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.
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