Most real world dynamic networks are evolved very fast with time. It is not feasible to collect the entire network at any given time to study its characteristics. This creates the need to propose local algorithms to study various properties of the network. In the present work, we estimate degree rank of a node without having the entire network. The proposed methods are based on the power law degree distribution characteristic or sampling techniques. The proposed methods are simulated on synthetic networks, as well as on real world social networks. The efficiency of the proposed methods is evaluated using absolute and weighted error functions. Results show that the degree rank of a node can be estimated with high accuracy using only @math samples of the network size. The accuracy of the estimation decreases from high ranked to low ranked nodes. We further extend the proposed methods for random networks and validate their efficiency on synthetic random networks, that are generated using Erdős-Renyi model. Results show that the proposed methods can be efficiently used for random networks as well.
The node or edge sampling methods are not feasible in real world networks as the structure of social networks is not known in advance. So, these networks can be sampled using graph traversal techniques like breadth first search (BFS) @cite_19 , depth first search (DFS) @cite_19 , forest fire sampling (FFS) @cite_32 , snowball sampling @cite_10 , or random walk based methods like simple random walk (RW) @cite_54 , Metropolis Hastings random walk (MHRW) @cite_48 , reweighted random walk (RWRW) @cite_36 , respondent driven sampling (RDS) @cite_14 , supervised random walk @cite_25 , Modified TOpology Sampling (MTO) @cite_69 , walk-estimate @cite_35 , Frontier sampling (m-dimensional random walk) @cite_64 , Rank Degree sampling based on edge selection @cite_46 , preferential random walk @cite_42 , and so on.
@cite_35 @cite_69 @cite_14 @cite_64 @cite_36 @cite_48 @cite_54 @cite_42 @cite_32 @cite_19 @cite_46 @cite_10 @cite_25
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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. 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. Recently, there has been a great deal of research focusing on the development of sampling algorithms for networks with small-world and or power-law structure. The peerto-peer research community (e.g., [7]) have used sampling to quickly explore and obtain a good representative sample of the network topology, as these networks are hard to explore completely and have significant amounts of churn in their topology. For collecting data from social networks, researchers often use snowball sampling (e.g., [2]) due to the lack of access to the complete graph. have developed Forest Fire Sampling, which uses a hybrid combination of snowball sampling and random-walk sampling to produce samples that match the temporal evolution of the underlying social network [5]. have developed a Metropolis algorithm which samples in a manner designed to match desired properties in the original network [3]. Although there has been a great deal of research focusing on the the development of sampling algorithms, much of this work is based on empirical study and evaluation (i.e., measuring the similarity between sampled and original network properties). There has been some work (e.g., [4, 8, 6]) that has studied the statistical properties of samples of complex networks produced by traditional sampling algorithms such as node sampling, edge sampling and random walks. However, there has been relatively little attention paid to the development of a theoretical foundation for sampling from networks—including a formal framework for sampling, an understanding of various network characteristics and their dependencies, and an analysis of their impact on the accuracy of sampling algorithms. In this paper, we reconsider the foundations of network sampling and attempt to formalize the goals, and process of, sampling, in order to frame future development and analysis of sampling algorithms. 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.