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 @cite_9 , it is shown that when the sample size is relatively small, the RDS method may generate relatively large biases and errors if the studied networks are directed, indicating that estimates from previous RDS studies should be interpreted and generalized with caution. In @cite_40 , a random walk sampling algorithm with jumps, called Directed Unbiased Random Walk (DURW), is proposed that achieves asymptotically unbiased estimates of the outdegree distribution of a directed graph. The authors construct an undirected graph using the nodes that are selected by the random walker on the directed graph. The undirected graph is built to allow the walker to traverse known outgoing links backwards, and guarantees that the probability of sampling a node can be approximated, even though incoming links are not observed. However, this method is based on the assumption that nodes can be sampled uniformly at random from the original graph (i.e., there is some means of obtaining a list of nodes in the network) which is not always feasible. For example, while it is feasible for Wikipedia and Twitter, it is not feasible for the WWW graph.
@cite_40 @cite_9
2081082600 1547810917 2110117460 2168380307 2065222371 1630098887 2103799649 2336754337 1994312128 2147256592 2097938283 2139964991 2062833795 1823866642 2107648668 1540064387 2042359627 2005098805 2150516767 2124450885
Despite recent efforts to characterize complex networks such as citation graphs or online social networks (OSNs), little attention has been given to developing tools that can be used to characterize directed graphs in the wild, where no pre-processed data is available. The presence of hidden incoming edges but observable outgoing edges poses a challenge to characterize large directed graphs through crawling, as existing sampling methods cannot cope with hidden incoming links. The driving principle behind our random walk (RW) sampling method is to construct, in real-time, an undirected graph from the directed graph such that the random walk on the directed graph is consistent with one on the undirected graph. We then use the RW on the undirected graph to estimate the outdegree distribution. Our algorithm accurately estimates outdegree distributions of a variety of real world graphs. We also study the hardness of indegree distribution estimation when indegrees are latent (i.e., incoming links are only observed as outgoing edges). We observe that, in the same scenarios, indegree distribution estimates are highly innacurate unless the directed graph is highly symmetrical. Researchers in many scientific fields make inferences from individuals to larger groups. For many groups however, there is no list of members from which to take a random sample. Respondent-driven sampling (RDS) is a relatively new sampling methodology that circumvents this difficulty by using the social networks of the groups under study. The RDS method has been shown to provide unbiased estimates of population proportions given certain conditions. The method is now widely used in the study of HIV-related high-risk populations globally. In this paper, we test the RDS methodology by simulating RDS studies on the social networks of a large LGBT web community. The robustness of the RDS method is tested by violating, one by one, the conditions under which the method provides unbiased estimates. Results reveal that the risk of bias is large if networks are directed, or respondents choose to invite persons based on characteristics that are correlated with the study outcomes. If these two problems are absent, the RDS method shows strong resistance to low response rates and certain errors in the participants' reporting of their network sizes. Other issues that might affect the RDS estimates, such as the method for choosing initial participants, the maximum number of recruitments per participant, sampling with or without replacement and variations in network structures, are also simulated and discussed. Graph sampling via crawling has been actively considered as a generic and important tool for collecting uniform node samples so as to consistently estimate and uncover various characteristics of complex networks. The so-called simple random walk with re-weighting (SRW-rw) and Metropolis-Hastings (MH) algorithm have been popular in the literature for such unbiased graph sampling. However, an unavoidable downside of their core random walks -- slow diffusion over the space, can cause poor estimation accuracy. In this paper, we propose non-backtracking random walk with re-weighting (NBRW-rw) and MH algorithm with delayed acceptance (MHDA) which are theoretically guaranteed to achieve, at almost no additional cost, not only unbiased graph sampling but also higher efficiency (smaller asymptotic variance of the resulting unbiased estimators) than the SRW-rw and the MH algorithm, respectively. In particular, a remarkable feature of the MHDA is its applicability for any non-uniform node sampling like the MH algorithm, but ensuring better sampling efficiency than the MH algorithm. We also provide simulation results to confirm our theoretical findings. short-lived or high degree peers due to the dynamics of peer participation or the heterogeneity of peer degrees, respectively. This paper presents Respondent-Driven Sampling (RDS) as a promising technique for sampling unstructured P2P overlays. This allows one to accurately estimate the distribution of a desired peer property without capturing the entire overlay structure. RDS is a variant of snowball sampling that has been proposed and used in the social sciences to characterize hidden population in a society [9], [13]. We apply the RDS technique to unstructured P2P network and evaluate its performance over a wide range of static and dynamic graphs as well as a widely deployed P2P system. Throughout our evaluation, we compare and contrast the performance of the RDS technique with another sampling technique, namely Metropolized Random Walk (MRW), that we developed in our earlier work [16]. Our main findings can be summarized as follows: First, RDS outperforms MRW across all scenarios. In particular, RDS exhibits a significantly better performance than MRW when the overlay structure exhibits a combination of highly skewed node degrees and highly skewed (local) clustering coefficients. Second, our simulation and empirical evaluations reveal that both the RDS and MRW techniques can accurately estimate key peer properties over dynamic unstructured overlays. Third, our empirical evaluations suggest that the efficiency of the two sampling techniques in practice is lower than in our simulations involving synthetic graphs. We attribute this to our inability to capture accurate reference snapshots. The rest of the paper is organized as follows: Section II presents an overview of both the RDS and MRW techniques, and sketches our evaluation methodology. We examine both techniques over variety of static and dynamic graphs in Section III and IV, respectively. Section V presents the empirical evaluation of the two sampling techniques over Gnutella network. 