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Daniel Valesin - One of the best experts on this subject based on the ideXlab platform.
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Graph Constructions for the Contact Process with a Prescribed Critical Rate
Journal of Theoretical Probability, 2021Co-Authors: Stein Andreas Bethuelsen, Gabriel Baptista Silva, Daniel ValesinAbstract:We construct graphs (trees of bounded degree) on which the Contact Process has critical rate (which will be the same for both global and local survival) equal to any prescribed value between zero and $$\lambda _c({\mathbb {Z}})$$ λ c ( Z ) , the critical rate of the one-dimensional Contact Process. We exhibit both graphs in which the Process at this target critical value survives (locally) and graphs where it dies out (globally).
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The asymmetric multitype Contact Process
Stochastic Processes and their Applications, 2019Co-Authors: Thomas Mountford, Pedro Luis Barrios Pantoja, Daniel ValesinAbstract:Abstract We study the multitype Contact Process on Z d under the assumption that one of the types has a birth rate that is larger than that of the other type, and larger than the critical value of the standard Contact Process. We prove that, if initially present, the stronger type has a positive probability of never going extinct. Conditionally on this event, it takes over a ball of radius growing linearly in time. We also completely characterize the set of stationary distributions of the Process and prove a complete convergence theorem.
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Exponential rate for the Contact Process extinction time
2018Co-Authors: Bruno Schapira, Daniel ValesinAbstract:We consider the extinction time of the Contact Process on increasing sequences of finite graphs obtained from a variety of random graph models. Under the assumption that the infection rate is above the critical value for the Process on the integer line, in each case we prove that the logarithm of the extinction time divided by the size of the graph converges in probability to a (model-dependent) positive constant. The graphs we treat include various percolation models on increasing boxes of Z d or R d in their supercritical or percolative regimes (Bernoulli bond and site percolation, the occupied and vacant sets of random interlacements, excursion sets of the Gaussian free field, random geometric graphs) as well as supercritical Galton-Watson trees grown up to finite generations.
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The asymmetric multitype Contact Process
arXiv: Probability, 2018Co-Authors: Thomas Mountford, Pedro Luis Barrios Pantoja, Daniel ValesinAbstract:In the multitype Contact Process, vertices of a graph can be empty or occupied by a type 1 or a type 2 individual; an individual of type $i$ dies with rate 1 and sends a descendant to a neighboring empty site with rate $\lambda_i$. We study this Process on $\Z^d$ with $\lambda_1 > \lambda_2$ and $\lambda_1$ larger than the critical value of the (one-type) Contact Process. We prove that, if there is at least one type 1 individual in the initial configuration, then type 1 has a positive probability of never going extinct. Conditionally on this event, type 1 takes over a ball of radius growing linearly in time. We also completely characterize the set of stationary distributions of the Process and prove that the Process started from any initial configuration converges to a convex combination of distributions in this set.
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Uniformity of the hitting time of the Contact Process
Latin American Journal of Probability and Mathematical Statistics, 2018Co-Authors: Markus Heydenreich, Christian Hirsch, Daniel ValesinAbstract:For the supercritical Contact Process on the hyper-cubic lattice started from a single infection at the origin and conditioned on survival, we establish two uniformity results for the hitting times t(x), defined for each site x as the first time at which it becomes infected. First, the family of random variables (t(x) - t(y))/vertical bar x-y vertical bar, indexed by x not equal y in Z(d) , is stochastically tight. Second, for each epsilon > 0 there exists x such that, for infinitely many integers n, t(nx) < t((n + 1)x) with probability larger than 1-epsilon. A key ingredient in our proofs is a tightness result concerning the essential hitting times of the supercritical Contact Process introduced by Garet and Marchanc (2012).
Stein Andreas Bethuelsen - One of the best experts on this subject based on the ideXlab platform.
