An upper bound on the lower tail of the number of connected subgraphs in random geometric graphs
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Kim, Phillip
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Abstract
Random Graphs are well studied within Combinatorics, Probability Theory, and Computer Science. One particular class of Random Graphs are Random Geometric Graphs
which are constructed as follows: take n randomly and independently placed objects in some d−dimensional space Rd, and create an edge between two points xi, xj if and only if ∥xi − xj ∥ ≤ r for some r where ∥ · ∥ is a norm on Rd. In this paper we use Janson’s Inequalities to provide an upper bound on the upper tail of the number of cliques and induced connected subgraphs for random geometric graphs. That is, if T counts the number of cliques or induced connected subgraphs, and λ > 0, we give an upper bound for Pr[T ≤ (1 − λ)E[T ]].
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Undergraduate Research Option Thesis