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. A recurrent challenge for modern applications is the processing of large graphs. The ability to generate representative samples of smaller size is useful not only to circumvent scalability issues but also, per se, for statistical analysis and other data mining tasks. For such purposes adequate sampling techniques must be devised. We are interested, in this paper, in the uniform random sampling of a connected subgraph from a graph. We require that the sample contains a prescribed number of vertices. The sampled graph is the corresponding induced graph. We devise, present and discuss several algorithms that leverage three different techniques: Rejection Sampling, Random Walk and Markov Chain Monte Carlo. We empirically evaluate and compare the performance of the algorithms. We show that they are effective and efficient but that there is a trade-off, which depends on the density of the graphs and the sample size. We propose one novel algorithm, which we call Neighbour Reservoir Sampling (NRS), that very successfully realizes the trade-off between effectiveness and efficiency. Estimating characteristics of large graphs via sampling is a vital part of the study of complex networks. Current sampling methods such as (independent) random vertex and random walks are useful but have drawbacks. Random vertex sampling may require too many resources (time, bandwidth, or money). Random walks, which normally require fewer resources per sample, can suffer from large estimation errors in the presence of disconnected or loosely connected graphs. In this work we propose a new m-dimensional random walk that uses m dependent random walkers. We show that the proposed sampling method, which we call Frontier sampling, exhibits all of the nice sampling properties of a regular random walk. At the same time, our simulations over large real world graphs show that, in the presence of disconnected or loosely connected components, Frontier sampling exhibits lower estimation errors than regular random walks. We also show that Frontier sampling is more suitable than random vertex sampling to sample the tail of the degree distribution of the graph. Random walk is an important tool in many graph mining applications including estimating graph parameters, sampling portions of the graph, and extracting dense communities. In this paper we consider the problem of sampling nodes from a large graph according to a prescribed distribution by using random walk as the basic primitive. Our goal is to obtain algorithms that make a small number of queries to the graph but output a node that is sampled according to the prescribed distribution. Focusing on the uniform distribution case, we study the query complexity of three algorithms and show a near-tight bound expressed in terms of the parameters of the graph such as average degree and the mixing time. Both theoretically and empirically, we show that some algorithms are preferable in practice than the others. We also extend our study to the problem of sampling nodes according to some polynomial function of their degrees; this has implications for designing efficient algorithms for applications such as triangle counting. 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. 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. The adoption of everyday decisions in public affairs, fashion, movie-going, and consumer behavior is now thoroughly believed to migrate in a population through an influential network. The same diffusion process when being imitated by intention is called viral marketing. This process can be modeled by a (directed) graph G=(V,E) with a threshold t(v) for every vertex v?V, where v becomes active once at least t(v) of its (in-)neighbors are already active. A Perfect Target Set is a set of vertices whose activation will eventually activate the entire graph, and the Perfect Target Set Selection Problem (PTSS) asks for the minimum such initial set. It is known (Chen (2008) 6) that PTSS is hard to approximate, even for some special cases such as bounded-degree graphs, or majority thresholds.We propose a combinatorial model for this dynamic activation process, and use it to represent PTSS and its variants by linear integer programs. This allows one to use standard integer programming solvers for solving small-size PTSS instances. We also show combinatorial lower and upper bounds on the size of the minimum Perfect Target Set. Our upper bound implies that there are always Perfect Target Sets of size at most |V| 2 and 2|V| 3 under majority and strict majority thresholds, respectively, both in directed and undirected graphs. This improves the bounds of 0.727|V| and 0.7732|V| found recently by Chang and Lyuu (2010) 5 for majority and strict majority thresholds in directed graphs, and matches their bound under majority thresholds in undirected graphs. Furthermore, our proof is much simpler, and we observe that some of these bounds are tight. One interesting and perhaps surprising implication of our lower bound for undirected graphs, is that it is easy to get a constant factor approximation for PTSS for “relatively balanced” graphs (e.g., bounded-degree graphs, nearly regular graphs) with a “more than majority” threshold (that is, t(v)=??deg(v), for every v?V and some constant ?