42,362 research outputs found
Topology of random simplicial complexes: a survey
This expository article is based on a lecture from the Stanford Symposium on
Algebraic Topology: Application and New Directions, held in honor of Gunnar
Carlsson, Ralph Cohen, and Ib Madsen.Comment: After revisions, now 21 pages, 5 figure
Multivariate central limit theorems for random clique complexes
Motivated by open problems in applied and computational algebraic topology, we establish multivariate normal approximation theorems for three random vectors which arise organically in the study of random clique complexes. These are:
(1) the vector of critical simplex counts attained by a lexicographical Morse matching,
(2) the vector of simplex counts in the link of a fixed simplex, and
(3) the vector of total simplex counts.
The first of these random vectors forms a cornerstone of modern homology algorithms, while the second one provides a natural generalisation for the notion of vertex degree, and the third one may be viewed from the perspective of U-statistics. To obtain distributional approximations for these random vectors, we extend the notion of dissociated sums to a multivariate setting and prove a new central limit theorem for such sums using Stein’s method
Homological Domination in Large Random Simplicial Complexes
In this paper we state the homological domination principle for random
multi-parameter simplicial complexes, claiming that the Betti number in one
specific dimension (which is explicitly determined by the probability
multi-parameter) significantly dominates the Betti numbers in all other
dimensions. We also state and discuss evidence for two interesting conjectures
which would imply a stronger version of the homological domination principle,
namely that generically homology of a random simplicial complex coincides with
that of a wedges of k-dimensional spheres. These two conjectures imply that
under an additional assumption (specified in the paper) a random simplicial
complex collapses to a k-dimensional complex homotopy equivalent to a wedge of
spheres of dimension k.Comment: 8 pages, 1 figur
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