Fitting the linear preferential attachment model

Abstract

Preferential attachment is an appealing mechanism for modeling power-law behavior of the degree distributions in directed social networks. In this paper, we consider methods for fitting a 5-parameter linear preferential model to network data under two data scenarios. In the case where full history of the network formation is given, we derive the maximum likelihood estimator of the parameters and show that it is strongly consistent and asymptotically normal. In the case where only a single-time snapshot of the network is available, we propose an estimation method which combines method of moments with an approximation to the likelihood. The resulting estimator is also strongly consistent and performs quite well compared to the MLE estimator. We illustrate both estimation procedures through simulated data, and explore the usage of this model in a real data example.

On a Minimum Distance Procedure for Threshold Selection in Tail Analysis

“…Facebook wall posts (http://konect.uni-koblenz.de/networks/facebook-wosn-wall). Using the snap shot methodology described in [41] we estimate In Example I the RMSE of the Hill estimatorα n,k * n for the tail index of the in-degree is just 6.8% larger than the minimal RMSE over all deterministic choices of k ∈ {10, . .…”

“…Theoretically, the linear PA network models generate power-law degree distributions, and the consistency of Hill estimators based on the non-iid degree sequences in linear PA models has been justified in [44,45]. Limit theory for degree counts in a linear PA model can be found in [4,27,26,34,35,36,42,43,44,45], and statistical inferences on the linear PA models are given in [15,40,41].…”

A preferential attachment model for the stellar initial mass function

“…Most of the work done on estimation of the attachment function makes the assumption that observations are available regarding the full or partial evolution of the network. This includes the nonparametric methods of Jeong, Néda and Barabási (2003), Newman (2001), and Pham, Sheridan and Shimodaira (2015); the maximum likelihood approaches of Gómez, Kappen and Kaltenbrunner (2011), Wan et al (2017a), Onodera and Sheridan (2014); and the Bayesian approach (using MCMC) taken by Sheridan, Yagahara and Shimodaira (2012). Wan et al (2017a) also describes an approximation to the MLE that can be utilized when only a snapshot view of the network is available.…”

Are extreme value estimation methods useful for network data?

“…Marginal degree power laws were established in [3,16,17], while joint power-law behavior, also known as joint regular variation, was proved in [21,22,26] for the directed linear PA model. Given observed network data, [25] proposed parametric inference procedures for the model in two data scenarios. For the case where the history of network growth is available, the MLE estimators of model parameters were derived and shown to be strongly consistent, asymptotically normal and efficient.…”