Department of Computer Science | Institute of Theoretical Computer Science | CADMO

Prof. Emo Welzl and Prof. Bernd Gärtner

Mittagsseminar Talk Information |

**Date and Time**: Wednesday, September 25, 2013, 12:15 pm

**Duration**: 30 minutes

**Location**: OAT S15/S16/S17

**Speaker**: Matan Harel (Courant Institute of Mathematical Sciences)

Consider a random geometric graph G(n, r(n)), given by a connecting two vertices of a Poisson Point Process of intensity n on the unit torus whenever their distance is smaller than the parameter r(n). This model is conditioned on the rare event that the number of edges observed, |E|, is greater than [1 + delta (n)] times its expectation, producing a three parameter model. We show that, for delta constant or vanishing sufficiently slowly in n, with high probability, there exists a "giant clique" with almost all of the excess edges. Furthermore, if delta vanishes sufficiently quickly, the largest clique will be, at most, a constant multiple of the usual clique number of the unconditioned random geometric graph; roughly, all excess edges will result from the "standard" entropy-like effects of a vanishing Radon-Nykodyn derivative. Finally, we discuss progress in finding a phase transition function delta_{0}(n), so that when delta is much bigger than delta_{0}, the giant clique scenario holds, while delta much smaller than delta_{0} implies the entropy scenario.

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