An official website of the United States government
Bootstrap percolation is a process defined on a graph which begins with an initial set of infected vertices. In each subsequent round, an uninfected vertex becomes infected if it is adjacent to at least r previously infected vertices. If an initially infected set of vertices, A0, begins a process in which every vertex of the graph eventually becomes infected, then we say that A0 percolates. In this paper we investigate bootstrap percolation as it relates to graph distance and connectivity. We find a sufficient condition for the existence of cardinality 2 percolating sets in diameter 2 graphs when r = 2. We also investigate connections between connectivity and bootstrap percolation and lower and upper bounds on the number of rounds to percolation in terms of invariants related to graph distance.
more »
« less
A trust model for bootstrap percolation
This paper is dedicated to the study of the interaction between dynamical systems and percolation models, with views towards the study of viral infections whose virus mutate with time. Recall that r-bootstrap percolation describes a deterministic process where vertices of a graph are infected once r neighbors of it are infected. We generalize this by introducing F(t)-bootstrap percolation, a time-dependent process where the number of neighbouring vertices which need to be infected for a disease to be transmitted is determined by a percolation function F(t) at each time t. After studying some of the basic properties of the model, we consider smallest percolating sets and construct a polynomial-timed algorithm to nd one smallest minimal percolating set on nite trees for certain F(t)-bootstrap percolation models.
more »
« less
- Award ID(s):
- 1749013
- PAR ID:
- 10138123
- Date Published:
- Journal Name:
- Proceedings
- ISSN:
- 1471-2946
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
null (Ed.)We introduce here a multi-type bootstrap percolation model, which we call T -Bootstrap Percolation ( T -BP), and apply it to study information propagation in social networks. In this model, a social network is represented by a graph G whose vertices have different labels corresponding to the type of role the person plays in the network (e.g. a student, an educator etc.). Once an initial set of vertices of G is randomly selected to be carrying a gossip (e.g. to be infected), the gossip propagates to a new vertex provided it is transmitted by a minimum threshold of vertices with different labels. By considering random graphs, which have been shown to closely represent social networks, we study different properties of the T -BP model through numerical simulations, and describe its implications when applied to rumour spread, fake news and marketing strategies.more » « less
-
We consider a dynamical process on a graphG, in which vertices are infected (randomly) at a rate which depends on the number of their neighbors that are already infected. This model includes bootstrap percolation and first‐passage percolation as its extreme points. We give a precise description of the evolution of this process on the graph, significantly sharpening results of Dehghanpour and Schonmann. In particular, we determine the typical infection time up to a constant factor for almost all natural values of the parameters, and in a large range we obtain a stronger, sharp threshold.more » « less
-
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 $$K_{r,s}$$-bootstrap percolation on the Erdős-Rényi random graph, and determine the percolation threshold for balanced $$K_{r,s}$$ up to a logarithmic factor. This partially answers a question raised by Balogh, Bollobás, and Morris. We also establish a general lower bound of the percolation threshold for all $$K_{r,s}$$, with $$r\geq s \geq 3$$.more » « less
-
Abstract We study a stochastic compartmental susceptible–infected (SI) epidemic process on a configuration model random graph with a given degree distribution over a finite time interval. We split the population of graph vertices into two compartments, namely, S and I, denoting susceptible and infected vertices, respectively. In addition to the sizes of these two compartments, we keep track of the counts of SI-edges (those connecting a susceptible and an infected vertex) and SS-edges (those connecting two susceptible vertices). We describe the dynamical process in terms of these counts and present a functional central limit theorem (FCLT) for them as the number of vertices in the random graph grows to infinity. The FCLT asserts that the counts, when appropriately scaled, converge weakly to a continuous Gaussian vector semimartingale process in the space of vector-valued càdlàg functions endowed with the Skorokhod topology. We discuss applications of the FCLT in percolation theory and in modelling the spread of computer viruses. We also provide simulation results illustrating the FCLT for some common degree distributions.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- The DOI is not currently available.
