14,311 research outputs found
Multi-dimensional Boltzmann Sampling of Languages
This paper addresses the uniform random generation of words from a
context-free language (over an alphabet of size ), while constraining every
letter to a targeted frequency of occurrence. Our approach consists in a
multidimensional extension of Boltzmann samplers \cite{Duchon2004}. We show
that, under mostly \emph{strong-connectivity} hypotheses, our samplers return a
word of size in and exact frequency in
expected time. Moreover, if we accept tolerance
intervals of width in for the number of occurrences of each
letters, our samplers perform an approximate-size generation of words in
expected time. We illustrate these techniques on the
generation of Tetris tessellations with uniform statistics in the different
types of tetraminoes.Comment: 12p
A Scalable Null Model for Directed Graphs Matching All Degree Distributions: In, Out, and Reciprocal
Degree distributions are arguably the most important property of real world
networks. The classic edge configuration model or Chung-Lu model can generate
an undirected graph with any desired degree distribution. This serves as a good
null model to compare algorithms or perform experimental studies. Furthermore,
there are scalable algorithms that implement these models and they are
invaluable in the study of graphs. However, networks in the real-world are
often directed, and have a significant proportion of reciprocal edges. A
stronger relation exists between two nodes when they each point to one another
(reciprocal edge) as compared to when only one points to the other (one-way
edge). Despite their importance, reciprocal edges have been disregarded by most
directed graph models.
We propose a null model for directed graphs inspired by the Chung-Lu model
that matches the in-, out-, and reciprocal-degree distributions of the real
graphs. Our algorithm is scalable and requires random numbers to
generate a graph with edges. We perform a series of experiments on real
datasets and compare with existing graph models.Comment: Camera ready version for IEEE Workshop on Network Science; fixed some
typos in tabl
Loom: Query-aware Partitioning of Online Graphs
As with general graph processing systems, partitioning data over a cluster of
machines improves the scalability of graph database management systems.
However, these systems will incur additional network cost during the execution
of a query workload, due to inter-partition traversals. Workload-agnostic
partitioning algorithms typically minimise the likelihood of any edge crossing
partition boundaries. However, these partitioners are sub-optimal with respect
to many workloads, especially queries, which may require more frequent
traversal of specific subsets of inter-partition edges. Furthermore, they
largely unsuited to operating incrementally on dynamic, growing graphs.
We present a new graph partitioning algorithm, Loom, that operates on a
stream of graph updates and continuously allocates the new vertices and edges
to partitions, taking into account a query workload of graph pattern
expressions along with their relative frequencies.
First we capture the most common patterns of edge traversals which occur when
executing queries. We then compare sub-graphs, which present themselves
incrementally in the graph update stream, against these common patterns.
Finally we attempt to allocate each match to single partitions, reducing the
number of inter-partition edges within frequently traversed sub-graphs and
improving average query performance.
Loom is extensively evaluated over several large test graphs with realistic
query workloads and various orderings of the graph updates. We demonstrate
that, given a workload, our prototype produces partitionings of significantly
better quality than existing streaming graph partitioning algorithms Fennel and
LDG
Quenched Random Graphs
Spin models on quenched random graphs are related to many important
optimization problems. We give a new derivation of their mean-field equations
that elucidates the role of the natural order parameter in these models.Comment: 9 pages, report CPTH-A264.109
Controlled non uniform random generation of decomposable structures
Consider a class of decomposable combinatorial structures, using different
types of atoms \Atoms = \{\At_1,\ldots ,\At_{|{\Atoms}|}\}. We address the
random generation of such structures with respect to a size and a targeted
distribution in of its \emph{distinguished} atoms. We consider two
variations on this problem. In the first alternative, the targeted distribution
is given by real numbers \TargFreq_1, \ldots, \TargFreq_k such that 0 <
\TargFreq_i < 1 for all and \TargFreq_1+\cdots+\TargFreq_k \leq 1. We
aim to generate random structures among the whole set of structures of a given
size , in such a way that the {\em expected} frequency of any distinguished
atom \At_i equals \TargFreq_i. We address this problem by weighting the
atoms with a -tuple \Weights of real-valued weights, inducing a weighted
distribution over the set of structures of size . We first adapt the
classical recursive random generation scheme into an algorithm taking
\bigO{n^{1+o(1)}+mn\log{n}} arithmetic operations to draw structures from
the \Weights-weighted distribution. Secondly, we address the analytical
computation of weights such that the targeted frequencies are achieved
asymptotically, i. e. for large values of . We derive systems of functional
equations whose resolution gives an explicit relationship between \Weights
and \TargFreq_1, \ldots, \TargFreq_k. Lastly, we give an algorithm in
\bigO{k n^4} for the inverse problem, {\it i.e.} computing the frequencies
associated with a given -tuple \Weights of weights, and an optimized
version in \bigO{k n^2} in the case of context-free languages. This allows
for a heuristic resolution of the weights/frequencies relationship suitable for
complex specifications. In the second alternative, the targeted distribution is
given by a natural numbers such that
where is the number of undistinguished atoms.
The structures must be generated uniformly among the set of structures of size
that contain {\em exactly} atoms \At_i (). We give
a \bigO{r^2\prod_{i=1}^k n_i^2 +m n k \log n} algorithm for generating
structures, which simplifies into a \bigO{r\prod_{i=1}^k n_i +m n} for
regular specifications
Critical Points for Random Boolean Networks
A model of cellular metabolism due to S. Kauffman is analyzed. It consists of
a network of Boolean gates randomly assembled according to a probability
distribution. It is shown that the behavior of the network depends very
critically on certain simple algebraic parameters of the distribution. In some
cases, the analytic results support conclusions based on simulations of random
Boolean networks, but in other cases, they do not.Comment: 19 page
Bayesian stochastic blockmodeling
This chapter provides a self-contained introduction to the use of Bayesian
inference to extract large-scale modular structures from network data, based on
the stochastic blockmodel (SBM), as well as its degree-corrected and
overlapping generalizations. We focus on nonparametric formulations that allow
their inference in a manner that prevents overfitting, and enables model
selection. We discuss aspects of the choice of priors, in particular how to
avoid underfitting via increased Bayesian hierarchies, and we contrast the task
of sampling network partitions from the posterior distribution with finding the
single point estimate that maximizes it, while describing efficient algorithms
to perform either one. We also show how inferring the SBM can be used to
predict missing and spurious links, and shed light on the fundamental
limitations of the detectability of modular structures in networks.Comment: 44 pages, 16 figures. Code is freely available as part of graph-tool
at https://graph-tool.skewed.de . See also the HOWTO at
https://graph-tool.skewed.de/static/doc/demos/inference/inference.htm
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