14,311 research outputs found

    Multi-dimensional Boltzmann Sampling of Languages

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    This paper addresses the uniform random generation of words from a context-free language (over an alphabet of size kk), 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 [(1−ε)n,(1+ε)n][(1-\varepsilon)n, (1+\varepsilon)n] and exact frequency in O(n1+k/2)\mathcal{O}(n^{1+k/2}) expected time. Moreover, if we accept tolerance intervals of width in Ω(n)\Omega(\sqrt{n}) for the number of occurrences of each letters, our samplers perform an approximate-size generation of words in expected O(n)\mathcal{O}(n) 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

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    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 O(m)O(m) random numbers to generate a graph with mm 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

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    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

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    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

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    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 nn and a targeted distribution in kk of its \emph{distinguished} atoms. We consider two variations on this problem. In the first alternative, the targeted distribution is given by kk real numbers \TargFreq_1, \ldots, \TargFreq_k such that 0 < \TargFreq_i < 1 for all ii and \TargFreq_1+\cdots+\TargFreq_k \leq 1. We aim to generate random structures among the whole set of structures of a given size nn, 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 kk-tuple \Weights of real-valued weights, inducing a weighted distribution over the set of structures of size nn. 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 mm 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 nn. 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 kk-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 kk natural numbers n1,…,nkn_1, \ldots, n_k such that n1+⋯+nk+r=nn_1+\cdots+n_k+r=n where r≥0r \geq 0 is the number of undistinguished atoms. The structures must be generated uniformly among the set of structures of size nn that contain {\em exactly} nin_i atoms \At_i (1≤i≤k1 \leq i \leq k). We give a \bigO{r^2\prod_{i=1}^k n_i^2 +m n k \log n} algorithm for generating mm structures, which simplifies into a \bigO{r\prod_{i=1}^k n_i +m n} for regular specifications

    {HyGen}: {G}enerating Random Graphs with Hyperbolic Communities

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    Critical Points for Random Boolean Networks

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    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

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    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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