Complex network view of evolving manifolds

Abstract

We study complex networks formed by triangulations and higher-dimensional simplicial complexes representing closed evolving manifolds. 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. Stochastic application of these operations leads to random networks with different architectures. We perform extensive numerical simulations and explore the geometries of growing and equilibrium complex networks generated by these transformations and their local structural properties. This characterization includes the Hausdorff and spectral dimensions of the resulting networks, their degree distributions, and various structural correlations. Our 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 Hausdorff dimension equal or higher than the original dimensionality of their simplices. The range of spectral dimensions of the evolving triangulations turns out to be from about 1.4 to infinity. Our models include simplicial complexes representing manifolds with evolving topologies, for example, an $h$-holed torus with progressively growing number of holes. This evolving graph demonstrates features of a small-world network and has a particularly heavy-tailed degree distribution.