  Topological study of existing random simplicial complexes is non-trivial and
has led to several seminal works. However, the applicability of such studies is
limited since the randomness there is usually governed by a single parameter.
With this in mind, we focus here on the topology of the recently proposed
multi-parameter random simplicial complex and, more importantly, of its dynamic
analogue that we introduce here. In this dynamic setup, the temporal evolution
of simplices is determined by stationary and possibly non-Markovian processes
with a renewal structure. The dynamic versions of the clique complex and the
Linial-Meshulum complex are special cases of our setup. Our key result concerns
the regime where face-counts of a particular dimension dominate. We show that
the Betti numbers corresponding to this dimension and the Euler characteristic
satisfy functional strong law of large numbers and functional central limit
theorems. Surprisingly, in the latter result, the limiting Gaussian process
depends only upon the dynamics in the smallest non-trivial dimension.
