### Abstract

We introduce the link-space formalism for analyzing network models with degree-degree correlations. The formalism is based on a statistical description of the fraction of links li,j connecting nodes of degrees i and j. To demonstrate its use, we apply the framework to some pedagogical network models, namely, random attachment, Barabási-Albert preferential attachment, and the classical Erdos and Rényi random graph. For these three models the link-space matrix can be solved analytically. We apply the formalism to a simple one-parameter growing network model whose numerical solution exemplifies the effect of degree-degree correlations for the resulting degree distribution. We also employ the formalism to derive the degree distributions of two very simple network decay models, more specifically, that of random link deletion and random node deletion. The formalism allows detailed analysis of the correlations within networks and we also employ it to derive the form of a perfectly nonassortative network for arbitrary degree distribution.

Original language | English (US) |
---|---|

Article number | 036112 |

Journal | Physical Review E - Statistical, Nonlinear, and Soft Matter Physics |

Volume | 77 |

Issue number | 3 |

DOIs | |

State | Published - Mar 11 2008 |

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### ASJC Scopus subject areas

- Physics and Astronomy(all)
- Condensed Matter Physics
- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Physical Review E - Statistical, Nonlinear, and Soft Matter Physics*,

*77*(3), [036112]. https://doi.org/10.1103/PhysRevE.77.036112

**Link-space formalism for network analysis.** / Smith, David M D; Lee, Chiu Fan; Onnela, Jukka Pekka; Johnson, Neil F.

Research output: Contribution to journal › Article

*Physical Review E - Statistical, Nonlinear, and Soft Matter Physics*, vol. 77, no. 3, 036112. https://doi.org/10.1103/PhysRevE.77.036112

}

TY - JOUR

T1 - Link-space formalism for network analysis

AU - Smith, David M D

AU - Lee, Chiu Fan

AU - Onnela, Jukka Pekka

AU - Johnson, Neil F

PY - 2008/3/11

Y1 - 2008/3/11

N2 - We introduce the link-space formalism for analyzing network models with degree-degree correlations. The formalism is based on a statistical description of the fraction of links li,j connecting nodes of degrees i and j. To demonstrate its use, we apply the framework to some pedagogical network models, namely, random attachment, Barabási-Albert preferential attachment, and the classical Erdos and Rényi random graph. For these three models the link-space matrix can be solved analytically. We apply the formalism to a simple one-parameter growing network model whose numerical solution exemplifies the effect of degree-degree correlations for the resulting degree distribution. We also employ the formalism to derive the degree distributions of two very simple network decay models, more specifically, that of random link deletion and random node deletion. The formalism allows detailed analysis of the correlations within networks and we also employ it to derive the form of a perfectly nonassortative network for arbitrary degree distribution.

AB - We introduce the link-space formalism for analyzing network models with degree-degree correlations. The formalism is based on a statistical description of the fraction of links li,j connecting nodes of degrees i and j. To demonstrate its use, we apply the framework to some pedagogical network models, namely, random attachment, Barabási-Albert preferential attachment, and the classical Erdos and Rényi random graph. For these three models the link-space matrix can be solved analytically. We apply the formalism to a simple one-parameter growing network model whose numerical solution exemplifies the effect of degree-degree correlations for the resulting degree distribution. We also employ the formalism to derive the degree distributions of two very simple network decay models, more specifically, that of random link deletion and random node deletion. The formalism allows detailed analysis of the correlations within networks and we also employ it to derive the form of a perfectly nonassortative network for arbitrary degree distribution.

UR - http://www.scopus.com/inward/record.url?scp=40949106789&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=40949106789&partnerID=8YFLogxK

U2 - 10.1103/PhysRevE.77.036112

DO - 10.1103/PhysRevE.77.036112

M3 - Article

AN - SCOPUS:40949106789

VL - 77

JO - Physical review. E

JF - Physical review. E

SN - 2470-0045

IS - 3

M1 - 036112

ER -