Uniform dispersion in growth models on homogeneous trees Valdivino V. Junior, Fábio P. Machado, Alejandro Roldán-Correa Alea, 2025 We consider the dynamics of a population spatially structured in colonies that are vulnerable to catastrophic events occurring at random times, which randomly reduce their population size and compel survivors to disperse to neighboring areas.The dispersion behavior of survivors is critically significant for the survival of the entire species.In this paper, we consider an uniform dispersion scheme, where all possible survivor groupings are equally probable.The aim of the survivors is to establish new colonies, with individuals who settle in empty sites potentially initiating a new colony by themselves.However, all other individuals succumb to the catastrophe.We consider the number of dispersal options for surviving individuals in the aftermath of a catastrophe to be a fixed value d within the neighborhood.In this context, we conceptualize the evolution of population dynamics occurring over a homogeneous tree.We investigate the conditions necessary for these populations to survive, presenting pertinent bounds for survival probability, the number of colonized vertices, the extent of dispersion within the population, and the mean time to extinction for the entire population.
The impact of effective participation in stopping misinformation: an approach based on branching processes Luz Marina Gomez, Valdivino V Junior, Pablo M Rodriguez Journal of Statistical Mechanics Theory and Experiment, 2024 The emergence of research that focuses on understanding the spreading and impact of disinformation is increasing year after year. Most of the time, the purpose of those who start the spreading of intentionally false information that is designed to cause harm is to catalyze its fast transformation into misinformation, which is the false content shared by people who do not realize it is false or misleading. Our interest is in discussing the role of people who decide to adopt an active role in stopping the propagation of information when they realize that it is false. For this, we formulate two simple probabilistic models to compare misinformation spreading in possible scenarios for which there is a passive or an active environment of aware individuals. With aware individuals, we mean those individuals who realize that a piece of given information is false or misleading. In the passive environment, we assume that if one of an aware individual is exposed to the misinformation then he/she will not spread it. In the active environment, we assume that if one of an aware individual is exposed to the misinformation then he/she will not spread it, but also he/she will stop the propagation to other individuals from the individual who contacted him/her. We appeal to the theory of branching processes to analyze propagation in both scenarios, and we discuss the role and the impact of effective participation in stopping misinformation. We show that the propagation reduces drastically, provided we assume an active environment. We also obtain theoretical and computational results to measure such a reduction, which in turn depends on the proportion of aware individuals and the number of potential contacts of each individual, which is assumed to be random.
Extinction time in growth models subject to binomial catastrophes F Duque, V V Junior, F P Machado, A Roldán-Correa Journal of Statistical Mechanics Theory and Experiment, 2023 Populations are often subject to catastrophes that lead to significant reductions in the number of individuals. Many stochastic growth models have been considered to explain such dynamics. Among the reported results, it has been considered whether dispersion strategies, at times of catastrophes, increase the survival probability of the population. In this paper, we contrast dispersion strategies by comparing the mean extinction times of a population under conditions of near-certain extinction. Specifically, we consider populations subject to binomial catastrophes, where the population size is reduced according to a binomial law when a catastrophe occurs. Our findings delineate the optimal strategy (dispersion or non-dispersion) based on variations in model parameter values.
Stochastic rumors on random trees Valdivino V Junior, Pablo M Rodriguez, Adalto Speroto Journal of Statistical Mechanics Theory and Experiment, 2021 The Maki–Thompson rumor model is defined by assuming that a population represented by a graph is subdivided into three classes of individuals; namely, ignorants, spreaders and stiflers. A spreader tells the rumor to any of its nearest ignorant neighbors at rate one. At the same rate, a spreader becomes a stifler after a contact with other nearest neighbor spreaders, or stiflers. In this work we study the model on random trees. As usual we define a critical parameter of the model as the critical value around which the rumor either becomes extinct almost-surely or survives with positive probability. We analyze the existence of phase-transition regarding the survival of the rumor, and we obtain estimates for the mean range of the rumor. The applicability of our results is illustrated with examples on random trees generated from some well-known discrete distributions.
The cone percolation model on galton–watson and on spherically symmetric trees Valdivino V. Junior, Fábio P. Machado, Krishnamurthi Ravishankar Brazilian Journal of Probability and Statistics, 2020 We study a rumor model from a percolation theory and branching process point of view. It is defined according to the following rules: (1) at time zero, only the root (a fixed vertex of the tree) is declared informed, (2) at time $n+1$, an ignorant vertex gets the information if it is, at a graph distance, at most $R_{v}$ of some its ancestral vertex $v$, previously informed. We present relevant lower and upper bounds for the probability of that event, according to the distribution of the random variables that defines the radius of influence of each individual. We work with (homogeneous and non-homogeneous) Galton–Watson branching trees and spherically symmetric trees which includes homogeneous and $k$-periodic trees. We also present bounds for the expected size of the connected component in the subcritical case for homogeneous trees and homogeneous Galton–Watson branching trees.
