85 research outputs found
Emergence of a Giant Component in Random Site Subgraphs of a d-Dimensional Hamming Torus
The d-dimensional Hamming torus is the graph whose vertices are all of the
integer points inside an a_1 n X a_2 n X ... X a_d n box in R^d (for constants
a_1, ..., a_d > 0), and whose edges connect all vertices within Hamming
distance one. We study the size of the largest connected component of the
subgraph generated by independently removing each vertex of the Hamming torus
with probability 1-p. We show that if p=\lambda / n, then there exists
\lambda_c > 0, which is the positive root of a degree d polynomial whose
coefficients depend on a_1, ..., a_d, such that for \lambda < \lambda_c the
largest component has O(log n) vertices (a.a.s. as n \to \infty), and for
\lambda > \lambda_c the largest component has (1-q) \lambda (\prod_i a_i)
n^{d-1} + o(n^{d-1}) vertices and the second largest component has O(log n)
vertices (a.a.s.). An implicit formula for q < 1 is also given. Surprisingly,
the value of \lambda_c that we find is distinct from the critical value for the
emergence of a giant component in the random edge subgraph of the Hamming
torus. Additionally, we show that if p = c log n / n, then when c < (d-1) /
(\sum a_i) the site subgraph of the Hamming torus is not connected, and when c
> (d-1) / (\sum a_i) the subgraph is connected (a.a.s.). We also show that the
subgraph is connected precisely when it contains no isolated vertices.Comment: 37 pages, 1 figur
Nucleation scaling in jigsaw percolation
Jigsaw percolation is a nonlocal process that iteratively merges connected
clusters in a deterministic "puzzle graph" by using connectivity properties of
a random "people graph" on the same set of vertices. We presume the
Erdos--Renyi people graph with edge probability p and investigate the
probability that the puzzle is solved, that is, that the process eventually
produces a single cluster. In some generality, for puzzle graphs with N
vertices of degrees about D (in the appropriate sense), this probability is
close to 1 or small depending on whether pD(log N) is large or small. The one
dimensional ring and two dimensional torus puzzles are studied in more detail
and in many cases the exact scaling of the critical probability is obtained.
The paper settles several conjectures posed by Brummitt, Chatterjee, Dey, and
Sivakoff who introduced this model.Comment: 39 pages, 3 figures. Moved main results to the introduction and
improved exposition of section
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
- …