Network Geometry aims at characterizing discrete network structures as geometrical objects. In this context simplicial complexes, i.e. network formed by triangles, tetrahedra, etc.., play an important role.
The advance in this field it is important for characterizing complex networks in the brain or in communication technology. At the same this emerging field is providing the ground for extablishing a crossfertilization between complex networks and quantum gravity.
I have recently worked on emergent network geometry obtain results in different directions:

In Emergent Complex Network Geometry an growing simplicial complex formed by triangles is able, just with two parameters to generate either 2d manifols or scalefree networks, with high clustering coefficient, smallworld diameter and significant modular structure.

Complex Quantum Network Manifolds are emerging from a non equilibrium dynamics of simplicial complexes in arbitrary diemnsions. They are scalefree for any dimension greater than two. Interestingly their statistical properties are described by generalized degrees. In dimension 3 the generalized degree of the tringular faces, of the link and of the nodes follow respectively the FermiDirac, the Boltzmann, and the BoseEinstein distribution.
The paper has been widely covered by the press.

Network Geometry with Flavor is a general framework for emergent network geometry that includes the BianconiBarabasi model, and Complex Quantum Network Manifolds. Interestingly the model can be related to Regge calculus.