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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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. 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. Standard statistical methods often provide no way to make accurate estimates about the characteristics of hidden populations such as injection drug users, the homeless, and artists. In this paper, we further develop a sampling and estimation technique called respondent-driven sampling, which allows researchers to make asymptotically unbiased estimates about these hidden populations. The sample is selected with a snowball-type design that can be done more cheaply, quickly, and easily than other methods currently in use. Further, we can show that under certain specified (and quite general) conditions, our estimates for the percentage of the population with a specific trait are asymptotically unbiased. We further show that these estimates are asymptotically unbiased no matter how the seeds are selected. We conclude with a comparison of respondent-driven samples of jazz musicians in New York and San Francisco, with corresponding institutional samples of jazz musicians from these cities. The results show that ... 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. A general method, suitable for fast computing machines, for investigating such properties as equations of state for substances consisting of interacting individual molecules is described. The method consists of a modified Monte Carlo integration over configuration space. Results for the two‐dimensional rigid‐sphere system have been obtained on the Los Alamos MANIAC and are presented here. These results are compared to the free volume equation of state and to a four‐term virial coefficient expansion. Many ant species use distributed population density estimation in applications ranging from quorum sensing, to task allocation, to appraisal of enemy colony strength. It has been shown that ants estimate local population density by tracking encounter rates: The higher the density, the more often the ants bump into each other. We study distributed density estimation from a theoretical perspective. We prove that a group of anonymous agents randomly walking on a grid are able to estimate their density within a small multiplicative error in few steps by measuring their rates of encounter with other agents. Despite dependencies inherent in the fact that nearby agents may collide repeatedly (and, worse, cannot recognize when this happens), our bound nearly matches what would be required to estimate density by independently sampling grid locations. From a biological perspective, our work helps shed light on how ants and other social insects can obtain relatively accurate density estimates via encounter rates. From a technical perspective, our analysis provides tools for understanding complex dependencies in the collision probabilities of multiple random walks. We bound the strength of these dependencies using local mixing properties of the underlying graph. Our results extend beyond the grid to more general graphs, and we discuss applications to size estimation for social networks, density estimation for robot swarms, and random walk-based sampling for sensor networks. Predicting the occurrence of links is a fundamental problem in networks. In the link prediction problem we are given a snapshot of a network and would like to infer which interactions among existing members are likely to occur in the near future or which existing interactions are we missing. Although this problem has been extensively studied, the challenge of how to effectively combine the information from the network structure with rich node and edge attribute data remains largely open. We develop an algorithm based on Supervised Random Walks that naturally combines the information from the network structure with node and edge level attributes. We achieve this by using these attributes to guide a random walk on the graph. We formulate a supervised learning task where the goal is to learn a function that assigns strengths to edges in the network such that a random walker is more likely to visit the nodes to which new links will be created in the future. We develop an efficient training algorithm to directly learn the edge strength estimation function. Our experiments on the Facebook social graph and large collaboration networks show that our approach outperforms state-of-the-art unsupervised approaches as well as approaches that are based on feature extraction.