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Many complex systems exhibit a natural hierarchy in which elements can beranked according to a notion of "influence". While the complete and accurateknowledge of the interactions between constituents is ordinarily required forthe computation of nodes' influence, using a low-rank approximation we showthat in a variety of contexts local information about the neighborhoods ofnodes is enough to reliably estimate how influential they are, without the needto infer or reconstruct the whole map of interactions. Our framework issuccessful in approximating with high accuracy different incarnations ofinfluence in systems as diverse as the WWW PageRank, trophic levels ofecosystems, upstreamness of industrial sectors in complex economies, andcentrality measures of social networks, as long as the underlying network isnot exceedingly sparse. We also discuss the implications of this "emerginglocality" on the approximate calculation of non-linear network observables.

Ranking influential nodes in networks from partial information