1Distributed cardinality estimation in anonymous networks

Abstract—We consider estimation of network cardinality by distributed anonymous strategies relying on statistical inference methods. In partic-ular, we focus on the relative Mean Square Error (MSE) of Maximum Likelihood (ML) estimators based on either the maximum or the average of M-dimensional vectors randomly generated at each node. In the case of continuous probability distributions, we show that the relative MSE achieved by the max-based strategy decreases as 1/M, independently of the used distribution, while that of the average-based estimator scales approximately as 2/M. We then introduce a novel strategy based on the average of M-dimensional vectors locally generated from Bernoulli random variables. In this case, the ML estimator, which is the Least Common Multiple (LCM) of the denominators of the irreducible fractions corresponding to the M elements of the average vector, leads to an MSE which goes exponentially to zero as M increases. We then discuss the effects of finite precision arithmetics in practical dynamic implementations. Numerical experiments reveal that the MSE of the strategy based on Bernoulli trials is two order of magnitude smaller than that based on continuous random variables, at a price of one order of magnitude slower convergence time. Index Terms—Size estimation, sensor networks, distributed estimation, privacy-preservation, number of nodes, number of agents, anonymous networks, consensus