Stratification in the Preferential Attachment Network

TL;DR

This paper analyzes the stratification of preferential attachment trees, revealing that node degree distributions exhibit depth-dependent power-law tails, with closer nodes to the root being more connected, and extends findings to redirection-based networks.

Contribution

It introduces a detailed classification of nodes by depth in preferential attachment trees and uncovers depth-dependent power-law degree distributions, extending to redirection network models.

Findings

Degree distribution tail exponent increases linearly with depth

Nodes near the root have higher connectivity

Power-law in-component size distribution also varies with depth

Abstract

We study structural properties of trees grown by preferential attachment. In this mechanism, nodes are added sequentially and attached to existing nodes at a rate that is strictly proportional to the degree. We classify nodes by their depth n, defined as the distance from the root of the tree, and find that the network is strongly stratified. Most notably, the distribution f_k^(n) of nodes with degree k at depth n has a power-law tail, f_k^(n) ~ k^{-\gamma(n)}. The exponent grows linearly with depth, gamma(n)=2+(n-1)/<n-1>, where the brackets denote an average over all nodes. Therefore, nodes that are closer to the root are better connected, and moreover, the degree distribution strongly varies with depth. Similarly, the in-component size distribution has a power-law tail and the characteristic exponent grows linearly with depth. Qualitatively, these behaviors extend to a class of networks that grow by a redirection mechanism.