arXiv subjects

Pablo M. Rodriguez

Publications and source records attributed to Pablo M. Rodriguez.

25 records · Page 2Linked to original sources

arXiv2018

Comment on "Nodal infection in Markovian susceptible-infected-susceptible and susceptible-infected-removed epidemics on networks are non-negatively correlated''

Cator and Van Mieghem [Cator E, Van Mieghem P., Phys. Rev. E 89, 052802 (2014)] stated that the correlation of infection at the same time between any pair of nodes in a network is non-negative for the Markovian SIS and SIR epidemic models. The arguments used to obtain this result rely strongly on the graphical construction of the stochastic process, as well as the FKG inequality. In this note we show that although the approach used by the authors applies to the SIS model, it cannot be used for the SIR model as stated in their work. In particular, we observe that monotonicity in the process is crucial for invoking the FKG inequality. Moreover, we provide an example of simple graph for which the nodal infection in the SIR Markovian model is negatively correlated.

physics.soc-ph↗

arXiv2018

On the existence of accessibility in a tree-indexed percolation model

We study the accessibility percolation model on infinite trees. The model is defined by associating an absolute continuous random variable $X_v$ to each vertex $v$ of the tree. The main question to be considered is the existence or not of an infinite path of nearest neighbors $v_1,v_2,v_3\ldots$ such that $X_{v_1} 0$ is a given constant. We show that there is a percolation threshold at $α_c =1$ such that there is percolation if $α> 1$ and there is absence of percolation if $α\leq 1$. Moreover, we study the event of percolation starting at any vertex, as well as the continuity of the percolation probability function. Finally, we provide a comparison between this model with the well known $F^α$ record model. We also discuss a number of open problems concerning the accessibility percolation model for further consideration in future research.

math.PR↗

arXiv2017

Galton-Watson processes in varying environment and accessibility percolation

This paper deals with branching processes in varying environment, namely, whose offspring distributions depend on the generations. We provide sufficient conditions for survival or extinction which rely only on the first and second moments of the offspring distributions. These results are then applied to branching processes in varying environment with selection where every particle has a real-valued label and labels can only increase along genealogical lineages; we obtain analogous conditions for survival or extinction. These last results can be interpreted in terms of accessibility percolation on Galton-Watson trees, which represents a relevant tool for modeling the evolution of biological populations.

math.PR↗

arXiv2017

Phase transition for the Maki-Thompson rumour model on a small-world network

We consider the Maki-Thompson model for the stochastic propagation of a rumour within a population. We extend the original hypothesis of homogenously mixed population by allowing for a small-world network embedding the model. This structure is realized starting from a $k$-regular ring and by inserting, in the average, $c$ additional links in such a way that $k$ and $c$ are tuneable parameter for the population architecture. We prove that this system exhibits a transition between regimes of localization (where the final number of stiflers is at most logarithmic in the population size) and propagation (where the final number of stiflers grows algebraically with the population size) at a finite value of the network parameter $c$. A quantitative estimate for the critical value of $c$ is obtained via extensive numerical simulations.

math.PR↗

arXiv2016

Evolution of a modified binomial random graph by agglomeration

In the classical Erdös-Rényi random graph G(n,p) there are n vertices and each of the possible edges is independently present with probability p. The random graph G(n,p) is homogeneous in the sense that all vertices have the same characteristics. On the other hand, numerous real-world networks are inhomogeneous in this respect. Such an inhomogeneity of vertices may influence the connection probability between pairs of vertices. The purpose of this paper is to propose a new inhomogeneous random graph model which is obtained in a constructive way from the Erdös-Rényi random graph G(n,p). Given a configuration of n vertices arranged in N subsets of vertices (we call each subset a super-vertex), we define a random graph with N super-vertices by letting two super-vertices be connected if and only if there is at least one edge between them in G(n,p). Our main result concerns the threshold for connectedness. We also analyze the phase transition for the emergence of the giant component and the degree distribution. Even though our model begins with G(n,p), it assumes the existence of some community structure encoded in the configuration. Furthermore, under certain conditions it exhibits a power law degree distribution. Both properties are important for real applications.

math.CO↗

arXiv2016

A stochastic two-stage innovation diffusion model on a lattice

We propose a stochastic model describing a process of awareness, evaluation and decision-making by agents on the d-dimensional integer lattice. Each agent may be in any of the three states belonging to the set {0, 1, 2}. In this model 0 stands for ignorants, 1 for aware and 2 for adopters. Aware and adopters inform its nearest ignorant neighbors about a new product innovation at rate lambda. At rate alpha an agent in aware state becomes an adopter due to the influence of adopters neighbors. Finally, aware and adopters forget the information about the new product, thus becoming ignorant, at rate one. Our purpose is to analyze the influence of the parameters on the qualitative behavior of the process. We obtain sufficient conditions under which the innovation diffusion (and adoption) either becomes extinct or propagates through the population with positive probability.

math.PR↗

arXiv2012

A spatial stochastic model for rumor transmission

We consider an interacting particle system representing the spread of a rumor by agents on the $d$-dimensional integer lattice. Each agent may be in any of the three states belonging to the set {0,1,2}. Here 0 stands for ignorants, 1 for spreaders and 2 for stiflers. A spreader tells the rumor to any of its (nearest) ignorant neighbors at rate λ. At rate αa spreader becomes a stifler due to the action of other (nearest neighbor) spreaders. Finally, spreaders and stiflers forget the rumor at rate one. We study sufficient conditions under which the rumor either becomes extinct or survives with positive probability.

math.PR↗