On the evolution of topology in dynamic clique complexes

Abstract

We consider a time varying analogue of the Erdős-Rényi graph and study the topological variations of its associated clique complex. The dynamics of the graph are stationary and are determined by the edges, which evolve independently as continuous time Markov chains. Our main result is that when the edge inclusion probability is of the form p = n α , where n is the number of vertices and α ∈ (−1/k, −1/(k + 1)), then the process of the normalized k−th Betti number of these dynamic clique complexes converges weakly to the Ornstein-Uhlenbeck process as n → ∞.

Limit theorems for topological invariants of the dynamic multi-parameter simplicial complex

“…Kahle and Meckes [14] did such a study in the context of random clique complexes and proved a central limit theorem for the dominating Betti number. To obtain an even deeper understanding, Thoppe et al [21] investigated the topological fluctuations in the dynamic variant of this model. Specifically, they considered the setup in which every edge can change its state between being ON and being OFF, i.e., between being present and being absent, at the transition times of a continuous-time Markov chain.…”

“…Within the context of the combinatorial simplicial complexes, few attempts have been made at deriving "process-level" limit theorems for topological invariants (with a few exceptions such as Thoppe et al [21], Skraba et al [20], and Fraiman et al [8]). Our work fills in this gap.…”

“…Both of our results are proved in the space D[0, ∞) of right continuous functions with left limits. Additionally, unlike Thoppe et al [21], we do not assume a Markovian structure for the process according to which the faces of the complex are switched on or off. Instead, the evolution here is determined by a stationary process with a renewal structure.…”

Dynamic Erdős-Rényi Graphs

“…Topics for future research may relate to other graph metrics than the total number of edges. In the introduction, we mentioned that [13] considers the behavior of the Betti number, but one could also think of e.g. the evolution of the number of wedges or triangles in the random graph.…”

“…The next step is to consider scaling limits; under a particular scaling, the process Y (t) satis es a functional central limit theorem. More speci cally, after centering and scaling it converges to an Ornstein-Uhlenbeck ( ) process; interestingly, in [13] it is shown that for certain dynamic Erdős-Rényi graphs that a particular clique-complex related quantity (the 'Betti number') is described by an process as well. Finally we discuss for both models the corresponding sample-path large deviations, characterizing the models' rare-event behavior.…”