The Spectrum of a Random Geometric Graph is Concentrated

Sanatan Rai ()
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Sanatan Rai: Stanford University

Journal of Theoretical Probability, 2007, vol. 20, issue 2, 119-132

Abstract: Consider n points, x 1,... , x n , distributed uniformly in [0, 1] d . Form a graph by connecting two points x i and x j if $$\Vert x_i - x_j\Vert \leq r(n)$$ . This gives a random geometric graph, $$G({\mathcal {X}}_n;r(n))$$ , which is connected for appropriate r(n). We show that the spectral measure of the transition matrix of the simple random walk on $$G({\mathcal {X}}_n; r(n))$$ is concentrated, and in fact converges to that of the graph on the deterministic grid.

Keywords: Random geometric graphs; Spectral measure; Primary 60D05; Secondary 34L20 (search for similar items in EconPapers)
Date: 2007
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DOI: 10.1007/s10959-006-0049-7

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