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Graph Constructions for the Contact Process with a Prescribed Critical Rate
Journal of Theoretical Probability, 2021Co-Authors: Stein Andreas Bethuelsen, Gabriel Baptista Silva, Daniel ValesinAbstract:We construct graphs (trees of bounded degree) on which the Contact Process has critical rate (which will be the same for both global and local survival) equal to any prescribed value between zero and $$\lambda _c({\mathbb {Z}})$$ λ c ( Z ) , the critical rate of the one-dimensional Contact Process. We exhibit both graphs in which the Process at this target critical value survives (locally) and graphs where it dies out (globally).
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Stochastic domination in space‐time for the Contact Process
Random Structures & Algorithms, 2018Co-Authors: Jacob Van Den Berg, Stein Andreas BethuelsenAbstract:Liggett and Steif (2006) proved that, for the supercritical Contact Process on certain graphs, the upper invariant measure stochastically dominates an i.i.d.\ Bernoulli product measure. In particular, they proved this for $\mathbb{Z}^d$ and (for infection rate sufficiently large) $d$-ary homogeneous trees $T_d$. In this paper we prove some space-time versions of their results. We do this by combining their methods with specific properties of the Contact Process and general correlation inequalities. One of our main results concerns the Contact Process on $T_d$ with $d\geq2$. We show that, for large infection rate, there exists a subset $\Delta$ of the vertices of $T_d$, containing a "positive fraction" of all the vertices of $T_d$, such that the following holds: The Contact Process on $T_d$ observed on $\Delta$ stochastically dominates an independent spin-flip Process. (This is known to be false for the Contact Process on graphs having subexponential growth.) We further prove that the supercritical Contact Process on $\mathbb{Z}^d$ observed on certain $d$-dimensional space-time slabs stochastically dominates an i.i.d.\ Bernoulli product measure, from which we conclude strong mixing properties important in the study of certain random walks in random environment.
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Random walks and the Contact Process
2016Co-Authors: Stein Andreas BethuelsenAbstract:This thesis concerns the mathematical analysis of certain random walks in a dynamic random environment. Such models are important in the understanding of various models in physics, chemistry and biology. The interest is in questions such as how to determine the average velocity of the random walker and how to control fluctuations and deviations thereof. This is in general a very challenging problem due to the possibility of strong dependence both in space and time, and many problems are still wide open. After a general introduction in Chapter 1, we present several approaches for determining the asymptotic behaviour for random walks in a dynamic random environment in Chapter 2-5 of this thesis. Our work improves on the existing literature for general models with strongly mixing dynamics and provides new insight for certain models with poorly mixing dynamics. One particular model is analysed in more detailed, namely the so-called Contact Process. This model is a prototype of a dynamic random environment with poor mixing properties. In addition to results for certain random walks with the Contact Process as dynamic random environment, we also provide new insight for the Contact Process itself, given in Chapter 5.
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the Contact Process as seen from a random walk
arXiv: Probability, 2016Co-Authors: Stein Andreas BethuelsenAbstract:We consider a random walk on top of the Contact Process on $\mathbb{Z}^d$ with $d\geq 1$. In particular, we focus on the "Contact Process as seen from the random walk". Under the assumption that the infection rate of the Contact Process is large or the jump rate of the random walk is small, we show that this Process has at most two extremal measures. Moreover, the convergence to these extremal measures is characterised by whether the Contact Process survives or dies out, similar to the complete convergence theorem known for the ordinary Contact Process. Using this, we furthermore provide a law of large numbers for the random walk which holds under general assumptions on the jump probabilities of the random walk.
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Stochastic Domination in Space-Time for the Contact Process
arXiv: Probability, 2016Co-Authors: Jacob Van Den Berg, Stein Andreas BethuelsenAbstract:Liggett and Steif (2006) proved that, for the supercritical Contact Process on certain graphs, the upper invariant measure stochastically dominates an i.i.d.\ Bernoulli product measure. In particular, they proved this for $\mathbb{Z}^d$ and (for infection rate sufficiently large) $d$-ary homogeneous trees $T_d$. In this paper we prove some space-time versions of their results. We do this by combining their methods with specific properties of the Contact Process and general correlation inequalities. One of our main results concerns the Contact Process on $T_d$ with $d\geq2$. We show that, for large infection rate, there exists a subset $\Delta$ of the vertices of $T_d$, containing a "positive fraction" of all the vertices of $T_d$, such that the following holds: The Contact Process on $T_d$ observed on $\Delta$ stochastically dominates an independent spin-flip Process. (This is known to be false for the Contact Process on graphs having subexponential growth.) We further prove that the supercritical Contact Process on $\mathbb{Z}^d$ observed on certain $d$-dimensional space-time slabs stochastically dominates an i.i.d.\ Bernoulli product measure, from which we conclude strong mixing properties important in the study of certain random walks in random environment.
