  Entangled quantum networks provide great flexibilities and scalabilities for
quantum information processing or quantum Internet. Most of results are focused
on the nonlocalities of quantum networks. Our goal in this work is to explore
new characterizations of any networks with theory-independent configurations.
We firstly prove the configuration inequality for any network using the
fractional independent set of the associated graph. These inequalities can be
built with polynomial-time complexity. The new result allows featuring
correlations of any classical network depending only on its network topology.
Similar inequalities hold for all entangled quantum networks with any local
measurements. This shows an inherent feature of quantum networks under local
unitary operations. It is then applied for verifying almost all multipartite
entangled pure states with linear complexity, and witnessing quantum network
topology without assumption of inputs. The configuration theory is further
extended for any no-signalling networks. These results may be interesting in
entanglement theory, quantum information processing, and quantum networks.
