  We propose a novel Bayesian methodology which uses random walks for rapid
inference of statistical properties of undirected networks with weighted or
unweighted edges. Our formalism yields high-accuracy estimates of the
probability distribution of any network node-based property, and of the network
size, after only a small fraction of network nodes has been explored. The
Bayesian nature of our approach provides rigorous estimates of all parameter
uncertainties. We demonstrate our framework on several standard examples,
including random, scale-free, and small-world networks, and apply it to study
epidemic spreading on a scale-free network. We also infer properties of the
large-scale network formed by hyperlinks between Wikipedia pages.
