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The Shannon and the Von Neumann entropy of random networks with heterogeneous expected degree

Entropic measures of complexity are able to quantify the information encoded in complex network structures. Several entropic measures have been proposed in this respect. Here we study the relation between the Shannon entropy and the Von Neumann entropy of networks with a given expected degree sequence. We find in different examples of network topologies that when the degree distribution contains some heterogeneity, an intriguing correlation emerges between the two entropies. This result seems to suggest that this kind of heterogeneity is implying an equivalence between a quantum and a classical description of networks, which respectively correspond to the Von Neumann and the Shannon entropy.