W911NF-12-R-0012-03: 3.0 Mathematical Sciences
Abstract
The project aims to answer the following fundamental question: What structural properties of a given sparse graph or simplicial complex ensure that this complex is geometric, meaning that there exists a latent space such that the complex is a typical element in the ensemble of sparse random geometric simplicial complexes in this space? Three main tasks. The first task uses methods from maximum-entropy theory of random graphs, calculus of variations, and functional analysis to identify graph properties that guarantee the existence of a solution to a variational problem maximizing the expected entropy density in the ensemble of soft random geometric graphs on the real line. This entropy is a functional that depends onthe connection probability function, also known as a graphon. The graphon in soft random geometric graphs is the probability of existence of an edge between a pair of vertices located at a given distance on the real line. The second task uses the graph properties identified in the first task, as well as methods from the exponential random graph formalism and the theory of large deviations, to derive the explicit form of the entropy-maximizing graphon. The third task relies on recent results on ensembles of random geometric graphs in non-Euclidean Riemannian and Lorentzian manifolds, and maximum-entropy models of random simplicial complexes, to extend the results of the first two tasks to broader and more realistic models of random geometric graphs and simplicial complexes with non-Poisson distributions of (generalized) degrees.
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- Jan 12, 2017
- Source ID
- W911NF1610391
Entities
People
- Dima Krioukov
Organizations
- Army Contracting Command
- Northeastern University
- United States Army