3,588 research outputs found
The threshold for combs in random graphs
For let denote the tree consisting of an
-vertex path with disjoint -vertex paths beginning at each of its
vertices. An old conjecture says that for any the threshold for the
random graph to contain is at .
Here we verify this for with any fixed . In a companion
paper, using very different methods, we treat the complementary range, proving
the conjecture for (with ).Comment: 9 page
Sharp threshold for embedding combs and other spanning trees in random graphs
When , the tree consists of a path containing
vertices, each of whose vertices has a disjoint path length
beginning at it. We show that, for any and , the binomial
random graph almost surely contains
as a subgraph. This improves a recent result of Kahn,
Lubetzky and Wormald. We prove a similar statement for a more general class of
trees containing both these combs and all bounded degree spanning trees which
have at least disjoint bare paths length .
We also give an efficient method for finding large expander subgraphs in a
binomial random graph. This allows us to improve a result on almost spanning
trees by Balogh, Csaba, Pei and Samotij.Comment: 20 page
Slow Encounters of Particle Pairs in Branched Structures
On infinite homogeneous structures, two random walkers meet with certainty if
and only if the structure is recurrent, i.e., a single random walker returns to
its starting point with probability 1. However, on general inhomogeneous
structures this property does not hold and, although a single random walker
will certainly return to its starting point, two moving particles may never
meet. This striking property has been shown to hold, for instance, on infinite
combs. Due to the huge variety of natural phenomena which can be modeled in
terms of encounters between two (or more) particles diffusing in comb-like
structures, it is fundamental to investigate if and, if so, to what extent
similar effects may take place in finite structures. By means of numerical
simulations we evidence that, indeed, even on finite structures, the
topological inhomogeneity can qualitatively affect the two-particle problem. In
particular, the mean encounter time can be polynomially larger than the time
expected from the related one particle problem.Comment: 8 pages, 12 figures; accepted for publication in Physical Review
Cycle factors and renewal theory
For which values of does a uniformly chosen -regular graph on
vertices typically contain vertex-disjoint -cycles (a -cycle
factor)? To date, this has been answered for and for ; the
former, the Hamiltonicity problem, was finally answered in the affirmative by
Robinson and Wormald in 1992, while the answer in the latter case is negative
since with high probability most vertices do not lie on -cycles.
Here we settle the problem completely: the threshold for a -cycle factor
in as above is with . Precisely, we prove a 2-point concentration result: if divides then contains a -cycle factor
w.h.p., whereas if then w.h.p. it
does not. As a byproduct, we confirm the "Comb Conjecture," an old problem
concerning the embedding of certain spanning trees in the random graph
.
The proof follows the small subgraph conditioning framework, but the
associated second moment analysis here is far more delicate than in any earlier
use of this method and involves several novel features, among them a sharp
estimate for tail probabilities in renewal processes without replacement which
may be of independent interest.Comment: 45 page
Network of Time-Multiplexed Optical Parametric Oscillators as a Coherent Ising Machine
Finding the ground states of the Ising Hamiltonian [1] maps to various
combinatorial optimization problems in biology, medicine, wireless
communications, artificial intelligence, and social network. So far no
efficient classical and quantum algorithm is known for these problems, and
intensive research is focused on creating physical systems - Ising machines -
capable of finding the absolute or approximate ground states of the Ising
Hamiltonian [2-6]. Here we report a novel Ising machine using a network of
degenerate optical parametric oscillators (OPOs). Spins are represented with
above-threshold binary phases of the OPOs and the Ising couplings are realized
by mutual injections [7]. The network is implemented in a single OPO ring
cavity with multiple trains of femtosecond pulses and configurable mutual
couplings, and operates at room temperature. We programed the smallest
non-deterministic polynomial time (NP)- hard Ising problem on the machine, and
in 1000 runs of the machine no computational error was detected
Distantly Labeling Data for Large Scale Cross-Document Coreference
Cross-document coreference, the problem of resolving entity mentions across
multi-document collections, is crucial to automated knowledge base construction
and data mining tasks. However, the scarcity of large labeled data sets has
hindered supervised machine learning research for this task. In this paper we
develop and demonstrate an approach based on ``distantly-labeling'' a data set
from which we can train a discriminative cross-document coreference model. In
particular we build a dataset of more than a million people mentions extracted
from 3.5 years of New York Times articles, leverage Wikipedia for distant
labeling with a generative model (and measure the reliability of such
labeling); then we train and evaluate a conditional random field coreference
model that has factors on cross-document entities as well as mention-pairs.
This coreference model obtains high accuracy in resolving mentions and entities
that are not present in the training data, indicating applicability to
non-Wikipedia data. Given the large amount of data, our work is also an
exercise demonstrating the scalability of our approach.Comment: 16 pages, submitted to ECML 201
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