Sampling properties of random graphs: The degree distribution

Stumpf, Michael; Wiuf, Carsten

doi:10.1103/physreve.72.036118

Cited by 97 publications

(91 citation statements)

References 25 publications

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“…where p(k) is the degree distribution of the original network and α is the sampled coverage [6,[30][31][32]. Equation (1) also describes the incoming and outgoing degree distribution of randomly sampled directed networks, where k and k 0 are replaced with k in and k 0,in (or k out and k 0,out respectively).…”

Section: Sampling Resultsmentioning

confidence: 99%

“…Stumpf et al [30,31] studied the degree distribution of two random networks -one that had been sampled "uniformly" by picking nodes at random, and one that was subject to connectivity-dependent sampling -both analytically and numerically. Lee et al [32] also studied, numerically, the effects of random sampling and snowball sampling [33], on statistical properties of real scale-free networks -including degree distribution exponents, betweenness centrality exponents, assortativity, and clustering coefficient -demonstrating that these quantities could be either overestimated or underestimated, depending on the fraction of the network sampled and the type of sampling method used.…”

Section: Introductionmentioning

confidence: 99%

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Sampling properties of directed networks

Son

Christensen²,

Bizhani³

et al. 2012

Phys. Rev. E

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For many real-world networks only a small "sampled" version of the original network may be investigated; those results are then used to draw conclusions about the actual system. Variants of breadth-first search (BFS) sampling, which are based on epidemic processes, are widely used. Although it is well established that BFS sampling fails, in most cases, to capture the IN-component(s) of directed networks, a description of the effects of BFS sampling on other topological properties are all but absent from the literature. To systematically study the effects of sampling biases on directed networks, we compare BFS sampling to random sampling on complete large-scale directed networks. We present new results and a thorough analysis of the topological properties of seven different complete directed networks (prior to sampling), including three versions of Wikipedia, three different sources of sampled World Wide Web data, and an Internet-based social network. We detail the differences that sampling method and coverage can make to the structural properties of sampled versions of these seven networks. Most notably, we find that sampling method and coverage affect both the bow-tie structure, as well as the number and structure of strongly connected components in sampled networks. In addition, at low sampling coverage (i.e. less than 40%), the values of average degree, variance of out-degree, degree auto-correlation, and link reciprocity are overestimated by 30% or more in BFS-sampled networks, and only attain values within 10% of the corresponding values in the complete networks when sampling coverage is in excess of 65%. These results may cause us to rethink what we know about the structure, function, and evolution of real-world directed networks.

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Section: Sampling Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Sampling properties of directed networks

Son

Christensen²,

Bizhani³

et al. 2012

Phys. Rev. E

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show abstract

“…It has previously been shown [31,30,16,35,19] that the properties of subnets can differ quite considerably from those of the true network. Such differences will, in fact, be of a qualitative nature [31,35] for most types of networks, even when sampling of nodes is essentially uniform and random.…”

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confidence: 99%

Multi-model inference of network properties from incomplete data

Stumpf

Thorne

2006

Journal of Integrative Bioinformatics

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SummaryIt has previously been shown that subnets differ from global networks from which they are sampled for all but a very limited number of theoretical network models. These differences are of qualitative as well as quantitative nature, and the properties of subnets may be very different from the corresponding properties in the true, unobserved network. Here we propose a novel approach which allows us to infer aspects of the true network from incomplete network data in a multi-model inference framework. We develop the basic theoretical framework, including procedures for assessing confidence intervals of our estimates and evaluate the performance of this approach in simulation studies and against subnets drawn from the presently available PIN network data in Saccaromyces cerevisiae. We then illustrate the potential power of this new approach by estimating the number of interactions that will be detectable with present experimental approaches in sfour eukaryotic species, inlcuding humans. Encouragingly, where independent datasets are available we obtain consistent estimates from different partial protein interaction networks. We conclude with a discussion of the scope of this approaches and areas for further research.

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“…The impact of sampling methods on the discovery of graph properties has also been studied in [19,36,35,18]. They cover a wide range of network properties, and focus on the properties of the derived sub-graph, instead of the estimation of the properties of the original graph.…”

Section: Pertinent Workmentioning

confidence: 99%

“…The degree sampling of networks, which is the focus of this paper, has also received special attention. Stump et al studied the sampling of degree distribution [35] for two sampling schemes, i.e., random sampling and the degree dependent sampling of the nodes. For average degree estimation, both [8] and [10] used uniform random sampling of the nodes.…”

Section: Pertinent Workmentioning

confidence: 99%

Variance reduction in large graph sampling

Lü

Wang

2014

Information Processing & Management

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The norm of practice in estimating graph properties is to use uniform random node (RN) samples whenever possible. Many graphs are large and scale-free, inducing large degree variance and estimator variance. This paper shows that random edge (RE) sampling and the corresponding harmonic mean estimator for average degree can reduce the estimation variance significantly. First, we demonstrate that the degree variance, and consequently the variance of the RN estimator, can grow almost linearly with data size for typical scale-free graphs. Then we prove that the RE estimator has a variance bounded from above. Therefore, the variance ratio between RN and RE samplings can be very large for big data. The analytical result is supported by both simulation studies and 18 real networks. We observe that the variance reduction ratio can be more than a hundred for some real networks such as Twitter. Furthermore, we show that random walk (RW) sampling is always worse than RE sampling, and it can reduce the variance of RN method only when its performance is close to that of RE sampling.

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Sampling properties of random graphs: The degree distribution

Cited by 97 publications

References 25 publications

Sampling properties of directed networks

Sampling properties of directed networks

Multi-model inference of network properties from incomplete data

Variance reduction in large graph sampling

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