Comparison dynamics in population protocols

There has recently been a surge of interest in the computational and complexity properties of the population model, which assumes n anonymous, computationally-bounded nodes, interacting at random, with the goal of jointly computing global predicates. Significant work has gone towards investigating majority or consensus dynamics in this model: that is, assuming that every node is initially in one of two states X or Y, determine which state had higher initial count. In this paper, we consider a natural generalization of majority/consensus, which we call comparison : in its simplest formulation, we are given two baseline states, X and Y, present in any initial configuration in fixed, but possibly small counts. One of these states has higher count than the other: we will assume |X_0| > C |Y_0| for some constant C > 1. The challenge is to design a protocol by which nodes can quickly and reliably decide on which of the baseline states X_0 and Y_0 has higher initial count. We begin by analyzing a simple and general dynamics solving the above comparison problem, which uses O( log n ) states per node, and converges in O(log n) (parallel) time, with high probability, to a state where the whole population votes on opinions X or Y at rates proportional to the initial concentrations of |X_0| vs. |Y_0|. We then describe how this procedure can be bootstrapped to solve comparison, i.e. have every node in the population reach the "correct'' decision, with probability 1 - o(1), at the cost of O (log log n) additional states. Further, we prove that this dynamics is self-stabilizing, in the sense that it converges to the correct decision from arbitrary initial states, and leak-robust, in the sense that it can withstand spurious faulty reactions, which are known to occur in practical implementations of population protocols. Our analysis is based on a new martingale concentration result relating the discrete-time evolution of a population protocol to its expected (steady-state) analysis, which should be a useful tool when analyzing opinion dynamics and epidemic dissemination in the population model.