Paul Jung - One of the best experts on this subject based on the ideXlab platform.
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Two phase transitions for the Contact Process on small worlds
Stochastic Processes and their Applications, 2007Co-Authors: Rick Durrett, Paul JungAbstract:Abstract In our version of Watts and Strogatz’s small world model, space is a d -dimensional torus in which each individual has in addition exactly one long-range neighbor chosen at random from the grid. This modification is natural if one thinks of a town where an individual’s interactions at school, at work, or in social situations introduce long-range connections. However, this change dramatically alters the behavior of the Contact Process, producing two phase transitions. We establish this by relating the small world to an infinite “big world” graph where the Contact Process behavior is similar to the Contact Process on a tree. We then consider the Contact Process on a slightly modified small world model in order to show that its behavior is decidedly different from that of the Contact Process on a tree.
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Two Phase Transitions for the Contact Process on Small Worlds
arXiv: Probability, 2005Co-Authors: Rick Durrett, Paul JungAbstract:In our version of Watts and Strogatz's small world model, space is a d-dimensional torus in which each individual has in addition exactly one long-range neighbor chosen at random from the grid. This modification is natural if one thinks of a town where an individual's interactions at school, at work, or in social situations introduces long-range connections. However, this change dramatically alters the behavior of the Contact Process, producing two phase transitions. We establish this by relating the small world to an infinite "big world" graph where the Contact Process behavior is similar to the Contact Process on a tree.
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The Critical Value of the Contact Process with Added and Removed Edges
Journal of Theoretical Probability, 2005Co-Authors: Paul JungAbstract:We show that the critical value for the Contact Process on a vertex-transitive graph $$\mathcal{G}$$ with finitely many edges added and/or removed is the same as the critical value for the Contact Process on $$\mathcal{G}$$ . This gives a partial answer to a conjecture of Pemantle and Stacey.
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The Critical Value of the Contact Process with Added and Removed Edges
arXiv: Probability, 2004Co-Authors: Paul JungAbstract:We show that the critical value for the Contact Process on a vertex-transitive graph G with finitely many edges added and/or removed is the same as the critical value for the Contact Process on G.
Jose M. F. Moura - One of the best experts on this subject based on the ideXlab platform.
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Contact Process with exogenous infection and the scaled SIS Process
Journal of Complex Networks, 2017Co-Authors: June Zhang, Jose M. F. MouraAbstract:Abstract Propagation of contagion in networks depends on the graph topology. This article is concerned with studying the time-asymptotic behaviour of the extended Contact Processes on static, undirected, finite-size networks. This is a Contact Process with nonzero exogenous infection rate (also known as the $\epsilon$-susceptible-infected-susceptible model). The only known analytical characterization of the equilibrium distribution of this Process is for complete networks. For large networks with arbitrary topology, it is infeasible to numerically solve for the equilibrium distribution since it requires solving the eigenvalue-eigenvector problem of a matrix that is exponential in $N$, the size of the network. We derive a condition on the infection rates under which, depending on the degree distribution of the network, the equilibrium distribution of extended Contact Processes on arbitrary, finite-size networks is well approximated by a closed-form formulation. We confirm the goodness of the approximation with small networks answering inference questions like the distribution of the percentage of infected individuals and the most-probable equilibrium configuration. We then use the approximation to analyse the equilibrium distribution of the extended Contact Process on the 4941-node US Western power grid.
