Description
Probability seminar.
Speaker: Kartick Adhikari.
Affiliation: I.I.T Technion, Israel.
Date and Time: Thursday 13 February, 04:00 pm - 05:00 pm.
Venue: Room 105, Department of Mathematics.
Title: The Spectrum of Dense Random Geometric Graphs.
Abstract: We study the spectrum of Laplacian of a random geometric graph,
in a regime where the graph is dense and highly connected. As opposed to
other random graph models (e.g. the Erdos-Renyi random graph), even when
the graph is dense, not all the eigenvalues are concentrated around 1. In
the case where the vertices are generated uniformly in a unit
d-dimensional box, we show that for every $0\le k \le d$ there are
$\binom{d}{k}$ eigenvalues at $1-2^{-k}$. The rest of the eigenvalues are
indeed close to 1. The spectrum of the graph Laplacian plays a key role in
both theory and applications. Aside from the interesting mathematical
phenomenon we reveal here, the results of this paper can also be used to
analyze the homology of the random Vietoris-Rips complex via spectral
methods.
The talk will be based on a joint work with R. Adler, O. Bobrowski and R.
Rosenthal.