Partially-observed network data collected by link-tracing based sampling methods is often being studied to obtain the characteristics of a large complex network. However, little attention has been paid to sampling from directed networks such as WWW and Peer-to-Peer networks. In this paper, we propose a novel two-step (sampling estimation) framework to measure nodal characteristics which can be defined by an average target function in an arbitrary directed network. To this end, we propose a personalized PageRank-based algorithm to visit and sample nodes. This algorithm only uses already visited nodes as local information without any prior knowledge about the latent structure of the network. Moreover, we introduce a new estimator based on the approximate importance sampling to estimate average target functions. The proposed estimator utilizes calculated PageRank value of each sampled node as an approximation for the exact visiting probability. To the best of our knowledge, this is the first study on correcting the bias of a sampling method by re-weighting of measured values that considers the effect of approximation of visiting probabilities. Comprehensive theoretical and empirical analysis of the estimator demonstrate that it is asymptotically unbiased even in situations where stationary distribution of PageRank is poorly approximated.
In general, the probability of selecting the next node determines the probability that nodes are visited in sampling procedure. Metropolis-Hastings technique @cite_2 can be used to modify the probabilities of the node selection in random walk in order to have the uniform stationary distribution for visiting each node. This technique is a general Markov Chain Monte Carlo (MCMC) method @cite_20 for sampling from probability distributions based on constructing a Markov Chain that has the desired distribution as its stationary distribution. This approach, known as Metropolis-Hastings Random Walk (MHRW), has been applied to Peer-to-Peer networks @cite_13 , Facebook @cite_10 , and Twitter @cite_15 .
@cite_13 @cite_2 @cite_15 @cite_10 @cite_20
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Abstract This article describes a new Metropolis-like transition rule, the multiple-try Metropolis, for Markov chain Monte Carlo (MCMC) simulations. By using this transition rule together with adaptive direction sampling, we propose a novel method for incorporating local optimization steps into a MCMC sampler in continuous state-space. Numerical studies show that the new method performs significantly better than the traditional Metropolis-Hastings (M-H) sampler. With minor tailoring in using the rule, the multiple-try method can also be exploited to achieve the effect of a griddy Gibbs sampler without having to bear with griddy approximations, and the effect of a hit-and-run algorithm without having to figure out the required conditional distribution in a random direction. Twitter as one of the most important microblogging online social networks has attracted more than 200 million users in recent years. Although there have been several attempts on characterizing the Twitter by using incomplete sampled data, they have not been very successful to estimate the characteristics of the whole network. In this paper, we characterize Twitter by sampling from its social graph and user behaviors through a random walk based sampling technique called Respondent-Driven Sampling (RDS). To the best of our knowledge, for the first time RDS method and its estimator are used in order to obtain uniform unbiased estimation of several key structural and behavioral properties of Twitter. We compared the performance of the proposed method with other sampling methods such as Metropolis-Hasting Random Walk (MHRW) and sampling from active users (Timeline) against the uniform sampling (UNI). In order to gather the required data, we have implemented four independent crawlers. Our experimental results indicate that the RDS method exhibits lower estimation errors to the sample in- and out-degree distribution compared to MHRW and Timeline. We also show that RDS is more suitable to sample the followers vs. followings ratio, and the correlation between followers followings vs. tweets. Perturbation theory for Markov chains addresses the question how small differences in the transitions of Markov chains are reflected in differences between their distributions. We prove powerful and flexible bounds on the distance of the @math th step distributions of two Markov chains when one of them satisfies a Wasserstein ergodicity condition. Our work is motivated by the recent interest in approximate Markov chain Monte Carlo (MCMC) methods in the analysis of big data sets. By using an approach based on Lyapunov functions, we provide estimates for geometrically ergodic Markov chains under weak assumptions. In an autoregressive model, our bounds cannot be improved in general. We illustrate our theory by showing quantitative estimates for approximate versions of two prominent MCMC algorithms, the Metropolis-Hastings and stochastic Langevin algorithms. We propose Adaptive Incremental Mixture Markov chain Monte Carlo (AIMM), a novel approach to sample from challenging probability distributions defined on a general state-space. While adaptive MCMC methods usually update a parametric proposal kernel with a global rule, AIMM locally adapts a semiparametric kernel. AIMM is based on an independent Metropolis-Hastings proposal distribution which takes the form of a finite mixture of Gaussian distributions. Central to this approach is the idea that the proposal distribution adapts to the target by locally adding a mixture component when the discrepancy between the proposal mixture and the target is deemed to be too large. As a result, the number of components in the mixture proposal is not fixed in advance. Theoretically, we prove that there exists a process that can be made arbitrarily close to AIMM and that converges to the correct target distribution. We also illustrate that it performs well in practice in a variety of challenging situations, including