### Uittreksel

Originele taal-2 | Engels |
---|---|

Uitgeverij | s.n. |

Aantal pagina's | 15 |

Status | Gepubliceerd - 2012 |

### Publicatie series

Naam | arXiv.org [math.PR] |
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Volume | 1202.3071 |

### Vingerafdruk

### Citeer dit

*Degree-degree correlations in random graphs with heavy-tailed degrees*. (arXiv.org [math.PR]; Vol. 1202.3071). s.n.

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*Degree-degree correlations in random graphs with heavy-tailed degrees*. arXiv.org [math.PR], vol. 1202.3071, s.n.

**Degree-degree correlations in random graphs with heavy-tailed degrees.** / Litvak, N.; Hofstad, van der, R.W.

Onderzoeksoutput: Boek/rapport › Rapport › Academic

TY - BOOK

T1 - Degree-degree correlations in random graphs with heavy-tailed degrees

AU - Litvak, N.

AU - Hofstad, van der, R.W.

PY - 2012

Y1 - 2012

N2 - We investigate degree-degree correlations for scale-free graph sequences. The main conclusion of this paper is that the assortativity coefficient is not the appropriate way to describe degree-dependences in scale-free random graphs. Indeed, we study the infinite volume limit of the assortativity coefficient, and show that this limit is always non-negative when the degrees have finite first but infinite third moment, i.e., when the degree exponent $\gamma+1$ of the density satisfies $\gamma\in (1,3)$. More generally, our results show that the correlation coefficient is inappropriate to describe dependencies between random variables having infinite variance. We start with a simple model of the sample correlation of random variables $X$ and $Y$, which are linear combinations with non-negative coefficients of the \emph{same} infinite variance random variables. In this case, the correlation coefficient of $X$ and $Y$ is not defined, and the sample covariance converges to a proper random variable with support that is a subinterval of $(-1,1)$. Further, for any joint distribution $(X, Y)$ with equal marginals being non-negative power-law distributions with infinite variance (as in the case of degree-degree correlations), we show that the limit is non-negative. We next adapt these results to the assortativity in networks as described by the degree-degree correlation coefficient, and show that it is non-negative in the large graph limit when the degree distribution has an infinite third moment. We illustrate these results with several examples where the assortativity behaves in a non-sensible way. We further discuss alternatives for describing assortativity in networks based on rank correlations that are appropriate for infinite variance variables. We support these mathematical results by simulations.

AB - We investigate degree-degree correlations for scale-free graph sequences. The main conclusion of this paper is that the assortativity coefficient is not the appropriate way to describe degree-dependences in scale-free random graphs. Indeed, we study the infinite volume limit of the assortativity coefficient, and show that this limit is always non-negative when the degrees have finite first but infinite third moment, i.e., when the degree exponent $\gamma+1$ of the density satisfies $\gamma\in (1,3)$. More generally, our results show that the correlation coefficient is inappropriate to describe dependencies between random variables having infinite variance. We start with a simple model of the sample correlation of random variables $X$ and $Y$, which are linear combinations with non-negative coefficients of the \emph{same} infinite variance random variables. In this case, the correlation coefficient of $X$ and $Y$ is not defined, and the sample covariance converges to a proper random variable with support that is a subinterval of $(-1,1)$. Further, for any joint distribution $(X, Y)$ with equal marginals being non-negative power-law distributions with infinite variance (as in the case of degree-degree correlations), we show that the limit is non-negative. We next adapt these results to the assortativity in networks as described by the degree-degree correlation coefficient, and show that it is non-negative in the large graph limit when the degree distribution has an infinite third moment. We illustrate these results with several examples where the assortativity behaves in a non-sensible way. We further discuss alternatives for describing assortativity in networks based on rank correlations that are appropriate for infinite variance variables. We support these mathematical results by simulations.

UR - http://arxiv.org/pdf/1202.3071

M3 - Report

T3 - arXiv.org [math.PR]

BT - Degree-degree correlations in random graphs with heavy-tailed degrees

PB - s.n.

ER -