6 research outputs found
Linear Programming Bounds for Randomly Sampling Colorings
Here we study the problem of sampling random proper colorings of a bounded
degree graph. Let be the number of colors and let be the maximum
degree. In 1999, Vigoda showed that the Glauber dynamics is rapidly mixing for
any . It turns out that there is a natural barrier at
, below which there is no one-step coupling that is contractive,
even for the flip dynamics.
We use linear programming and duality arguments to guide our construction of
a better coupling. We fully characterize the obstructions to going beyond
. These examples turn out to be quite brittle, and even starting
from one, they are likely to break apart before the flip dynamics changes the
distance between two neighboring colorings. We use this intuition to design a
variable length coupling that shows that the Glauber dynamics is rapidly mixing
for any where . This is the first improvement to Vigoda's analysis that
holds for general graphs.Comment: 30 pages, 3 figures; fixed some typo
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
Notes on Randomized Algorithms
Lecture notes for the Yale Computer Science course CPSC 469/569 Randomized
Algorithms. Suitable for use as a supplementary text for an introductory
graduate or advanced undergraduate course on randomized algorithms. Discusses
tools from probability theory, including random variables and expectations,
union bound arguments, concentration bounds, applications of martingales and
Markov chains, and the Lov\'asz Local Lemma. Algorithmic topics include
analysis of classic randomized algorithms such as Quicksort and Hoare's FIND,
randomized tree data structures, hashing, Markov chain Monte Carlo sampling,
randomized approximate counting, derandomization, quantum computing, and some
examples of randomized distributed algorithms