Consistency of Hill estimators in a linear preferential attachment model

Abstract

Preferential attachment is widely used to model power-law behavior of degree distributions in both directed and undirected networks. Practical analyses on the tail exponent of the power-law degree distribution use Hill estimator as one of the key summary statistics, whose consistency is justified mostly for iid data. The major goal in this paper is to answer the question whether the Hill estimator is still consistent when applied to non-iid network data. To do this, we first derive the asymptotic behavior of the degree sequence via embedding the degree growth of a fixed node into a birth immigration process. We also need to show the convergence of the tail empirical measure, from which the consistency of Hill estimators is obtained. This step requires checking the concentration of degree counts. We give a proof for a particular linear preferential attachment model and use simulation results as an illustration in other choices of models.

Scale-free networks well done

“…In particular, their consistency has been recently proven for a wide range of preferentialattachment models, in which degree sequences are not exactly i.i.d. [60]. In case of the configuration model [61][62][63], it is known that a degree sequence sampled i.i.d.…”

Degree growth rates and index estimation in a directed preferential attachment model

“…Remark 3.2. The form of σ in (3.2) and its Gamma density is justified in [21,27]. For a BI process {Z ′ (t)} t≥0 with Z ′ (0) = j ≥ 1, modifying the representation in (3.1) gives…”

“…This paper only considers the directed model where at each stage, a new node is born and either it points to one of the existing nodes or one of the existing nodes attaches to the new node. Results on the degree growth in the undirected case are investigated in [1,27].…”

“…What is the justification for interest in Hill estimation of power-law indices for network data? Repositories of large network datasets such as KONECT (http://konect.uni-koblenz.de/, [14]) provide summary statistics for all the archived network datasets and among the summary statistics are estimates of degree indices computed with Hill estimators, despite the fact that evidence for Hill estimator consistency is scant for network data [27].…”

“…Using convergence results for BI processes (cf. [17,Chapter 5.11], [21,27]), we give the joint limits of the in-and out-degrees of a fixed node as well as the joint maximal degree growth. Proving the convergence of the tail empirical joint measure in the second question requires showing concentration results for degree counts compared with expected degree counts.…”