7,187 research outputs found
Approximating the monomer-dimer constants through matrix permanent
The monomer-dimer model is fundamental in statistical mechanics. However, it
is #P-complete in computation, even for two dimensional problems. A
formulation in matrix permanent for the partition function of the monomer-dimer
model is proposed in this paper, by transforming the number of all matchings of
a bipartite graph into the number of perfect matchings of an extended bipartite
graph, which can be given by a matrix permanent. Sequential importance sampling
algorithm is applied to compute the permanents. For two-dimensional lattice
with periodic condition, we obtain , where the exact value is
. For three-dimensional lattice with periodic condition,
our numerical result is , {which agrees with the best known
bound .}Comment: 6 pages, 2 figure
Recommendation Subgraphs for Web Discovery
Recommendations are central to the utility of many websites including
YouTube, Quora as well as popular e-commerce stores. Such sites typically
contain a set of recommendations on every product page that enables visitors to
easily navigate the website. Choosing an appropriate set of recommendations at
each page is one of the key features of backend engines that have been deployed
at several e-commerce sites.
Specifically at BloomReach, an engine consisting of several independent
components analyzes and optimizes its clients' websites. This paper focuses on
the structure optimizer component which improves the website navigation
experience that enables the discovery of novel content.
We begin by formalizing the concept of recommendations used for discovery. We
formulate this as a natural graph optimization problem which in its simplest
case, reduces to a bipartite matching problem. In practice, solving these
matching problems requires superlinear time and is not scalable. Also,
implementing simple algorithms is critical in practice because they are
significantly easier to maintain in production. This motivated us to analyze
three methods for solving the problem in increasing order of sophistication: a
sampling algorithm, a greedy algorithm and a more involved partitioning based
algorithm.
We first theoretically analyze the performance of these three methods on
random graph models characterizing when each method will yield a solution of
sufficient quality and the parameter ranges when more sophistication is needed.
We complement this by providing an empirical analysis of these algorithms on
simulated and real-world production data. Our results confirm that it is not
always necessary to implement complicated algorithms in the real-world and that
very good practical results can be obtained by using heuristics that are backed
by the confidence of concrete theoretical guarantees
Construction of and efficient sampling from the simplicial configuration model
Simplicial complexes are now a popular alternative to networks when it comes
to describing the structure of complex systems, primarily because they encode
multi-node interactions explicitly. With this new description comes the need
for principled null models that allow for easy comparison with empirical data.
We propose a natural candidate, the simplicial configuration model. The core of
our contribution is an efficient and uniform Markov chain Monte Carlo sampler
for this model. We demonstrate its usefulness in a short case study by
investigating the topology of three real systems and their randomized
counterparts (using their Betti numbers). For two out of three systems, the
model allows us to reject the hypothesis that there is no organization beyond
the local scale.Comment: 6 pages, 4 figure
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