1,558 research outputs found
Forman's Ricci curvature - From networks to hypernetworks
Networks and their higher order generalizations, such as hypernetworks or
multiplex networks are ever more popular models in the applied sciences.
However, methods developed for the study of their structural properties go
little beyond the common name and the heavy reliance of combinatorial tools. We
show that, in fact, a geometric unifying approach is possible, by viewing them
as polyhedral complexes endowed with a simple, yet, the powerful notion of
curvature - the Forman Ricci curvature. We systematically explore some aspects
related to the modeling of weighted and directed hypernetworks and present
expressive and natural choices involved in their definitions. A benefit of this
approach is a simple method of structure-preserving embedding of hypernetworks
in Euclidean N-space. Furthermore, we introduce a simple and efficient manner
of computing the well established Ollivier-Ricci curvature of a hypernetwork.Comment: to appear: Complex Networks '18 (oral presentation
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
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