5,497 research outputs found

    On the Convergence of Population Protocols When Population Goes to Infinity

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    Population protocols have been introduced as a model of sensor networks consisting of very limited mobile agents with no control over their own movement. A population protocol corresponds to a collection of anonymous agents, modeled by finite automata, that interact with one another to carry out computations, by updating their states, using some rules. Their computational power has been investigated under several hypotheses but always when restricted to finite size populations. In particular, predicates stably computable in the original model have been characterized as those definable in Presburger arithmetic. We study mathematically the convergence of population protocols when the size of the population goes to infinity. We do so by giving general results, that we illustrate through the example of a particular population protocol for which we even obtain an asymptotic development. This example shows in particular that these protocols seem to have a rather different computational power when a huge population hypothesis is considered.Comment: Submitted to Applied Mathematics and Computation. 200

    On the algebraic numbers computable by some generalized Ehrenfest urns

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    This article deals with some stochastic population protocols, motivated by theoretical aspects of distributed computing. We modelize the problem by a large urn of black and white balls from which at every time unit a fixed number of balls are drawn and their colors are changed according to the number of black balls among them. When the time and the number of balls both tend to infinity the proportion of black balls converges to an algebraic number. We prove that, surprisingly enough, not every algebraic number can be "computed" this way

    Generalized Communicating P Systems Working in Fair Sequential Model

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    In this article we consider a new derivation mode for generalized communicating P systems (GCPS) corresponding to the functioning of population protocols (PP) and based on the sequential derivation mode and a fairness condition. We show that PP can be seen as a particular variant of GCPS. We also consider a particular stochastic evolution satisfying the fairness condition and obtain that it corresponds to the run of a Gillespie's SSA. This permits to further describe the dynamics of GCPS by a system of ODEs when the population size goes to the infinity.Comment: Presented at MeCBIC 201

    Passively Mobile Communicating Logarithmic Space Machines

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    We propose a new theoretical model for passively mobile Wireless Sensor Networks. We call it the PALOMA model, standing for PAssively mobile LOgarithmic space MAchines. The main modification w.r.t. the Population Protocol model is that agents now, instead of being automata, are Turing Machines whose memory is logarithmic in the population size n. Note that the new model is still easily implementable with current technology. We focus on complete communication graphs. We define the complexity class PLM, consisting of all symmetric predicates on input assignments that are stably computable by the PALOMA model. We assume that the agents are initially identical. Surprisingly, it turns out that the PALOMA model can assign unique consecutive ids to the agents and inform them of the population size! This allows us to give a direct simulation of a Deterministic Turing Machine of O(nlogn) space, thus, establishing that any symmetric predicate in SPACE(nlogn) also belongs to PLM. We next prove that the PALOMA model can simulate the Community Protocol model, thus, improving the previous lower bound to all symmetric predicates in NSPACE(nlogn). Going one step further, we generalize the simulation of the deterministic TM to prove that the PALOMA model can simulate a Nondeterministic TM of O(nlogn) space. Although providing the same lower bound, the important remark here is that the bound is now obtained in a direct manner, in the sense that it does not depend on the simulation of a TM by a Pointer Machine. Finally, by showing that a Nondeterministic TM of O(nlogn) space decides any language stably computable by the PALOMA model, we end up with an exact characterization for PLM: it is precisely the class of all symmetric predicates in NSPACE(nlogn).Comment: 22 page

    Branching Feller diffusion for cell division with parasite infection

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    We describe the evolution of the quantity of parasites in a population of cells which divide in continuous-time. The quantity of parasites in a cell follows a Feller diffusion, which is splitted randomly between the two daughter cells when a division occurs. The cell division rate may depend on the quantity of parasites inside the cell and we are interested in the cases of constant or monotone division rate. We first determine the asymptotic behavior of the quantity of parasites in a cell line, which follows a Feller diffusion with multiplicative jumps. We then consider the evolution of the infection of the cell population and give criteria to determine whether the proportion of infected cells goes to zero (recovery) or if a positive proportion of cells becomes largely infected (proliferation of parasites inside the cells)

    Some stochastic models for structured populations : scaling limits and long time behavior

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    The first chapter concerns monotype population models. We first study general birth and death processes and we give non-explosion and extinction criteria, moment computations and a pathwise representation. We then show how different scales may lead to different qualitative approximations, either ODEs or SDEs. The prototypes of these equations are the logistic (deterministic) equation and the logistic Feller diffusion process. The convergence in law of the sequence of processes is proved by tightness-uniqueness argument. In these large population approximations, the competition between individuals leads to nonlinear drift terms. We then focus on models without interaction but including exceptional events due either to demographic stochasticity or to environmental stochasticity. In the first case, an individual may have a large number of offspring and we introduce the class of continuous state branching processes. In the second case, catastrophes may occur and kill a random fraction of the population and the process enjoys a quenched branching property. We emphasize on the study of the Laplace transform, which allows us to classify the long time behavior of these processes. In the second chapter, we model structured populations by measure-valued stochastic differential equations. Our approach is based on the individual dynamics. The individuals are characterized by parameters which have an influence on their survival or reproduction ability. Some of these parameters can be genetic and are inheritable except when mutations occur, but they can also be a space location or a quantity of parasites. The individuals compete for resources or other environmental constraints. We describe the population by a point measure-valued Markov process. We study macroscopic approximations of this process depending on the interplay between different scalings and obtain in the limit either integro-differential equations or reaction-diffusion equations or nonlinear super-processes. In each case, we insist on the specific techniques for the proof of convergence and for the study of the limiting model. The limiting processes offer different models of mutation-selection dynamics. Then, we study two-level models motivated by cell division dynamics, where the cell population is discrete and characterized by a trait, which may be continuous. In 1 particular, we finely study a process for parasite infection and the trait is the parasite load. The latter grows following a Feller diffusion and is randomly shared in the two daughter cells when the cell divides. Finally, we focus on the neutral case when the rate of division of cells is constant but the trait evolves following a general Markov process and may split in a random number of cells. The long time behavior of the structured population is then linked and derived from the behavior a well chosen SDE (monotype population)
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