@misc{9226480,
  abstract     = {{This thesis studies a geometric random graph model on the discrete circle Z/N Z, in which an
edge between distinct vertices u and v is present independently with probability
p(u, v) = min{cN^{-(1-α)}ρ(u, v)^{-α},1}, 0 < α < 1,
where ρ(u, v) is the graph distance. The main focus is on the number of cycles of a fixed
length, which provides a natural measure of the deviation from a tree-like structure. The
expected number of k-cycles is shown to exhibit different asymptotic regimes depending on
the relationship between k and the decay parameter α. More precisely, it converges to a finite constant when k(1 − α) > 1, grows logarithmically in the critical case k(1 − α) = 1, and grows polynomially when k(1 − α) < 1. In addition, when α < 2/3, the number of k-cycles is asymptotically Poisson distributed.}},
  author       = {{Yao, Xinxu}},
  issn         = {{1404-6342}},
  language     = {{eng}},
  note         = {{Student Paper}},
  series       = {{Master's Theses in Mathematical Sciences}},
  title        = {{THE CYCLES IN GEOMETRIC RANDOM GRAPHS ON DISCRETE CIRCLE}},
  year         = {{2026}},
}