>1 2), whereas no polylogarithmic approximation exists for “more than majority” graphs. We consider a finite random walk on a weighted graph G; we show that the fraction of time spent in a set of vertices A converges to the stationary probability @math with error probability exponentially small in the length of the random walk and the square of the size of the deviation from @math . The exponential bound is in terms of the expansion of G and improves previous results of [D. Aldous, Probab. Engrg. Inform. Sci., 1 (1987), pp. 33--46], [L. Lovasz and M. Simonovits, Random Structures Algorithms , 4 (1993), pp. 359--412], [M. Ajtai, J. Komlos, and E. Szemeredi, Deterministic simulation of logspace, in Proc. 19th ACM Symp. on Theory of Computing, 1987]. We show that taking the sample average from one trajectory gives a more efficient estimate of @math than the standard method of generating independent sample points from several trajectories. Using this more efficient sampling method, we improve the algorithms of Jerrum and Sinclair for approximating the number of perfect matchings in a dense graph and for approximating the partition function of a ferromagnetic Ising system, and we give an efficient algorithm to estimate the entropy of a random walk on an unweighted graph. Sampling from large graphs is an area of great interest, especially since the emergence of huge structures such as Online Social Networks (OSNs) and the World Wide Web (WWW). These networks, when viewed as graphs, often contain hundreds of millions of vertices and billions of edges. The large scale properties of a network can be summarized in terms of parameters of the underlying graph, such as the total number of vertices, edges and triangles. The large size of these networks makes it computationally expensive to obtain such structural properties of the underlying graph by exhaustive search. If we can estimate these properties by taking small but representative samples from the network, then size is no longer such a problem. In this paper we present a general framework to estimate network properties using random walks. These methods work under the assumption we are able to obtain local characteristics of a vertex during each step of the random walk, for example the number and labels of the neighboring vertices of a specific vertex These assumptions are relatively reasonable in practice, but may add some additional query cost to each step of the random walk. We also present some practical methods to estimate the total number of edges links m, number of vertices nodes n and number of connected triads of vertices (triangles) t in graphs with degree distributions which follow a power-law and higher number of triangles higher than expected in random graphs. We use these graphs since they tend to better correspond to the structure of large online networks, and in fact some of the data used are taken from such a network. Additionally we present experimental estimates for n, m, t we obtained using our methods on real or manufactured networks. In order to make the methods practical, the total number of steps made by the walk was limited to at most the size n of the network. In fact the results appear to converge for a lower number of steps, indicating that our proposed methods are feasible in practice. 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. Considerable attention has been focused on the properties of graphs derived from Internet measurements. Router-level topologies collected via traceroute-like methods have led some to conclude that the router graph of the Internet is well modeled as a power-law random graph. In such a graph, the degree distribution of nodes follows a distribution with a power-law tail. We argue that the evidence to date for this conclusion is at best insufficient We show that when graphs are sampled using traceroute-like methods, the resulting degree distribution can differ sharply from that of the underlying graph. For example, given a sparse Erdos-Renyi random graph, the subgraph formed by a collection of shortest paths from a small set of random sources to a larger set of random destinations can exhibit a degree distribution remarkably like a power-law. We explore the reasons for how this effect arises, and show that in such a setting, edges are sampled in a highly biased manner. This insight allows us to formulate tests for determining when sampling bias is present. When we apply these tests to a number of well-known datasets, we find strong evidence for sampling bias. Mapping the Internet generally consists in sampling the network from a limited set of sources by using "traceroute"-like probes. This methodology, akin to the merging of different spanning trees to a set of destinations, has been argued to introduce uncontrolled sampling biases that might produce statistical properties of the sampled graph which sharply differ from the original ones. Here we explore these biases and provide a statistical analysis of their origin. We derive a mean-field analytical approximation for the probability of edge and vertex detection that exploits the role of the number of sources and targets and allows us to relate the global topological properties of the underlying network with the statistical accuracy of the sampled graph. In particular we find that the edge and vertex detection probability is depending on the betweenness centrality of each element. This allows us to show that shortest path routed sampling provides a better characterization of underlying graphs with scale-free topology. We complement the analytical discussion with a throughout numerical investigation of simulated mapping strategies in different network models. We show that sampled graphs provide a fair qualitative characterization of the statistical properties of the original networks in a fair range of different strategies and exploration parameters. The numerical study also allows the identification of intervals of the exploration parameters that optimize the fraction of nodes and edges discovered in the sampled graph. This finding might hint the steps toward more efficient mapping strategies. Breadth First Search (BFS) and other graph traversal techniques are widely used for measuring large unknown graphs, such as online social networks. It has been empirically observed