A trick: why \hat\gamma < \gamma (=3) in [1=arXiv:cond-mat/0106096]?

TL;DR

This paper explains why the estimated degree exponent in Barabási-Albert networks is biased and rises with the number of links added, due to simulation inconsistencies and initial node differences.

Contribution

It provides a theorem on series sums and clarifies the origin of simulation deviations in BA model degree exponents, proposing corrections for initial node effects.

Findings

Estimated  is always smaller than 3 in simulations.

 rises monotonically with the number of links added.

Simulation errors stem from incompatible conditions and initial node differences.

Abstract

In this paper, first a theorem on the partial sum of a particular series is given. Then, based on it, the origin of obvious simulation deviation from theory is explained: i) why the numerically estimated \hat\gamma (degree exponent) in [1=arXiv:cond-mat/0106096] is always smaller than \gamma (=3) that is predicted by theory; ii) and why \hat\gamma rises monotonically as m (the links added at each step in Barabasi-Albert (BA) model [1]) increases. Strictly, it declares such errors are basically from the inconsistence of simulation with the theoretical model, which is caused by an additional incompatible condition used in simulation. In addition, noticing the evolving differences between the initial m_0 nodes and those after, we correct the derived BA model which unfairly omitted such differences.