Complex quantum network geometries: Evolution and phase transitions

Ginestra Bianconi, Christoph Rahmede, and Zhihao Wu
Phys. Rev. E 92, 022815 – Published 24 August 2015

Abstract

Networks are topological and geometric structures used to describe systems as different as the Internet, the brain, or the quantum structure of space-time. Here we define complex quantum network geometries, describing the underlying structure of growing simplicial 2-complexes, i.e., simplicial complexes formed by triangles. These networks are geometric networks with energies of the links that grow according to a nonequilibrium dynamics. The evolution in time of the geometric networks is a classical evolution describing a given path of a path integral defining the evolution of quantum network states. The quantum network states are characterized by quantum occupation numbers that can be mapped, respectively, to the nodes, links, and triangles incident to each link of the network. We call the geometric networks describing the evolution of quantum network states the quantum geometric networks. The quantum geometric networks have many properties common to complex networks, including small-world property, high clustering coefficient, high modularity, and scale-free degree distribution. Moreover, they can be distinguished between the Fermi-Dirac network and the Bose-Einstein network obeying, respectively, the Fermi-Dirac and Bose-Einstein statistics. We show that these networks can undergo structural phase transitions where the geometrical properties of the networks change drastically. Finally, we comment on the relation between quantum complex network geometries, spin networks, and triangulations.

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  • Received 16 March 2015

DOI:https://doi.org/10.1103/PhysRevE.92.022815

©2015 American Physical Society

Authors & Affiliations

Ginestra Bianconi1, Christoph Rahmede2, and Zhihao Wu3

  • 1School of Mathematical Sciences, Queen Mary University of London, London E1 4NS, United Kingdom
  • 2Institute for Theoretical Physics, Karlsruhe Institute of Technology, 76128 Karlsruhe, Germany
  • 3School of Computer and Information Technology, Beijing Jiaotong University, Beijing 100044, China

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Issue

Vol. 92, Iss. 2 — August 2015

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