  Efficient and robust algorithms for decentralized estimation in networks are
essential to many distributed systems. Whereas distributed estimation of sample
mean statistics has been the subject of a good deal of attention, computation
of $U$-statistics, relying on more expensive averaging over pairs of
observations, is a less investigated area. Yet, such data functionals are
essential to describe global properties of a statistical population, with
important examples including Area Under the Curve, empirical variance, Gini
mean difference and within-cluster point scatter. This paper proposes new
synchronous and asynchronous randomized gossip algorithms which simultaneously
propagate data across the network and maintain local estimates of the
$U$-statistic of interest. We establish convergence rate bounds of $O(1/t)$ and
$O(\log t / t)$ for the synchronous and asynchronous cases respectively, where
$t$ is the number of iterations, with explicit data and network dependent
terms. Beyond favorable comparisons in terms of rate analysis, numerical
experiments provide empirical evidence the proposed algorithms surpasses the
previously introduced approach.
