18,339 research outputs found
Generalizations of the Kolmogorov-Barzdin embedding estimates
We consider several ways to measure the `geometric complexity' of an
embedding from a simplicial complex into Euclidean space. One of these is a
version of `thickness', based on a paper of Kolmogorov and Barzdin. We prove
inequalities relating the thickness and the number of simplices in the
simplicial complex, generalizing an estimate that Kolmogorov and Barzdin proved
for graphs. We also consider the distortion of knots. We give an alternate
proof of a theorem of Pardon that there are isotopy classes of knots requiring
arbitrarily large distortion. This proof is based on the expander-like
properties of arithmetic hyperbolic manifolds.Comment: 45 page
SPIDA: Abstracting and generalizing layout design cases
Abstraction and generalization of layout design cases generate new knowledge that is more widely applicable to use than specific design cases. The abstraction and generalization of design cases into hierarchical levels of abstractions provide the designer with the flexibility to apply any level of abstract and generalized knowledge for a new layout design problem. Existing case-based layout learning (CBLL) systems abstract and generalize cases into single levels of abstractions, but not into a hierarchy. In this paper, we propose a new approach, termed customized viewpoint - spatial (CV-S), which supports the generalization and abstraction of spatial layouts into hierarchies along with a supporting system, SPIDA (SPatial Intelligent Design Assistant)
Asymptotic Delsarte cliques in distance-regular graphs
We give a new bound on the parameter (number of common neighbors of
a pair of adjacent vertices) in a distance-regular graph , improving and
generalizing bounds for strongly regular graphs by Spielman (1996) and Pyber
(2014). The new bound is one of the ingredients of recent progress on the
complexity of testing isomorphism of strongly regular graphs (Babai, Chen, Sun,
Teng, Wilmes 2013). The proof is based on a clique geometry found by Metsch
(1991) under certain constraints on the parameters. We also give a simplified
proof of the following asymptotic consequence of Metsch's result: if then each edge of belongs to a unique maximal clique of size
asymptotically equal to , and all other cliques have size
. Here denotes the degree and the number of common
neighbors of a pair of vertices at distance 2. We point out that Metsch's
cliques are "asymptotically Delsarte" when , so families
of distance-regular graphs with parameters satisfying are
"asymptotically Delsarte-geometric."Comment: 10 page
Random Graph Models with Hidden Color
We demonstrate how to generalize two of the most well-known random graph
models, the classic random graph, and random graphs with a given degree
distribution, by the introduction of hidden variables in the form of extra
degrees of freedom, color, applied to vertices or stubs (half-edges). The color
is assumed unobservable, but is allowed to affect edge probabilities. This
serves as a convenient method to define very general classes of models within a
common unifying formalism, and allowing for a non-trivial edge correlation
structure.Comment: 17 pages, 2 figures; contrib. to the Workshop on Random Geometry in
Krakow, May 200
Ollivier-Ricci Curvature for Hypergraphs: A Unified Framework
Bridging geometry and topology, curvature is a powerful and expressiveinvariant. While the utility of curvature has been theoretically andempirically confirmed in the context of manifolds and graphs, itsgeneralization to the emerging domain of hypergraphs has remained largelyunexplored. On graphs, Ollivier-Ricci curvature measures differences betweenrandom walks via Wasserstein distances, thus grounding a geometric concept inideas from probability and optimal transport. We develop ORCHID, a flexibleframework generalizing Ollivier-Ricci curvature to hypergraphs, and prove thatthe resulting curvatures have favorable theoretical properties. Throughextensive experiments on synthetic and real-world hypergraphs from differentdomains, we demonstrate that ORCHID curvatures are both scalable and useful toperform a variety of hypergraph tasks in practice.<br
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