that incomplete BFS is biased toward high degree nodes. In contrast to more studied sampling techniques, such as random walks, the bias of BFS has not been characterized to date. In this paper, we quantify the degree bias of BFS sampling. In particular, we calculate the node degree distribution expected to be observed by BFS as a function of the fraction of covered nodes, in a random graph RG(p k ) with a given (and arbitrary) degree distribution p k . Furthermore, we also show that, for RG(p k ), all commonly used graph traversal techniques (BFS, DFS, Forest Fire, and Snowball Sampling) lead to the same bias, and we show how to correct for this bias. To give a broader perspective, we compare this class of exploration techniques to random walks that are well-studied and easier to analyze. Next, we study by simulation the effect of graph properties not captured directly by our model. We find that the bias gets amplified in graphs with strong positive assortativity. Finally, we demonstrate the above results by sampling the Facebook social network, and we provide some practical guidelines for graph sampling in practice. Many recommendation and retrieval tasks can be represented as proximity queries on a labeled directed graph, with typed nodes representing documents, terms, and metadata, and labeled edges representing the relationships between them. Recent work has shown that the accuracy of the widely-used random-walk-based proximity measures can be improved by supervised learning - in particular, one especially effective learning technique is based on Path-Constrained Random Walks (PCRW), in which similarity is defined by a learned combination of constrained random walkers, each constrained to follow only a particular sequence of edge labels away from the query nodes. The PCRW based method significantly outperformed unsupervised random walk based queries, and models with learned edge weights. Unfortunately, PCRW query systems are expensive to evaluate. In this study we evaluate the use of approximations to the computation of the PCRW distributions, including fingerprinting, particle filtering, and truncation strategies. In experiments on several recommendation and retrieval problems using two large scientific publications corpora we show speedups of factors of 2 to 100 with little loss in accuracy. We use random sampling as a tool for solving undirected graph problems. We show that the sparse graph, or skeleton, that arises when we randomly sample a graph's edges will accurately approximate the value of all cuts in the original graph with high probability. This makes sampling effective for problems involving cuts in graphs. We present fast randomized (Monte Carlo and Las Vegas) algorithms for approximating and exactly finding minimum cuts and maximum flows in unweighted, undirected graphs. Our cut-approximation algorithms extend unchanged to weighted graphs while our weighted-graph flow algorithms are somewhat slower. Our approach gives a general paradigm with potential applications to any packing problem. It has since been used in a near-linear time algorithm for finding minimum cuts, as well as faster cut and flow algorithms. Our sampling theorems also yield faster algorithms for several other cut-based problems, including approximating the best balanced cut of a graph, finding a k-connected orientation of a 2k-connected graph, and finding integral multicommodity flows in graphs with a great deal of excess capacity. Our methods also improve the efficiency of some parallel cut and flow algorithms. Our methods also apply to the network design problem, where we wish to build a network satisfying certain connectivity requirements between vertices. We can purchase edges of various costs and wish to satisfy the requirements at minimum total cost. Since our sampling theorems apply even when the sampling probabilities are different for different edges, we can apply randomized rounding to solve network design problems. This gives approximation algorithms that guarantee much better approximations than previous algorithms whenever the minimum connectivity requirement is large. As a particular example, we improve the best approximation bound for the minimum k-connected subgraph problem from 1.85 to [math not displayed]. Sampling is a standard approach in big-graph analytics; the goal is to efficiently estimate the graph properties by consulting a sample of the whole population. A perfect sample is assumed to mirror every property of the whole population. Unfortunately, such a perfect sample is hard to collect in complex populations such as graphs (e.g. web graphs, social networks), where an underlying network connects the units of the population. Therefore, a good sample will be representative in the sense that graph properties of interest can be estimated with a known degree of accuracy. While previous work focused particularly on sampling schemes to estimate certain graph properties (e.g. triangle count), much less is known for the case when we need to estimate various graph properties with the same sampling scheme. In this paper, we pro- pose a generic stream sampling framework for big-graph analytics, called Graph Sample and Hold (gSH), which samples from massive graphs sequentially in a single pass, one edge at a time, while maintaining a small state in memory. We use a Horvitz-Thompson construction in conjunction with a scheme that samples arriving edges without adjacencies to previously sampled edges with probability p and holds edges with adjacencies with probability q. Our sample and hold framework facilitates the accurate estimation of subgraph patterns by enabling the dependence of the sampling process to vary based on previous history. Within our framework, we show how to produce statistically unbiased estimators for various graph properties from the sample. Given that the graph analytics will run on a sample instead of the whole population, the runtime complexity is kept under control. Moreover, given that the estimators are unbiased, the approximation error is also kept under control. Finally, we test the performance of the proposed framework (gSH) on various types of graphs, showing that from a sample with -- 40K edges, it produces estimates with relative errors