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
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. 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. 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. 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.