### Abstract

We consider a general class of preferential attachment schemes evolving by a reinforcement rule with respect to certain sublinear weights. In these schemes, which grow a random network, the sequence of degree distributions is an object of interest which sheds light on the evolving structures. In this article, we use a fluid limit approach to prove a functional law of large numbers for the degree structure in this class, starting from a variety of initial conditions. The method appears robust and applies in particular to 'non-tree' evolutions where cycles may develop in the network. A main part of the argument is to show that there is a unique nonnegative solution to an infinite system of coupled ODEs, corresponding to a rate formulation of the law of large numbers limit, through C_{0}-semigroup/dynamical systems methods. These results also resolve a question in Chung, Handjani and Jungreis (2003).

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

Pages (from-to) | 703-731 |

Number of pages | 29 |

Journal | Random Structures and Algorithms |

Volume | 48 |

Issue number | 4 |

DOIs | |

State | Published - Jul 1 2016 |

### Fingerprint

### Keywords

- Degree distribution
- Dynamical system
- Fluid limit
- Infinite
- Law of large numbers
- ODE
- Preferential attachment
- Random graphs
- Semigroup
- Sublinear weights
- Uniqueness

### ASJC Scopus subject areas

- Computer Graphics and Computer-Aided Design
- Software
- Mathematics(all)
- Applied Mathematics

### Cite this

**A scaling limit for the degree distribution in sublinear preferential attachment schemes.** / Choi, Jihyeok; Sethuraman, Sunder; Venkataramani, Shankar C.

Research output: Contribution to journal › Article

*Random Structures and Algorithms*, vol. 48, no. 4, pp. 703-731. https://doi.org/10.1002/rsa.20615

}

TY - JOUR

T1 - A scaling limit for the degree distribution in sublinear preferential attachment schemes

AU - Choi, Jihyeok

AU - Sethuraman, Sunder

AU - Venkataramani, Shankar C

PY - 2016/7/1

Y1 - 2016/7/1

N2 - We consider a general class of preferential attachment schemes evolving by a reinforcement rule with respect to certain sublinear weights. In these schemes, which grow a random network, the sequence of degree distributions is an object of interest which sheds light on the evolving structures. In this article, we use a fluid limit approach to prove a functional law of large numbers for the degree structure in this class, starting from a variety of initial conditions. The method appears robust and applies in particular to 'non-tree' evolutions where cycles may develop in the network. A main part of the argument is to show that there is a unique nonnegative solution to an infinite system of coupled ODEs, corresponding to a rate formulation of the law of large numbers limit, through C0-semigroup/dynamical systems methods. These results also resolve a question in Chung, Handjani and Jungreis (2003).

AB - We consider a general class of preferential attachment schemes evolving by a reinforcement rule with respect to certain sublinear weights. In these schemes, which grow a random network, the sequence of degree distributions is an object of interest which sheds light on the evolving structures. In this article, we use a fluid limit approach to prove a functional law of large numbers for the degree structure in this class, starting from a variety of initial conditions. The method appears robust and applies in particular to 'non-tree' evolutions where cycles may develop in the network. A main part of the argument is to show that there is a unique nonnegative solution to an infinite system of coupled ODEs, corresponding to a rate formulation of the law of large numbers limit, through C0-semigroup/dynamical systems methods. These results also resolve a question in Chung, Handjani and Jungreis (2003).

KW - Degree distribution

KW - Dynamical system

KW - Fluid limit

KW - Infinite

KW - Law of large numbers

KW - ODE

KW - Preferential attachment

KW - Random graphs

KW - Semigroup

KW - Sublinear weights

KW - Uniqueness

UR - http://www.scopus.com/inward/record.url?scp=84969786193&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84969786193&partnerID=8YFLogxK

U2 - 10.1002/rsa.20615

DO - 10.1002/rsa.20615

M3 - Article

AN - SCOPUS:84969786193

VL - 48

SP - 703

EP - 731

JO - Random Structures and Algorithms

JF - Random Structures and Algorithms

SN - 1042-9832

IS - 4

ER -