high-dimensional and multimodal target distributions. Tasks such as record linkage and multi-target tracking, which involve reconstructing the set of objects that underlie some observed data, are particularly challenging for probabilistic inference. Recent work has achieved efficient and accurate inference on such problems using Markov chain Monte Carlo (MCMC) techniques with customized proposal distributions. Currently, implementing such a system requires coding MCMC state representations and acceptance probability calculations that are specific to a particular application. An alternative approach, which we pursue in this paper, is to use a general-purpose probabilistic modeling language (such as BLOG) and a generic Metropolis-Hastings MCMC algorithm that supports user-supplied proposal distributions. Our algorithm gains flexibility by using MCMC states that are only partial descriptions of possible worlds; we provide conditions under which MCMC over partial worlds yields correct answers to queries. We also show how to use a context-specific Bayes net to identify the factors in the acceptance probability that need to be computed for a given proposed move. Experimental results on a citation matching task show that our general-purpose MCMC engine compares favorably with an application-specific system. We consider a symmetric random walk on a connected graph, where each edge is labeled with the probability of transition between the two adjacent vertices. The associated Markov chain has a uniform equilibrium distribution; the rate of convergence to this distribution, i.e., the mixing rate of the Markov chain, is determined by the second largest eigenvalue modulus (SLEM) of the transition probability matrix. In this paper we address the problem of assigning probabilities to the edges of the graph in such a way as to minimize the SLEM, i.e., the problem of finding the fastest mixing Markov chain on the graph. We show that this problem can be formulated as a convex optimization problem, which can in turn be expressed as a semidefinite program (SDP). This allows us to easily compute the (globally) fastest mixing Markov chain for any graph with a modest number of edges (say, @math ) using standard numerical methods for SDPs. Larger problems can be solved by exploiting various types of symmetry and structure in the problem, and far larger problems (say, 100,000 edges) can be solved using a subgradient method we describe. We compare the fastest mixing Markov chain to those obtained using two commonly used heuristics: the maximum-degree method, and the Metropolis--Hastings algorithm. For many of the examples considered, the fastest mixing Markov chain is substantially faster than those obtained using these heuristic methods. We derive the Lagrange dual of the fastest mixing Markov chain problem, which gives a sophisticated method for obtaining (arbitrarily good) bounds on the optimal mixing rate, as well as the optimality conditions. Finally, we describe various extensions of the method, including a solution of the problem of finding the fastest mixing reversible Markov chain, on a fixed graph, with a given equilibrium distribution. Preface 1. Events and probability 2. Discrete random variables and expectation 3. Moments and deviations 4. Chernoff bounds 5. Balls, bins and random graphs 6. The probabilistic method 7. Markov chains and random walks 8. Continuous distributions and the Poisson process 9. Entropy, randomness and information 10. The Monte Carlo method 11. Coupling of Markov chains 12. Martingales 13. Pairwise independence and universal hash functions 14. Balanced allocations References. With more than 250 million active users, Facebook (FB) is currently one of the most important online social networks. Our goal in this paper is to obtain a representative (unbiased) sample of Facebook users by crawling its social graph. In this quest, we consider and implement several candidate techniques. Two approaches that are found to perform well are the Metropolis-Hasting random walk (MHRW) and a re-weighted random walk (RWRW). Both have pros and cons, which we demonstrate through a comparison to each other as well as to the "ground-truth" (UNI - obtained through true uniform sampling of FB userIDs). In contrast, the traditional Breadth-First-Search (BFS) and Random Walk (RW) perform quite poorly, producing substantially biased results. In addition to offline performance assessment, we introduce online formal convergence diagnostics to assess sample quality during the data collection process. We show how these can be used to effectively determine when a random walk sample is of adequate size and quality for subsequent use (i.e., when it is safe to cease sampling). Using these methods, we collect the first, to the best of our knowledge, unbiased sample of Facebook. Finally, we use one of our representative datasets, collected through MHRW, to characterize several key properties of Facebook. Our goal in this paper is to develop a practical framework for obtaining a uniform sample of users in an online social network (OSN) by crawling its social graph. Such a sample allows to estimate any user property and some topological properties as well. To this end, first, we consider and compare several candidate crawling techniques. Two approaches that can produce approximately uniform samples are the Metropolis-Hasting random walk (MHRW) and a re-weighted random walk (RWRW). Both have pros and cons, which we demonstrate through a comparison to each other as well as to the "ground truth." In contrast, using Breadth-First-Search (BFS) or an unadjusted Random Walk (RW) leads to substantially biased results. Second, and in addition to offline performance assessment, we introduce online formal convergence diagnostics to assess sample quality during the data collection process. We show how these diagnostics can be used to effectively determine when a random walk sample is of adequate size and quality. Third, as a case study, we apply the above methods to Facebook and we collect the first, to the best of our knowledge, representative