Bipartite quantum states and random complex networks

Abstract

We introduce a mapping between graphs and pure quantum bipartite states and show that the associated entanglement entropy conveys non-trivial information about the structure of the graph. Our primary goal is to investigate the family of random graphs known as complex networks. In the case of classical random graphs we derive an analytic expression for the averaged entanglement entropy $\bar S$ while for general complex networks we rely on numerics. For large number of nodes $n$ we find a scaling $\bar{S} \sim c \log n +g_e$ where both the prefactor $c$ and the sub-leading O(1) term $g_e$ are a characteristic of the different classes of complex networks. In particular, $g_e$ encodes topological features of the graphs and is named network topological entropy. Our results suggest that quantum entanglement may provide a powerful tool in the analysis of large complex networks with non-trivial topological properties.