Computing with Large Populations Using Interactions

Abstract. We define a general model capturing the behavior of a pop-ulation 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 Large-Population Protocol, or LPP. We are interested in the design of LPPs enforcing, for every ν ∈ [0, 1], a proportion ν 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 computes ν. We prove that, for every ν ∈ [0, 1], ν is computable by a LPP if and only if ν is algebraic. Our positive result is constructive. That is, we show how to construct, for every algebraic number ν ∈ [0, 1], a protocol which computes ν.