Date & Time: Wednesday, August 27, 2014, 16:00-17:00.
Venue: Ramanujan Hall
Speaker: Gugan Thoppe, TIFR Mumbai
Title: Topology of a Randomly Evolving Erdos Renyi Graph
Abstract: Mathew Kahle and Elizabeth Meckes recently established interesting results concerning the topology of the clique complex $X(n,p)$ on an Erdos Renyi graph $G(n,p)$ when $p = n^{\alpha}$ with $\alpha \in (-1/k, -1/(k + 1)).$ They showed that as the number of vertices $n$ tends to infinity, every Betti number $\beta_j$ of $X(n, p)$ vanishes to zero except for the $k-$th one. In fact, $\beta_k$ follows a central limit theorem, i.e., $(\beta_k - \mathbb{E}[\beta_k])/\sqrt{Var(\beta_k)}$ is asymptotically Gaussian.
In this talk, we will extend the above result to the case of a randomly evolving Erdos Renyi graph $G(n, p, t).$ We will show that if p is chosen as above, then the process $(\beta_k(t) -\mathbb{E}[\beta_k(t)])/\sqrt{Var[\beta_k(t)]}$ is asymptotically an Ornstein-Uhlenbeck process. That is, the k-th Betti number asymptotically behaves like a stationary Gaussian Markov process with an exponentially decaying covariance function.
This is joint work with Prof. Robert Adler