48,683 research outputs found
Random geometric complexes
We study the expected topological properties of Cech and Vietoris-Rips
complexes built on i.i.d. random points in R^d. We find higher dimensional
analogues of known results for connectivity and component counts for random
geometric graphs. However, higher homology H_k is not monotone when k > 0. In
particular for every k > 0 we exhibit two thresholds, one where homology passes
from vanishing to nonvanishing, and another where it passes back to vanishing.
We give asymptotic formulas for the expectation of the Betti numbers in the
sparser regimes, and bounds in the denser regimes. The main technical
contribution of the article is in the application of discrete Morse theory in
geometric probability.Comment: 26 pages, 3 figures, final revisions, to appear in Discrete &
Computational Geometr
Random geometric complexes and graphs on Riemannian manifolds in the thermodynamic limit
We investigate some topological properties of random geometric complexes and
random geometric graphs on Riemannian manifolds in the thermodynamic limit. In
particular, for random geometric complexes we prove that the normalized
counting measure of connected components, counted according to isotopy type,
converges in probability to a deterministic measure. More generally, we also
prove similar convergence results for the counting measure of types of
components of each -skeleton of a random geometric complex. As a
consequence, in the case of the -skeleton (i.e. for random geometric graphs)
we show that the empirical spectral measure associated to the normalized
Laplace operator converges to a deterministic measure
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
Large random simplicial complexes, I
In this paper we introduce a new model of random simplicial complexes
depending on multiple probability parameters. This model includes the
well-known Linial - Meshulam random simplicial complexes and random clique
complexes as special cases. Topological and geometric properties of a
multi-parameter random simplicial complex depend on the whole combination of
the probability parameters and the thresholds for topological properties are
convex sets rather than numbers (as in all previously known models). We discuss
the containment properties, density domains and dimension of the random
simplicial complexes.Comment: 21 pages, 6 figure
Central limit theorems for Soft random simplicial complexes
A soft random graph can be obtained from the random geometric
graph by keeping every edge in with probability . The soft
random simplicial complexes is a model for random simplicial complexes built
over the soft random graph . This new model depends on a probability
vector which allows the simplicial complexes to present randomness in
all dimensions. In this article, we use a normal approximation theorem to prove
central limit theorems for the number of -faces and for the Euler's
characteristic for soft random simplicial complexes.Comment: 28 page
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