sample of Facebook users. We make it publicly available and employ it to characterize several key properties of Facebook. A proper choice of a proposal distribution for Markov chain Monte Carlo methods, for example for the Metropolis-Hastings algorithm, is well known to be a crucial factor for the convergence of the algorithm. In this paper we introduce an adaptive Metropolis (AM) algorithm, where the Gaussian proposal distribution is updated along the process using the full information cumulated so far. Due to the adaptive nature of the process, the AM algorithm is non-Markovian, but we establish here that it has the correct ergodic properties. We also include the results of our numerical tests, which indicate that the AM algorithm competes well with traditional Metropolis-Hastings algorithms, and demonstrate that the AM algorithm is easy to use in practical computation. In this paper, we introduce a novel, general purpose, technique for faster sampling of nodes over an online social network. Specifically, unlike traditional random walks which wait for the convergence of sampling distribution to a predetermined target distribution - a waiting process that incurs a high query cost - we develop WALK-ESTIMATE, which starts with a much shorter random walk, and then proactively estimate the sampling probability for the node taken before using acceptance-rejection sampling to adjust the sampling probability to the predetermined target distribution. We present a novel backward random walk technique which provides provably unbiased estimations for the sampling probability, and demonstrate the superiority of WALK-ESTIMATE over traditional random walks through theoretical analysis and extensive experiments over real world online social networks. Hamiltonian Monte Carlo (HMC) sampling methods provide a mechanism for defining distant proposals with high acceptance probabilities in a Metropolis-Hastings framework, enabling more efficient exploration of the state space than standard random-walk proposals. The popularity of such methods has grown significantly in recent years. However, a limitation of HMC methods is the required gradient computation for simulation of the Hamiltonian dynamical system--such computation is infeasible in problems involving a large sample size or streaming data. Instead, we must rely on a noisy gradient estimate computed from a subset of the data. In this paper, we explore the properties of such a stochastic gradient HMC approach. Surprisingly, the natural implementation of the stochastic approximation can be arbitrarily bad. To address this problem we introduce a variant that uses second-order Langevin dynamics with a friction term that counteracts the effects of the noisy gradient, maintaining the desired target distribution as the invariant distribution. Results on simulated data validate our theory. We also provide an application of our methods to a classification task using neural networks and to online Bayesian matrix factorization. Many online social networks feature restrictive web interfaces which only allow the query of a user's local neighborhood through the interface. To enable analytics over such an online social network through its restrictive web interface, many recent efforts reuse the existing Markov Chain Monte Carlo methods such as random walks to sample the social network and support analytics based on the samples. The problem with such an approach, however, is the large amount of queries often required (i.e., a long “mixing time”) for a random walk to reach a desired (stationary) sampling distribution. In this paper, we consider a novel problem of enabling a faster random walk over online social networks by “rewiring” the social network on-the-fly. Specifically, we develop Modified TOpology (MTO)-Sampler which, by using only information exposed by the restrictive web interface, constructs a “virtual” overlay topology of the social network while performing a random walk, and ensures that the random walk follows the modified overlay topology rather than the original one. We show that MTO-Sampler not only provably enhances the efficiency of sampling, but also achieves significant savings on query cost over real-world online social networks such as Google Plus, Epinion etc. Estimation of parameters of a diffusion based on discrete time observations poses a difficult problem due to the lack of a closed form expression for the likelihood. From a Bayesian computational perspective it can be casted as a missing data problem where the diffusion bridges in between discrete-time observations are missing. The computational problem can then be dealt with using a Markov-chain Monte-Carlo method known as data-augmentation. If unknown parameters appear in the diffusion coefficient, direct implementation of data-augmentation results in a Markov chain that is reducible. Furthermore, data-augmentation requires efficient sampling of diffusion bridges, which can be difficult, especially in the multidimensional case. We present a general framework to deal with with these problems that does not rely on discretisation. The construction generalises previous approaches and sheds light on the assumptions necessary to make these approaches work. We define a random-walk type Metropolis-Hastings sampler for updating diffusion bridges. Our methods are illustrated using guided proposals for sampling diffusion bridges. These are Markov processes obtained by adding a guiding term to the drift of the diffusion. We give general guidelines on the construction of these proposals and introduce a time change and scaling of the guided proposal that reduces discretisation error. Numerical examples demonstrate the performance of our methods. One of the most influential recent results in network analysis is that many natural networks exhibit a power-law or log-normal degree distribution. This has inspired numerous generative models that match this property. However, more recent work has shown that while these generative models do have the right degree distribution, they are not good models for real-life networks due to