z}^d$ with $d \geq 2$. We prove a lower bound of order $n^{- d^2}$ for point-to-point connection probabilities, where $n$ is the distance between the points. Most of the work in our proof concerns a `construction' which finally reduces the problem to a topological one. This is then solved by applying a topological fact, which follows from Brouwer's fixed point theorem. Our bound improves the lower bound with exponent $2 d (d-1)$, used by Cerf in 2015 to obtain an upper bound for the so-called two-arm probabilities. Apart from being of interest in itself, our result gives a small improvement of the bound on the two-arm exponent found by Cerf.

math.PR↗

arXiv2017

Estimating the covariance structure of heterogeneous SIS epidemics on networks

Heterogeneous Markovian Susceptible-Infected-Susceptible (SIS) epidemics with a general infection rate matrix $\widetilde{A}$ are considered. Using a non-negative matrix factorization to approximate $\widetilde{A}$, we are able to identify when a metastable state can be expected, and that the metastable distribution, under certain conditions, will feature a normal distribution with known expectation and covariance. Furthermore, we model a heterogeneous Markovian SIS epidemic, that starts with a fraction of initially infected nodes different from that in the metastable state, by approximating its behaviour by a standard linear stochastic differential equation (SDE) in sufficiently high dimensions. By exploiting the knowledge of the covariance matrix from the SDE, we demonstrate significant accuracy improvements over the first-order mean-field approximation NIMFA.

math.PR↗