On Laplacians of random complexes Conference Paper

Author(s): Gundert, Anna; Wagner, Uli
Title: On Laplacians of random complexes
Abstract: Eigenvalues associated to graphs are a well-studied subject. In particular the spectra of the adjacency matrix and of the Laplacian of random graphs G(n, p) are known quite precisely. We consider generalizations of these matrices to simplicial complexes of higher dimensions and study their eigenvalues for the Linial-Meshulam model X k(n, p) of random k-dimensional simplicial complexes on n vertices. We show that for p = Ω(log n/n), the eigenvalues of both, the higher-dimensional adjacency matrix and the Laplacian, are a.a.s. sharply concentrated around two values. In a second part of the paper, we discuss a possible higherdimensional analogue of the Discrete Cheeger Inequality. This fundamental inequality expresses a close relationship between the eigenvalues of a graph and its combinatorial expansion properties; in particular, spectral expansion (a large eigenvalue gap) implies edge expansion. Recently, a higher-dimensional analogue of edge expansion for simplicial complexes was introduced by Gromov, and independently by Linial, Meshulam and Wallach and by Newman and Rabinovich. It is natural to ask whether there is a higher-dimensional version of Cheeger's inequality. We show that the most straightforward version of a higher-dimensional Cheeger inequality fails: for every k > 1, there is an infinite family of k-dimensional complexes that are spectrally expanding (there is a large eigenvalue gap for the Laplacian) but not combinatorially expanding.
Keywords: Simplicial complexes; Eigenvalues; Hypergraphs; Laplacians; Measure Concentration; Random Complexes
Conference Title: SGC: Symposuim on Computational Geometry
Conference Dates: June 17-20, 2012
Conference Location: Chapel Hill, NC, United States
Publisher: ACM  
Date Published: 2012-06-01
Start Page: 151
End Page: 160
DOI: 10.1145/2261250.2261272
Open access: no
IST Austria Authors
  1. Uli Wagner
    50 Wagner
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