### Abstract

Original language | English |
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Article number | 1710.02027v1 |

Number of pages | 25 |

Journal | arXiv |

Issue number | 1710.02027 |

Publication status | Published - 5 Oct 2017 |

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*arXiv*, (1710.02027), [1710.02027v1].

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*arXiv*, no. 1710.02027, 1710.02027v1.

**Triadic closure in configuration models with unbounded degree fluctuations.** / van der Hofstad, R.W.; van Leeuwaarden, J.S.H.; Stegehuis, C.

Research output: Contribution to journal › Article › Academic

TY - JOUR

T1 - Triadic closure in configuration models with unbounded degree fluctuations

AU - van der Hofstad, R.W.

AU - van Leeuwaarden, J.S.H.

AU - Stegehuis, C.

PY - 2017/10/5

Y1 - 2017/10/5

N2 - The configuration model generates random graphs with any given degree distribution, and thus serves as a null model for scale-free networks with power-law degrees and unbounded degree fluctuations. For this setting, we study the local clustering c(k) , i.e., the probability that two neighbors of a degree-k node are neighbors themselves. We show that c(k) progressively falls off with k and eventually for k=Ω(n − − √ ) settles on a power law c(k)∼k −2(3−τ) with τ∈(2,3) the power-law exponent of the degree distribution. This fall-off has been observed in the majority of real-world networks and signals the presence of modular or hierarchical structure. Our results agree with recent results for the hidden-variable model and also give the expected number of triangles in the configuration model when counting triangles only once despite the presence of multi-edges. We show that only triangles consisting of triplets with uniquely specified degrees contribute to the triangle counting.

AB - The configuration model generates random graphs with any given degree distribution, and thus serves as a null model for scale-free networks with power-law degrees and unbounded degree fluctuations. For this setting, we study the local clustering c(k) , i.e., the probability that two neighbors of a degree-k node are neighbors themselves. We show that c(k) progressively falls off with k and eventually for k=Ω(n − − √ ) settles on a power law c(k)∼k −2(3−τ) with τ∈(2,3) the power-law exponent of the degree distribution. This fall-off has been observed in the majority of real-world networks and signals the presence of modular or hierarchical structure. Our results agree with recent results for the hidden-variable model and also give the expected number of triangles in the configuration model when counting triangles only once despite the presence of multi-edges. We show that only triangles consisting of triplets with uniquely specified degrees contribute to the triangle counting.

KW - math.PR

M3 - Article

JO - arXiv

JF - arXiv

IS - 1710.02027

M1 - 1710.02027v1

ER -