### Abstract

Original language | English |
---|---|

Article number | 1709.03466 |

Number of pages | 26 |

Journal | arXiv |

Issue number | 1709.03466 |

Publication status | Published - 11 Sep 2017 |

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### Bibliographical note

26 pages, 6 figures### Cite this

*arXiv*, (1709.03466), [1709.03466].

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*arXiv*, no. 1709.03466, 1709.03466.

**Optimal subgraph structures in scale-free configuration models.** / van der Hofstad, R.W.; van Leeuwaarden, J.S.H.; Stegehuis, C.

Research output: Contribution to journal › Article › Academic

TY - JOUR

T1 - Optimal subgraph structures in scale-free configuration models

AU - van der Hofstad, R.W.

AU - van Leeuwaarden, J.S.H.

AU - Stegehuis, C.

N1 - 26 pages, 6 figures

PY - 2017/9/11

Y1 - 2017/9/11

N2 - Subgraphs reveal information about the geometry and functionalities of complex networks. For scale-free networks with unbounded degree fluctuations, we count the number of times a small connected graph occurs as a subgraph (motif counting) or as an induced subgraph (graphlet counting). We obtain these results by analyzing the configuration model with degree exponent $\tau\in(2,3)$ and introducing a novel class of optimization problems. For any given subgraph, the unique optimizer describes the degrees of the nodes that together span the subgraph. We find that every subgraph occurs typically between vertices with specific degree ranges. In this way, we can count and characterize {\it all} subgraphs. We refrain from double counting in the case of multi-edges, essentially counting the subgraphs in the {\it erased} configuration model.

AB - Subgraphs reveal information about the geometry and functionalities of complex networks. For scale-free networks with unbounded degree fluctuations, we count the number of times a small connected graph occurs as a subgraph (motif counting) or as an induced subgraph (graphlet counting). We obtain these results by analyzing the configuration model with degree exponent $\tau\in(2,3)$ and introducing a novel class of optimization problems. For any given subgraph, the unique optimizer describes the degrees of the nodes that together span the subgraph. We find that every subgraph occurs typically between vertices with specific degree ranges. In this way, we can count and characterize {\it all} subgraphs. We refrain from double counting in the case of multi-edges, essentially counting the subgraphs in the {\it erased} configuration model.

KW - math.PR

M3 - Article

JO - arXiv

JF - arXiv

IS - 1709.03466

M1 - 1709.03466

ER -