text: 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.
  @cite_13: This paper addresses the difficult problem of selecting representative samples of peer properties (eg degree, link bandwidth, number of files shared) in unstructured peer-to-peer systems. Due to the large size and dynamic nature of these systems, measuring the quantities of interest on every peer is often prohibitively expensive, while sampling provides a natural means for estimating system-wide behavior efficiently. However, commonly-used sampling techniques for measuring peer-to-peer systems tend to introduce considerable bias for two reasons. First, the dynamic nature of peers can bias results towards short-lived peers, much as naively sampling flows in a router can lead to bias towards short-lived flows. Second, the heterogeneous nature of the overlay topology can lead to bias towards high-degree peers.We present a detailed examination of the ways that the behavior of peer-to-peer systems can introduce bias and suggest the Metropolized Random Walk with Backtracking (MRWB) as a viable and promising technique for collecting nearly unbiased samples. We conduct an extensive simulation study to demonstrate that the proposed technique works well for a wide variety of common peer-to-peer network conditions. Using the Gnutella network, we empirically show that our implementation of the MRWB technique yields more accurate samples than relying on commonly-used sampling techniques. Furthermore, we provide insights into the causes of the observed differences. The tool we have developed, ion-sampler, selects peer addresses uniformly at random using the MRWB technique. These addresses may then be used as input to another measurement tool to collect data on a particular property.
  @cite_38: Partially-observed data collected by sampling methods is often being studied to obtain the characteristics of information diffusion networks. However, these methods usually do not consider the behavior of diffusion process. In this paper, we propose a novel two-step (sampling estimation) measurement framework by utilizing the diffusion process characteristics. To this end, we propose a link-tracing based sampling design which uses the infection times as local information without any knowledge about the latent structure of diffusion network. To correct the bias of sampled data, we introduce three estimators for different categories, link-based, node-based, and cascade-based. To the best of our knowledge, this is the first attempt to introduce a complete measurement framework for diffusion networks. We also show that the estimator plays an important role in correcting the bias of sampling from diffusion networks. Our comprehensive empirical analysis over large synthetic and real datasets demonstrates that in average, the proposed framework outperforms the common BFS and RW sampling methods in terms of link-based characteristics by about 37 and 35 , respectively.
  @cite_36: 
  @cite_21: 
  @cite_24: We consider the problem of sampling URLs uniformly at random from the Web. A tool for sampling URLs uniformly can be used to estimate various properties of Web pages, such as the fraction of pages in various Internet domains or written in various languages. Moreover, uniform URL sampling can be used to determine the sizes of various search engines relative to the entire Web. In this paper, we consider sampling approaches based on random walks of the Web graph. In particular, we suggest ways of improving sampling based on random walks to make the samples closer to uniform. We suggest a natural test bed based on random graphs for testing the effectiveness of our procedures. We then use our sampling approach to estimate the distribution of pages over various Internet domains and to estimate the coverage of various search engine indexes.
  @cite_10: 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.
  @cite_25: Various aspects of the theory of random walks on graphs are surveyed. In particular, estimates on the important parameters of access time, commute time, cover time and mixing time are discussed. Connections with the eigenvalues of graphs and with electrical networks, and the use of these connections in the study of random walks is described. We also sketch recent algorithmic applications of random walks, in particular to the problem of sampling.
summary: Link-tracing based sampling methods in the context of networks are somewhat distinct from classical sampling methods @cite_36 . Random walk (RW) @cite_25 is one of the most important and widely used sampling methods in different kind of network contexts such as uniformly sampling Web pages from the Internet @cite_24 , content density in Peer-to-Peer networks @cite_13 , degree distributions of the Facebook social graph @cite_10 , and collecting data from information diffusion networks @cite_38 @cite_21 . A classic RW, samples a graph by moving from a node, @math , to a neighboring node, @math , through an outgoing link, chosen uniformly at random from the neighbors of @math . By this process links and nodes are sampled. In any given connected and non-bipartite undirected graph, the classic RW is biased towards node with higher degree.