The degree distribution is one of the most fundamental properties used in the analysis of massive graphs. There is a large literature on graph sampling, where the goal is to estimate properties (especially the degree distribution) of a large graph through a small, random sample. The degree distribution estimation poses a significant challenge, due to its heavy-tailed nature and the large variance in degrees. We design a new algorithm, SADDLES, for this problem, using recent mathematical techniques from the field of sublinear algorithms. The SADDLES algorithm gives provably accurate outputs for all values of the degree distribution. For the analysis, we define two fatness measures of the degree distribution, called the h-index and the z-index. We prove that SADDLES is sublinear in the graph size when these indices are large. A corollary of this result is a provably sublinear algorithm for any degree distribution bounded below by a power law. We deploy our new algorithm on a variety of real datasets and demonstrate its excellent empirical behavior. In all instances, we get extremely accurate approximations for all values in the degree distribution by observing at most 1 of the vertices. This is a major improvement over the state-of-the-art sampling algorithms, which typically sample more than 10 of the vertices to give comparable results. We also observe that the h and z-indices of real graphs are large, validating our theoretical analysis.
We note that there is a large body of work on sampling graphs from a stream @cite_43 . This is quite different from our setting, since a streaming algorithm observes every edge at least once. The specific problem of estimating the degree distribution at all scales was considered by @cite_55 . They observe many of the challenges we mentioned earlier: the difficulty of estimating the tail accurately, finding vertices at all degree scales, and combining estimates from the head and the tail.
@cite_43 @cite_55
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Despite great effort spent measuring topological features of large networks like the Internet, it was recently argued that sampling based on taking paths through the network (e.g., traceroutes) introduces a fundamental bias in the observed degree distribution. We examine this bias analytically and experimentally. For classic random graphs with mean degree c, we show analytically that traceroute sampling gives an observed degree distribution P(k) 1 k for k < c, even though the underlying degree distribution is Poisson. For graphs whose degree distributions have power-law tails P(k) k^-alpha, the accuracy of traceroute sampling is highly sensitive to the population of low-degree vertices. In particular, when the graph has a large excess (i.e., many more edges than vertices), traceroute sampling can significantly misestimate alpha. The degree distribution is one of the most fundamental graph properties of interest for real-world graphs. It has been widely observed in numerous domains that graphs typically have a tailed or scale-free degree distribution. While the average degree is usually quite small, the variance is quite high and there are vertices with degrees at all scales. We focus on the problem of approximating the degree distribution of a large streaming graph, with small storage. We design an algorithm headtail, whose main novelty is a new estimator of infrequent degrees using truncated geometric random variables. We give a mathematical analysis of headtail and show that it has excellent behavior in practice. We can process streams will millions of edges with storage less than 1 and get extremely accurate approximations for all scales in the degree distribution. We also introduce a new notion of Relative Hausdorff distance between tailed histograms. Existing notions of distances between distributions are not suitable, since they ignore infrequent degrees in the tail. The Relative Hausdorff distance measures deviations at all scales, and is a more suitable distance for comparing degree distributions. By tracking this new measure, we are able to give strong empirical evidence of the convergence of headtail. Many complex networks in natural and social phenomena have often been characterized by heavy-tailed degree distributions. However, due to rapidly growing size of network data and concerns on privacy issues about using these data, it becomes more difficult to analyze complete data sets. Thus, it is crucial to devise effective and efficient estimation methods for heavy tails of degree distributions in large-scale networks only using local information of a small fraction of sampled nodes. Here we propose a tail-scope method based on local observational bias of the friendship paradox. We show that the tail-scope method outperforms the uniform node sampling for estimating heavy tails of degree distributions, while the opposite tendency is observed in the range of small degrees. In order to take advantages of both sampling methods, we devise the hybrid method that successfully recovers the whole range of degree distributions. Our tail-scope method shows how structural heterogeneities of large-scale complex networks can be used to effectively reveal the network structure only with limited local information. 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.