Computing with Large Populations Using Interactions

International audience; We define a general model capturing the behavior of a population of anonymous agents that interact in pairs. This model captures some of the main features of opportunistic networks, in which nodes (such as the ones of a mobile ad hoc networks) meet sporadically. For its reminiscence to Population Protocol, we call our model \emph{Large-Population Protocol}, or LPP. We are interested in the design of LPPs enforcing, for every $\nu\in[0,1]$, a proportion $\nu$ of the agents to be in a specific subset of marked states, when the size of the population grows to infinity; In which case, we say that the protocol \emph{computes} $\nu$. We prove that, for every $\nu\in[0,1]$, $\nu$ is computable by a LPP if and only if $\nu$ is algebraic. Our positive result is constructive. That is, we show how to construct, for every algebraic number $\nu\in[0,1]$, a protocol which computes $\nu$.