their differences on other important metrics like conductance. We believe this is, in part, because many of these real-world networks have very different joint degree distributions, that is, the probability that a randomly selected edge will be between nodes of degree k and l. Assortativity is a sufficient statistic of the joint degree distribution, and it has been previously noted that social networks tend to be assortative, while biological and technological networks tend to be disassortative. We suggest understanding the relationship between network structure and the joint degree distribution of graphs is an interesting avenue of further research. An important tool for such studies are algorithms that can generate random instances of graphs with the same joint degree distribution. This is the main topic of this article, and we study the problem from both a theoretical and practical perspective. We provide an algorithm for constructing simple graphs from a given joint degree distribution, and a Monte Carlo Markov chain method for sampling them. We also show that the state space of simple graphs with a fixed degree distribution is connected via endpoint switches. We empirically evaluate the mixing time of this Markov chain by using experiments based on the autocorrelation of each edge. These experiments show that our Markov chain mixes quickly on these real graphs, allowing for utilization of our techniques in practice. Estimating characteristics of large graphs via sampling is vital in the study of complex networks. In this work, we study the Mean Squared Error (MSE) associated with different sampling methods for the degree distribution. These sampling methods include independent random vertex (RV) and random edge (RE) sampling, and crawling methods such as random walks (RWs) and the widely used Metropolis-Hastings algorithm for uniformly sampling vertices (MHRWu). We see that the RW MSE is upper bounded by a quantity that is proportional to the RE MSE and inversely proportional to the spectral gap of the RW transition probability matrix. We also determine conditions under which RW is preferable to RV. Finally, we present an approximation of the MHRWu MSE. We evaluate the accuracy of our approximations and bounds through simulations on large real world graphs. Training energy-based probabilistic models is confronted with apparently intractable sums, whose Monte Carlo estimation requires sampling from the estimated probability distribution in the inner loop of training. This can be approximately achieved by Markov chain Monte Carlo methods, but may still face a formidable obstacle that is the difficulty of mixing between modes with sharp concentrations of probability. Whereas an MCMC process is usually derived from a given energy function based on mathematical considerations and requires an arbitrarily long time to obtain good and varied samples, we propose to train a deep directed generative model (not a Markov chain) so that its sampling distribution approximately matches the energy function that is being trained. Inspired by generative adversarial networks, the proposed framework involves training of two models that represent dual views of the estimated probability distribution: the energy function (mapping an input configuration to a scalar energy value) and the generator (mapping a noise vector to a generated configuration), both represented by deep neural networks. Abstract We provide a detailed, introductory exposition of the Metropolis-Hastings algorithm, a powerful Markov chain method to simulate multivariate distributions. A simple, intuitive derivation of this method is given along with guidance on implementation. Also discussed are two applications of the algorithm, one for implementing acceptance-rejection sampling when a blanketing function is not available and the other for implementing the algorithm with block-at-a-time scans. In the latter situation, many different algorithms, including the Gibbs sampler, are shown to be special cases of the Metropolis-Hastings algorithm. The methods are illustrated with examples. Statistical phylogenetic inference methods use tree rearrangement operations such as subtree–prune–regraft (SPR) to perform Markov chain Monte Carlo (MCMC) across tree topologies. The structure of the graph induced by tree rearrangement operations is an important determinant of the mixing properties of MCMC, motivating the study of the underlying SPR graph in greater detail. In this paper, we investigate the SPR graph of rooted trees (rSPR graph) in a new way: by calculating the Ricci–Ollivier curvature with respect to uniform and Metropolis–Hastings random walks. This value quantifies the degree to which a pair of random walkers from specified points move towards each other; negative curvature means that they move away from one another on average, while positive curvature means that they move towards each other. In order to calculate this curvature, we develop fast new algorithms for rSPR graph computation. We then develop formulas characterizing how the number of rSPR neighbors of a tree changes after an rSPR operation is applied to that tree. These give bounds on the curvature, as well as a flatness-in-the-limit theorem indicating that paths of small topology changes are easy to traverse. However, we find that large topology changes (i.e. moving a large subtree) give pairs of trees with negative curvature. We show using simulation that mean access time distributions depend on distance, degree, and curvature, demonstrating the relevance of these results to stochastic tree search. This chapter describes a sequence of Monte Carlo methods: importance sampling, rejection sampling, the Metropolis method, and Gibbs sampling. For each method, we discuss whether the method is expected to be useful for high—dimensional problems such as arise in inference with graphical models. After the methods have been described, the terminology of Markov chain Monte Carlo methods is presented. The chapter concludes with a discussion of advanced methods, including methods for reducing random walk behaviour.