Properties of stochastic Kronecker graphs

Kang, Mihyun; Karoński, Michał; Koch, Christoph; Makai, Tamás

doi:10.48550/arxiv.1410.6328

Cited by 1 publication

(2 citation statements)

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“…A directed graph can then be sampled from M k by flipping one biased coin per matrix entry to obtain an adjacency matrix. In keeping with prior work, we assume M 1 = a b c d ; such a 2 × 2 initiator matrix has been most widely adopted in the literature (including [5], [12], [13], [14]) after experiments found it generates synthetic graphs that most closely match real-world data [11]. R-MAT erasure and rethrow models: Independent of SKG, in 2004, Chakrabarti et al [3] introduced the Recursive MATrix (R-MAT) graph generator.…”

Section: Skgmentioning

confidence: 99%

See 1 more Smart Citation

Asymptotic Analysis of Equivalences and Core-Structures in Kronecker-Style Graph Models

Chin

Goodrich

O’Brien

et al. 2016

2016 IEEE 16th International Conference on Data Mining (ICDM)

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Abstract-Growing interest in modeling large, complex networks has spurred significant research into generative graph models. These models not only allow users to gain insight into the processes underlying networks, but also provide synthetic data which allows algorithm scalability testing and addresses privacy concerns. Kronecker-style models (e.g. SKG and R-MAT) are often used due to their scalability and ability to mimic key properties of real-world networks (e.g. diameter and degree distribution). Although a few papers theoretically establish these models' behavior for specific parameters, many claims used to justify the use of these models in various applications are supported only by empirical evaluations. In this work, we prove several results using asymptotic analysis which illustrate that empirical studies may not fully capture the true behavior of the models.Paramount to the widespread adoption of Kronecker-style models was the introduction of a linear-time edge-sampling variant (R-MAT), which existing literature typically treats as interchangeable with SKG. We prove that although several R-MAT formulations are asymptotically equivalent, their behavior diverges from that of SKG. Further, we show these results are experimentally observable even at relatively small graph sizes. Second, we consider a case where asymptotic analysis reveals unexpected behavior within a given model. One of the criticisms of using Kronecker-style models has been that they are unable to generate the deep core-structures commonly observed in real-world data. We prove that in fact, for some parameter values, all the Kronecker-style models generate graphs whose maximum core depth grows as a function of the size of the network-including in the region of the parameter space most commonly used in prior work. Our results also illustrate why this behavior may be difficult to observe for moderate graph sizes, and highlight the dangers of extrapolating model-wide claims from empirical results.

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Section: Skgmentioning

confidence: 99%

“…degree distribution [12] and number of isolated vertices [5]). Finally, a few papers have considered the limiting behavior of these models-characterizing the emergence of a giant component and proving constant diameter [13], and proving that SKG generally cannot generate graphs with a power-law degree distribution [14].…”

Section: Introductionmentioning

confidence: 99%