## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 28, Number 1 (2018), 35-78.

### Change point detection in network models: Preferential attachment and long range dependence

Shankar Bhamidi, Jimmy Jin, and Andrew Nobel

#### Abstract

Inspired by empirical data on real world complex networks, the last few years have seen an explosion in proposed generative models to understand and explain observed properties of real world networks, including power law degree distribution and “small world” distance scaling. In this context, a natural question is how to understand the effect of *change points*—how abrupt changes in parameters driving the network model change structural properties of the network. We study this phenomenon in one popular class of dynamically evolving networks: preferential attachment models. We derive asymptotic properties of various functionals of the network including the degree distribution as well as maximal degree asymptotics, in essence showing that the change point does effect the degree distribution but does *not* change the degree exponent. This provides evidence for long range dependence and sensitive dependence of the evolution of the network on the initial evolution of the process. We propose an estimator for the change point and prove consistency properties of this estimator. The methodology developed highlights the effect of the nonergodic nature of the evolution of the network on classical change point estimators.

#### Article information

**Source**

Ann. Appl. Probab., Volume 28, Number 1 (2018), 35-78.

**Dates**

Received: June 2016

Revised: December 2016

First available in Project Euclid: 3 March 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1520046083

**Digital Object Identifier**

doi:10.1214/17-AAP1297

**Mathematical Reviews number (MathSciNet)**

MR3770872

**Zentralblatt MATH identifier**

06873679

**Subjects**

Primary: 60C05: Combinatorial probability 05C80: Random graphs [See also 60B20]

**Keywords**

Branching process Jagers–Nerman stable age distribution point process preferential attachment change point functional central limit theorem statistical estimation

#### Citation

Bhamidi, Shankar; Jin, Jimmy; Nobel, Andrew. Change point detection in network models: Preferential attachment and long range dependence. Ann. Appl. Probab. 28 (2018), no. 1, 35--78. doi:10.1214/17-AAP1297. https://projecteuclid.org/euclid.aoap/1520046083