Local degree distribution in scale free random graphs

TL;DR

This paper investigates how the degree distribution in scale-free random graphs varies when focusing on subsets of vertices, providing conditions for the existence of constrained degree distributions and identifying their exponents.

Contribution

It offers new theoretical conditions for the existence of asymptotic degree distributions on subsets of vertices in scale-free graphs and determines their characteristic exponents.

Findings

Conditions for almost sure existence of constrained degree distributions

Identification of characteristic exponents for these distributions

Insights into how degree distributions change with vertex subsets

Abstract

In several scale free graph models the asymptotic degree distribution and the characteristic exponent change when only a smaller set of vertices is considered. Looking at the common properties of these models, we present sufficient conditions for the almost sure existence of asymptotic degree distribution constrained on the set of selected vertices, and identify the characteristic exponent belonging to it.