Related papers: Bootstrap percolation in inhomogeneous random grap…
One model of real-life spreading processes is First Passage Percolation (also called SI model) on random graphs. Social interactions often follow bursty patterns, which are usually modelled with i.i.d.~heavy-tailed passage times on edges.…
Human to human transmissible infectious diseases spread in a population using human interactions as its transmission vector. The early stages of such an outbreak can be modeled by a graph whose edges encode these interactions between…
In $H$-percolation, we start with an Erd\H{o}s--R\'enyi graph ${\mathcal G}_{n,p}$ and then iteratively add edges that complete copies of $H$. The process percolates if all edges missing from ${\mathcal G}_{n,p}$ are eventually added. We…
The $r$-neighbour bootstrap process on a graph $G$ begins with a set of infected vertices; subsequently, healthy vertices become infected once they have at least $r$ infected neighbours. The central extremal problem in bootstrap percolation…
We consider first passage percolation on sparse random graphs with prescribed degree distributions and general independent and identically distributed edge weights assumed to have a density. Assuming that the degree distribution satisfies a…
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…
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…
Let $\mathbb{G}=\left(\mathbb{V},\mathbb{E}\right)$ be the graph obtained by taking the cartesian product of an infinite and connected graph $G=(V,E)$ and the set of integers $\mathbb{Z}$. We choose a collection $\mathcal{C}$ of finite…
We consider a class of random, weighted networks, obtained through a redefinition of patterns in an Hopfield-like model and, by performing percolation processes, we get information about topology and resilience properties of the networks…
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 $\epsilon$…
We study the process of bootstrap percolation on n x n permutation matrices, inspired by the work of Shapiro and Stephens [5]. In this percolation model, cells mutate (from 0 to 1) if at least two of their cardinal neighbors contain a 1,…
We study a random graph model which is a superposition of the bond percolation model on $Z^d$ with probability $p$ of an edge, and a classical random graph $G(n, c/n)$. We show that this model, being a {\it homogeneous} random graph, has a…
We provide sufficient conditions for a regular graph $G$ of growing degree $d$, guaranteeing a phase transition in its random subgraph $G_p$ similar to that of $G(n,p)$ when $p\cdot d\approx 1$. These conditions capture several well-studied…
For $k$-graphs $F$ and $H_0$ the $F$-bootstrap percolation process (or $F$-process) starting with $H_0$ is a sequence $(H_i)_{i\geq0}$ of $k$-graphs such that $H_{i+1}$ is obtained from $H_i$ by adding all those $e\in V(H_0)^{(k)}\setminus…
We introduce a process where a connected rooted multigraph evolves by splitting events on its vertices, occurring randomly in continuous time. When a vertex splits, its incoming edges are randomly assigned between its offspring and a…
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…
In this paper, we study a model of long-range site percolation on graphs of bounded degree, namely the Boolean percolation model. In this model, each vertex of an infinite connected graph is the center of a ball of random radius, and…
Recent experimental studies of living neural networks reveal that their global activation induced by electrical stimulation can be explained using the concept of bootstrap percolation on a directed random network. The experiment consists in…
We address the question of understanding the effect of the underlying network topology on the spread of a virus and the dissemination of information when users are mobile performing independent random walks on a graph. To this end we…
K-core and bootstrap percolation are widely studied models that have been used to represent and understand diverse deactivation and activation processes in natural and social systems. Since these models are considerably similar, it has been…