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UID:www.tcs.tifr.res.in/event/518
DTSTAMP:20230914T125927Z
SUMMARY:Topology of a Randomly Evolving Erdos Renyi Graph
DESCRIPTION:Speaker: Gugan  Thoppe\n\nAbstract: \nAbstract: Mathew Kahle an
 d Elizabeth Meckes recently established interesting results concerning the
  topology of the clique complex $X(n\,p)$ on an Erdos Renyi graph $G(n\,p)
 .$ Specifically\, they showed that\, if $p = n^{\\alpha}$ with $\\alpha \\
 in (-1/k\, -1/(k + 1))$ for some positive integer $k\,$ then asymptoticall
 y\, i.e.\, as $n \\rightarrow \\infty\,$ every Betti number $\\beta_j$ of 
 $X(n\,p)\,$ except for the $k-$th one\, vanishes. Further\, for the choice
  of $p$ as above\, $\\beta_k$ of $X(n\,p)$ follows a central limit theorem
 \, i.e.\, $(\\beta_k - \\mathbb{E}[\\beta_k])/\\sqrt{Var(\\beta_k)}$ is as
 ymptotically Gaussian.\n\n\n \nIn this talk\, we will consider a randomly
  evolving Erdos Renyi graph $G(n\, p\, t)$ and study how its topology evol
 ves as the time $t$ varies. Specifically\, we will prove that if $p$ is ch
 osen 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\, it asymptotically behaves like a stationary Gaussian Markov proc
 ess with an exponentially decaying covariance function. \n \nThis talk w
 ill NOT assume any prerequisites.\n\n\n \n
URL:https://www.tcs.tifr.res.in/web/events/518
DTSTART;TZID=Asia/Kolkata:20140711T143000
DTEND;TZID=Asia/Kolkata:20140711T160000
LOCATION:D-405 (D-Block Seminar Room)
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