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Contact Process with Exogenous Infection and the Scaled SIS Process
arXiv: Physics and Society, 2015Co-Authors: June Zhang, Jose M. F. MouraAbstract:Propagation of contagion in networks depends on the graph topology. This paper is concerned with studying the time-asymptotic behavior of the extended Contact Processes on static, undirected, finite-size networks. This is a Contact Process with nonzero exogenous infection rate (also known as the {\epsilon}-SIS, {\epsilon} susceptible-infected-susceptible, model [1]). The only known analytical characterization of the equilibrium distribution of this Process is for complete networks. For large networks with arbitrary topology, it is infeasible to numerically solve for the equilibrium distribution since it requires solving the eigenvalue-eigenvector problem of a matrix that is exponential in N , the size of the network. We show that, for a certain range of the network Process parameters, the equilibrium distribution of the extended Contact Process on arbitrary, finite-size networks is well approximated by the equilibrium distribution of the scaled SIS Process, which we derived in closed-form in prior work. We confirm this result with numerical simulations comparing the equilibrium distribution of the extended Contact Process with that of a scaled SIS Process. We use this approximation to decide, in polynomial-time, which agents and network substructures are more susceptible to infection by the extended Contact Process.
Mário J. De Oliveira - One of the best experts on this subject based on the ideXlab platform.
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Critical properties of the Contact Process with quenched dilution
Journal of Statistical Mechanics: Theory and Experiment, 2017Co-Authors: Alexander H. O. Wada, Mário J. De OliveiraAbstract:We have studied the critical properties of the Contact Process on a square lattice with quenched site dilution by Monte Carlo simulations. This was achieved by generating in advance the percolating cluster, through the use of an appropriate epidemic model, and then by the simulation of the Contact Process on the top of the percolating cluster. The dynamic critical exponents were calculated by assuming an activated scaling relation and the static exponents by the usual power law behavior. Our results are in agreement with the prediction that the quenched diluted Contact Process belongs to the universality class of the random transverse-field Ising model. We have also analyzed the model and determined the phase diagram by the use of a mean-field theory that takes into account the correlation between neighboring sites.
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Critical discontinuous phase transition in the threshold Contact Process
Journal of Physics A: Mathematical and Theoretical, 2011Co-Authors: Evandro F Da Silva, Mário J. De OliveiraAbstract:We analyze a threshold Contact Process on a square lattice in which particles are created on empty sites with at least two neighboring particles and are annihilated spontaneously. We show by means of Monte Carlo simulations that the Process undergoes a discontinuous phase transition at a definite value of the annihilation parameter, in accordance with the Gibbs phase rule, and that the discontinuous transition exhibits critical behavior. The simulations were performed by using boundary conditions in which the sites of the border of the lattice are permanently occupied by particles.
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Conserved Contact Process in one to five dimensions.
Physical Review E, 2002Co-Authors: Munir M. S. Sabag, Mário J. De OliveiraAbstract:We analyze the conserved Contact Process in hypercubic lattices with dimensions ranging from one to five. In this Process particles jump around, falling down only on empty sites beside an existing particle. The model is a version of the ordinary Contact Process with a strictly conserved particle number and can be seen as the Contact Process in an ensemble of fixed particle number. By means of numerical simulations we determine the critical point, the critical exponent beta, and the fractal dimension d(F) at the critical point. In the case of just two particles, the stationary state is obtained exactly in any dimension.
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Nonequilibrium model for the Contact Process in an ensemble of constant particle number.
Physical review letters, 2001Co-Authors: Tânia Tomé, Mário J. De OliveiraAbstract:We introduce and analyze numerically a nonequilibrium model with a conserved dynamics which is a realization of the Contact Process in an ensemble of constant particle number. The model possesses just one Process in which particles jump around landing only on empty sites next to an existing particle. Particles are not allowed to land on a vacant site surrounded by empty sites. In contrast with the ordinary Contact Process, the present model does not have an absorbing state. In spite of lacking an absorbing state, the model displays properties that, in the thermodynamic limit, are identical to those of the ordinary Contact Process.