FisMat2017 - Submission - View

Abstract's title: Emergent hyperbolic geometry of growing simplicial complexes
Submitting author: Ginestra Bianconi
Affiliation: Queen Mary University of London
Affiliation Address: School of Mathematical Sciences Queen Mary University of London Mile End Road E1 4NS London UK
Country: United Kingdom
Oral presentation/Poster (Author's request): Oral presentation
Other authors and affiliations:

A large variety of interacting complex systems are characterized by  
interactions occurring between more than two nodes. 
These systems are  described by simplicial complexes. Simplicial complexes 
are formed by simplices (nodes, links, triangles, tetrahedra etc.) 
that have a natural geometric interpretation. As such simplicial complexes 
are widely used in quantum gravity approaches that involve a discretization of spacetime.  
Here, by extending our knowledge of growing complex networks to growing 
simplicial complexes we investigate the nature of the emergent  
geometry of complex networks and explore whether  this geometry is  hyperbolic.  
Specifically we show that an hyperbolic network geometry emerges spontaneously 
from models of growing simplicial complexes that are purely combinatorial. 
The statistical  and geometrical properties of the growing simplicial complexes strongly depend on 
their dimensionality and display  the major universal properties of real complex networks 
(scale-free degree distribution, small-world and communities) at the same time.

Interestingly, when the network dynamics includes an heterogeneous fitness of the faces,
the statistical properties of the faces of the simplicies are described by Fermi-Dirac, Boltzmann and
Bose-Einstein statitics. Additionally the simplicial complexes can undergo a
phase transitions that are reflected by relevant changes in the network geometry.