Bipartite quantum states and random complex networks

TL;DR

This paper establishes a link between graph structures and quantum entanglement, revealing how entanglement entropy can characterize complex network topologies.

Contribution

It introduces a novel mapping from graphs to quantum states and derives entanglement entropy expressions for complex networks, highlighting topological features.

Findings

Entanglement entropy scales as c log n + g_e for large networks.

The sub-leading term g_e encodes topological information.

Analytic results for classical random graphs and numerical analysis for 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.