56 research outputs found
Isoperimetric Inequalities in Simplicial Complexes
In graph theory there are intimate connections between the expansion
properties of a graph and the spectrum of its Laplacian. In this paper we
define a notion of combinatorial expansion for simplicial complexes of general
dimension, and prove that similar connections exist between the combinatorial
expansion of a complex, and the spectrum of the high dimensional Laplacian
defined by Eckmann. In particular, we present a Cheeger-type inequality, and a
high-dimensional Expander Mixing Lemma. As a corollary, using the work of Pach,
we obtain a connection between spectral properties of complexes and Gromov's
notion of geometric overlap. Using the work of Gunder and Wagner, we give an
estimate for the combinatorial expansion and geometric overlap of random
Linial-Meshulam complexes
On Eigenvalues of Random Complexes
We consider higher-dimensional generalizations of the normalized Laplacian
and the adjacency matrix of graphs and study their eigenvalues for the
Linial-Meshulam model of random -dimensional simplicial complexes
on vertices. We show that for , the eigenvalues of
these matrices are a.a.s. concentrated around two values. The main tool, which
goes back to the work of Garland, are arguments that relate the eigenvalues of
these matrices to those of graphs that arise as links of -dimensional
faces. Garland's result concerns the Laplacian; we develop an analogous result
for the adjacency matrix. The same arguments apply to other models of random
complexes which allow for dependencies between the choices of -dimensional
simplices. In the second part of the paper, we apply this to the question of
possible higher-dimensional analogues of the discrete Cheeger inequality, which
in the classical case of graphs relates the eigenvalues of a graph and its edge
expansion. It is very natural to ask whether this generalizes to higher
dimensions and, in particular, whether the higher-dimensional Laplacian spectra
capture the notion of coboundary expansion - a generalization of edge expansion
that arose in recent work of Linial and Meshulam and of Gromov. We show that
this most straightforward version of a higher-dimensional discrete Cheeger
inequality fails, in quite a strong way: For every and , there is a -dimensional complex on vertices that
has strong spectral expansion properties (all nontrivial eigenvalues of the
normalised -dimensional Laplacian lie in the interval
) but whose coboundary expansion is bounded
from above by and so tends to zero as ;
moreover, can be taken to have vanishing integer homology in dimension
less than .Comment: Extended full version of an extended abstract that appeared at SoCG
2012, to appear in Israel Journal of Mathematic
Visual perceptual load induces inattentional deafness
In this article, we establish a new phenomenon of “inattentional deafness” and highlight the level of load on visual attention as a critical determinant of this phenomenon. In three experiments, we modified an inattentional blindness paradigm to assess inattentional deafness. Participants made either a low- or high-load visual discrimination concerning a cross shape (respectively, a discrimination of line color or of line length with a subtle length difference). A brief pure tone was presented simultaneously with the visual task display on a final trial. Failures to notice the presence of this tone (i.e., inattentional deafness) reached a rate of 79% in the high-visual-load condition, significantly more than in the low-load condition. These findings establish the phenomenon of inattentional deafness under visual load, thereby extending the load theory of attention (e.g., Lavie, Journal of Experimental Psychology. Human Perception and Performance, 25, 596–616, 1995) to address the cross-modal effects of visual perceptual load
The average size of an independent set in graphs with a given chromatic number
AbstractLet G be a graph on n vertices, let χ(G) denote its chromatic number, and let α(G) denote the average size of an independent set of vertices of G. A simple argument shows that α(G) = Ω(nχG) log χ(G). Here we show that this is sharp by constructing, for every χ ≤ O(nlog n), a graph G with χ(G) = χ and with α(G) = θ(nχ log χ). This settles a problem of Linial and Saks
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