  Real-world network applications must cope with failing nodes, malicious
attacks, or nodes facing corrupted data - data classified as outliers. Our work
addresses these concerns in the scope of the sensor network localization
problem where, despite the abundance of technical literature, prior research
seldom considered outlier data. We propose robust, fast, and distributed
network localization algorithms, resilient to high-power noise, but also
precise under regular Gaussian noise. We use a Huber M-estimator, thus
obtaining a robust (but nonconvex) optimization problem. We convexify and
change the problem representation, to allow for distributed robust localization
algorithms: a synchronous distributed method that has optimal convergence rate
and an asynchronous one with proven convergence guarantees. A major highlight
of our contribution lies on the fact that we pay no price for provable
distributed computation neither in accuracy, nor in communication cost or
convergence speed. Simulations showcase the superior performance of our
algorithms, both in the presence of outliers and under regular Gaussian noise:
our method exceeds the accuracy of alternative approaches, distributed and
centralized, even under heavy additive and multiplicative outlier noise.
