90 research outputs found
Rectilinear approximation and volume estimates for hereditary bodies via [0,1]-decorated containers
We use the hypergraph container theory of Balogh--Morris--Samotij and
Saxton--Thomason to obtain general rectilinear approximations and volume
estimates for sequences of bodies closed under certain families of projections.
We give a number of applications of our results, including a multicolour
generalisation of a theorem of Hatami, Janson and Szegedy on the entropy of
graph limits. Finally, we raise a number of questions on geometric and analytic
approaches to containers.Comment: 25 pages, author accepted manuscript, to appear in Journal of Graph
Theor
Sampling random graph homomorphisms and applications to network data analysis
A graph homomorphism is a map between two graphs that preserves adjacency
relations. We consider the problem of sampling a random graph homomorphism from
a graph into a large network . We propose two complementary
MCMC algorithms for sampling a random graph homomorphisms and establish bounds
on their mixing times and concentration of their time averages. Based on our
sampling algorithms, we propose a novel framework for network data analysis
that circumvents some of the drawbacks in methods based on independent and
neigborhood sampling. Various time averages of the MCMC trajectory give us
various computable observables, including well-known ones such as homomorphism
density and average clustering coefficient and their generalizations.
Furthermore, we show that these network observables are stable with respect to
a suitably renormalized cut distance between networks. We provide various
examples and simulations demonstrating our framework through synthetic
networks. We also apply our framework for network clustering and classification
problems using the Facebook100 dataset and Word Adjacency Networks of a set of
classic novels.Comment: 51 pages, 33 figures, 2 table
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