Emerging locality of network influence.

Many complex systems exhibit a natural hierarchy in which elements can be ranked according to a notion of "influence". Examples include the position of preys and predators in a food chain (so called trophic levels), or of manufactured goods in a production chain (leading to the notion of upstreamness). Finding the most "influential" nodes is key to understand the functioning and robustness of networked systems. The influence a node exerts on its neighborhood is an intrinsically non-local concept: it depends self-consistently on the influence exerted by all other nodes on their respective neighborhoods. Therefore, the complete and accurate knowledge of the interactions between constituents is ordinarily required for its computation. Using a low-rank approximation, we show instead that in a variety of contexts, only 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. We show that 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, input-output tables of complex economies, and centrality measures of social networks. We also discuss the implications of this "emerging locality" on the approximate calculation of non-linear network observables.