Related papers: Spread of Infection over P.A. random graphs with e…
The dynamic behaviour of stochastic spreading processes on a network model based on k-regular graphs is investigated. The contact process and the susceptible-infected-susceptible model for the spread of epidemics are considered as prototype…
In the $r$-neighbour bootstrap process on a graph $G$, vertices are infected (in each time step) if they have at least $r$ already-infected neighbours. Motivated by its close connections to models from statistical physics, such as the Ising…
We introduce a model of epidemics among moving particles on any locally finite graph. At any time, each vertex is empty, occupied by a healthy particle, or occupied by an infected particle. Infected particles recover at rate $1$ and…
We study preferential attachment mechanisms in random graphs that are parameterized by (i) a constant bias affecting the degree-biased distribution on the vertex set and (ii) the distribution of times at which new vertices are created by…
We study a version of first passage percolation on $\mathbb{Z}^d$ where the random passage times on the edges are replaced by contact times represented by random closed sets on $\mathbb{R}$. Similarly to the contact process without…
The problem of continuum percolation in dispersions of rods is reformulated in terms of weighted random geometric graphs. Nodes (or sites or vertices) in the graph represent spatial locations occupied by the centers of the rods. The…
A graph $G$ percolates in the $K_{r,s}$-bootstrap process if we can add all missing edges of $G$ in some order such that each edge creates a new copy of $K_{r,s}$, where $K_{r,s}$ is the complete bipartite graph. We study…
Edge-based percolation methods can be used to analyze disease transmission on complex social networks. This allows us to include complex social heterogeneity in our models while maintaining tractability. Here we review the seminal works on…
We study the two most common types of percolation process on a sparse random graph with a given degree sequence. Namely, we examine first a bond percolation process where the edges of the graph are retained with probability p and afterwards…
Bootstrap percolation is a cellular automaton modelling the spread of an `infection' on a graph. In this note, we prove a family of lower bounds on the critical probability for $r$-neighbour bootstrap percolation on Galton--Watson trees in…
In this paper we establish a connection between epidemic models on random networks with general infection times considered in Barbour and Reinert 2013 and first passage percolation. Using techniques developed in Bhamidi, van der Hofstad,…
First passage percolation with recovery is a process aimed at modeling the spread of epidemics. On a graph $G$ place a red particle at a reference vertex $o$ and colorless particles (seeds) at all other vertices. The red particle starts…
We consider the following activation process in undirected graphs: a vertex is active either if it belongs to a set of initially activated vertices or if at some point it has at least $r$ active neighbors. A \emph{contagious set} is a set…
In this paper we study the interplay between epidemic spreading and risk perception on multiplex networks. The basic idea is that the effective infection probability is affected by the perception of the risk of being infected, which we…
We study the SIR epidemic model with infections carried by $k$ particles making independent random walks on a random regular graph. Here we assume $k\leq n^{\epsilon}$, where $n$ is the number of vertices in the random graph, and $\epsilon$…
In this paper we focus on the problem of the degree sequence for the following random graph process. At any time-step $t$, one of the following three substeps is executed: with probability $\alpha_1$, a new vertex $x_t$ and $m$ edges…
One-dependent first passage percolation is a spreading process on a graph where the transmission time through each edge depends on the direct surroundings of the edge. In particular, the classical iid transmission time $L_{xy}$ is…
A version of ``preferential attachment'' random graphs, corresponding to linear ``weights'' with random ``edge additions,'' which generalizes some previously considered models, is studied. This graph model is embedded in a continuous-time…
The paper is devoted to the spreading of a message within the random graph evolving by the Norros-Reittu preferential attachment model. The latter model forms random Poissonian numbers of edges between newly added nodes and existing ones.…
Random walk is one of the basic mechanisms found in many network applications. We study the epidemic spreading dynamics driven by biased random walks on complex networks. In our epidemic model, each time infected nodes constantly spread…