Condition numbers and scale free graphs

TL;DR

This paper investigates the ill-conditioning of matrices derived from scale-free networks, demonstrating that their high condition numbers hinder accurate power law exponent estimation using least squares.

Contribution

It provides a theoretical lower bound on the condition number of matrices from scale-free networks and highlights the practical difficulties in parameter estimation.

Findings

Matrices from scale-free networks are ill-conditioned.

High condition numbers impair accurate power law exponent estimation.

Theoretical lower bounds confirm severe numerical issues.

Abstract

In this work we study the condition number of the least square matrix corresponding to scale free networks. We compute a theoretical lower bound of the condition number which proves that they are ill conditioned. Also, we analyze several matrices from networks generated with the linear preferential attachment model showing that it is very difficult to compute the power law exponent by the least square method due to the severe lost of accuracy expected from the corresponding condition numbers.