Degree Correlation in Scale-Free Graphs

TL;DR

This paper derives exact formulas for degree correlations and distributions in scale-free networks generated by preferential attachment models, enhancing understanding of their structural properties.

Contribution

It provides closed-form expressions for the expected conditional and joint degree distributions in linear and shifted-linear preferential attachment models.

Findings

Derived formulas for joint degree distribution p(k, l)

Obtained steady-state degree distribution expressions

Analyzed degree correlation properties in scale-free graphs

Abstract

We obtain closed form expressions for the expected conditional degree distribution and the joint degree distribution of the linear preferential attachment model for network growth in the steady state. We consider the multiple-destination preferential attachment growth model, where incoming nodes at each timestep attach to $\beta$ existing nodes, selected by degree-proportional probabilities. By the conditional degree distribution $p(\ell| k)$, we mean the degree distribution of nodes that are connected to a node of degree $k$. By the joint degree distribution $p(k,\ell)$, we mean the proportion of links that connect nodes of degrees $k$ and $\ell$. In addition to this growth model, we consider the shifted-linear preferential growth model and solve for the same quantities, as well as a closed form expression for its steady-state degree distribution.