In a quantum network, distant observers sharing physical resources emitted by independent sources can establish strong correlations, which defy any classical explanation in terms of local variables. We discuss the characterization of nonlocal correlations in such a situation, when compared to those that can be generated in networks distributing independent local variables. We present an iterative procedure for constructing Bell inequalities tailored for networks: starting from a given network, and a corresponding Bell inequality, our technique provides new Bell inequalities for a more complex network, involving one additional source and one additional observer. We illustrate the relevance of our method on a variety of networks, demonstrating significant quantum violations, which could not have been detected using standard Bell inequalities.Received 8 July 2015DOI:https://doi.org/10.1103/PhysRevLett.116.010403© 2016 American Physical SocietyPhysics Subject Headings (PhySH)Research AreasNetwork structureNonlocalityGeneral PhysicsNetworks