Distributed size estimation in anonymous networks

The knowledge of the size of a network, i.e. of the number of nodes composing it, is important for maintenance and organization purposes. In networks where the identity of the nodes or is not unique or cannot be disclosed for privacy reasons, the size-estimation problem is particularly challenging since the exchanged messages cannot be uniquely associated with a specific node. In this work, we propose a totally distributed anonymous strategy based on statistical inference concepts. In our approach, each node starts generating a vector of independent random numbers from a known distribution. Then nodes compute a common function via some distributed consensus algorithms, and finally they compute the Maximum Likelihood (ML) estimate of the network size exploiting opportune statistical inferences. In this work we study the performance that can be obtained following this computational scheme when the consensus strategy is either the maximum or the average. In the max-consensus scenario, when data come from absolutely continuous distributions, we provide a complete characterization of the ML estimator. In particular, we show that the squared estimation error decreases as 1/M , where M is the amount of random numbers locally generated by each node, independently of the chosen probability distribution. Differently, in the average-consensus scenario, we show that if the locally generated data are independent Bernoulli trials, then the probability for the ML estimator to return a wrong answer decreases exponentially in M . Finally, we provide a discussion as how the numerical errors may affect the estimators performance under different scenarios. Index Terms Size estimation, sensor networks, distributed estimation, privacy-preservation, number of nodes, number of agents, anonymous networks, consensus.