# Robust Comparison 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, and attempting to jointly compute global predicates. In particular, a significant amount of work, has gone towards investigating majority and consensus dynamics in this model: assuming that each 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. We are given two baseline states, X_0 and Y_0, present in any initial configuration in fixed, possibly small counts. Importantly, one of these states has higher count than the other: we will assume |X_0| > C |Y_0| for some constant C. The challenge is to design a protocol which can quickly and reliably decide on which of the baseline states X_0 and Y_0 has higher initial count. We propose a simple algorithm solving comparison: the baseline algorithm uses O(log n) states per node, and converges in O(log n) (parallel) time, with high probability, to a state where whole population votes on opinions X or Y at rates proportional to initial |X_0| vs. |Y_0| concentrations. We then describe how such output can be then used to solve comparison. The algorithm is self-stabilizing, in the sense that it converges to the correct decision even if the relative counts of baseline states X_0 and Y_0 change dynamically during the execution, and leak-robust, in the sense that it can withstand spurious faulty reactions. Our analysis relies on a new martingale concentration result which relates the evolution of a population protocol to its expected (steady-state) analysis, which should be broadly applicable in the context of population protocols and opinion dynamics.

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