NDLI: On laplacians of random complexes

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On laplacians of random complexes

Content Provider	ACM Digital Library
Author	Wagner, Uli Gundert, Anna
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 higher-dimensional 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.
Starting Page	151
Ending Page	160
Page Count	10
File Format	PDF
ISBN	9781450312998
DOI	10.1145/2261250.2261272
Language	English
Publisher	Association for Computing Machinery (ACM)
Publisher Date	2012-06-17
Publisher Place	New York
Access Restriction	Subscribed
Subject Keyword	Measure concentration Eigenvalues Laplacians Simplicial complexes Hypergraphs Random complexes
Content Type	Text
Resource Type	Article

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