Statistical mechanics of quantum complex networks
The field of complex networks has been flourishing in the last fifteen years and has shed light on the underlying structure of complex systems as different as the Internet, or the protein interaction in the cell. This extensive body of work has defined a new paradigm for looking at interacting systems (described by a network) and nowadays complex networks structures constitute the new “lattices for complex systems” where dynamical processes can present a completely novel phase diagrams.
In this talk I will present an overview of a series of results at the frontier between complex network theory and statistical quantum mechanics including non-equilibrium and equilibrium processes.
First of all I will show how symmetries in growing networks models can be used to construct on one side scale-free complex networks with heterogeneous properties of the nodes described by the Bose-Einstein distribution and on the other side Cayley trees with heterogeneous properties of the nodes described by the Fermi distribution. These results demonstrate that complex networks topologies of non-equilibrium growing networks can be characterized by quantum statistics. Moreover the mapping between the complex networks and the corresponding quantum gases is able to predict that growing complex networks might undergo a structural phase transition called the Bose-Einstein condensation of complex networks in which one node grabs a finite fraction of all the links.
Then I will consider how complex networks and quantum statistical mechanics can be related by looking at dynamical processes defined on complex networks. I will show that complex scale-free network topologies can change the phase diagram of quantum critical phenomena such as the Transverse Ising model, the Bose-Hubbard model or the Jaynes-Cumming model, offering the possibility to explore the interplay between quantum critical phenomena and the topology of complex networks.
The Colloquium will be followed by coffee and biscuits from 2:00pm.