On the Convergence of Population Protocols When Population Goes to Infinity

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

Playing With Population Protocols

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 collection of anonymous agents, modeled by finite automata, interact in pairs according to some rules. Predicates on the initial configurations that can be computed by such protocols have been characterized under several hypotheses. We discuss here whether and when the rules of interactions between agents can be seen as a game from game theory. We do so by discussing several basic protocols

Protocoles de populations, jeux, et grandes populations

Population protocols were introduced to capture the specifics of opportunistic networks oftny mobile agents with limited memory and capable of wireless communication in pairs. Thisthesis aims at extending the understanding and analysis of population protocols as well astheir links to other models of population dynamics including ones from game theory.The first contribution of this thesis is to translate in terms of population protocols thedynamics of a population of agents playing a game repeatedly against each-other and adaptingtheir strategy according to the PAVLOV behaviour. We show that protocols born fromgames are exactly as powerful as general population protocols.The second contribution consists in the study of the impact of symmetry on games andin the transitions of a population protocol to show that, if symmetric population protocolsare equivalent to general protocols, symmetric games are significantly less powerful.The third contribution is to show how the dynamic of a population protocol can beapproximated by an ordinary differential equation when the population grows to infinity. Wethen define a computation by a large population to be the convergence of this differentialequation to a stable equilibrium.The fourth and final contribution of this thesis is the characterisation of the numberscomputable in the above sense as exactly the algebraic real numbers in [0; 1].Le modèle des population protocols a été proposé pour capturer les spécificités de réseauxopportunistes constitués d’une population d’agents mobiles à la mémoire limitée capables decommunications sans fil par paires. L’objet de cette thèse est d’étendre la compréhension etl’analyse des population protocols ainsi que leur liens avec d’autres modèles de dynamiquesde populations.La première contribution de cette thèse est l’étude de la traduction en terme de protocolesde population de la dynamique d’une population d’agents jouant à un jeu de manière répétéeles uns contre les autres et adaptant leur stratégie selon le comportement de PAVLOV. Nousmontrons que les protocoles issus de tels jeux sont aussi puissants que les protocoles depopulation généraux.La deuxième contribution consiste à étudier des hypothèse de symétrie dans les jeux etdans les transitions d’un protocole de population, pour montrer que, si les protocoles depopulation symétriques sont équivalents aux protocoles généraux, les jeux symétriques sont,eux, significativement moins puissants.La troisième contribution est de montrer comment étudier le comportement d’une protocolede population lorsque la taille de la population tend vers l’infini en approchant ladynamique résultante à l’aide d’une équation différentielle ordinaire et de définir un calculpar grande population comme la convergence de cette équation différentielle vers un équilibrestable.La quatrième et dernière contribution de la thèse est la caractérisation des nombrescalculables en ce sens comme étant très exactement les réels algébriques des [0; 1]

Protocoles de population, jeux et grandes populations

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