Evolution of triangulations: Hausdorff and spectral dimensions

How complex networks formed by triangulations and higher-dimensional simplicial complexes can represent closed evolving manifolds [1]. In particular, for triangulations, the set of possible transformations of these networks is restricted by the condition that at each step, all the faces must be triangles, which is the key constraint in this theory. Stochastic application of these operations leads to random networks with different architectures. I will show how geometries of growing and equilibrium complex networks generated by these transformations and their local structural properties can be described. This characterisation includes the Hausdorff and spectral dimensions of the resulting networks, their degree distributions, and various structural correlations. The results reveal a rich zoo of architectures and geometries of these networks, some of which appear to be small worlds while others are finite-dimensional with a wide spectrum of Hausdorff and spectral dimensions. 

[1] D. C. da Silva, G. Bianconi, R. A. da Costa, S. N. Dorogovtsev, and J. F. F. Mendes, Complex network view of evolving manifolds, Phys. Rev. E 97, 032316 (2018).