Zero Pearson Coefficient for Strongly Correlated Growing Trees

TL;DR

This paper investigates Pearson's correlation coefficient in growing recursive networks, revealing it is zero in the infinite limit for trees and depends on network size and parameters, questioning its usefulness for network comparison.

Contribution

It provides an exact analysis of Pearson's coefficient in strongly correlated growing trees and shows its dependence on network size and degree distribution parameters.

Findings

Pearson's coefficient is exactly zero for recursive trees ($m=1$) in the infinite limit.

For $m>1$, the coefficient is zero only if the degree exponent $$ does not exceed 4.

Finite networks show a slow, power-law approach to the infinite network limit.

Abstract

We obtained Pearson's coefficient of strongly correlated recursive networks growing by preferential attachment of every new vertex by $m$ edges. We found that the Pearson coefficient is exactly zero in the infinite network limit for the recursive trees ($m=1$). If the number of connections of new vertices exceeds one ($m>1$), then the Pearson coefficient in the infinite networks equals zero only when the degree distribution exponent $\gamma$ does not exceed 4. We calculated the Pearson coefficient for finite networks and observed a slow, power-law like approach to an infinite network limit. Our findings indicate that Pearson's coefficient strongly depends on size and details of networks, which makes this characteristic virtually useless for quantitative comparison of different networks.