  Many complex systems exhibit a natural hierarchy in which elements can be
ranked according to a notion of "influence". While the complete and accurate
knowledge of the interactions between constituents is ordinarily required for
the computation of nodes' influence, using a low-rank approximation we show
that in a variety of contexts local information about the neighborhoods of
nodes is enough to reliably estimate how influential they are, without the need
to infer or reconstruct the whole map of interactions. Our framework is
successful in approximating with high accuracy different incarnations of
influence in systems as diverse as the WWW PageRank, trophic levels of
ecosystems, upstreamness of industrial sectors in complex economies, and
centrality measures of social networks, as long as the underlying network is
not exceedingly sparse. We also discuss the implications of this "emerging
locality" on the approximate calculation of non-linear network observables.
