Dynamical Models for Random Simplicial Complexes

Abstract

We study a general model of random dynamical simplicial complexes and derive a formula for the asymptotic degree distribution. This asymptotic formula encompasses results for a number of existing models, including random Apollonian networks and the weighted random recursive tree. It also confirms results on the scale-free nature of Complex Quantum Network Manifolds in dimensions $d>2$, and special types of Network Geometry with Flavour models studied in the physics literature by Bianconi, Rahmede [$Sci.Rep.5,\text{ 13979 (2015) and }Phys.Rev.E93,\text{ 032315 (2016)}$].

53 pages (main body 45 pages), 4 figures. Accepted version, to appear in Annals of Applied Probability