  The correlations in quantum networks have attracted strong interest with new
types of violations of the locality. The standard Bell inequalities cannot
characterize the multipartite correlations that are generated by multiple
sources. The main problem is that no computationally efficient method is
available for constructing useful Bell inequalities for general quantum
networks. In this work, we show a significant improvement by presenting new,
explicit Bell-type inequalities for general networks including cyclic networks.
These nonlinear inequalities are related to the matching problem of an
equivalent unweighted bipartite graph that allows constructing a
polynomial-time algorithm. For the quantum resources consisting of bipartite
entangled pure states and generalized Greenberger-Horne-Zeilinger (GHZ) states,
we prove the generic non-multilocality of quantum networks with multiple
independent observers using new Bell inequalities. The violations are maximal
with respect to the presented Tsirelson's bound for Einstein-Podolsky-Rosen
(EPR) states and GHZ states. Moreover, these violations hold for Werner states
or some general noisy states. Our results suggest that the presented Bell
inequalities can be used to characterize experimental quantum networks.
