Characterizing the intrinsic correlations of scale-free networks

TL;DR

This paper investigates the intrinsic correlations in scale-free networks caused by structural constraints, revealing their impact on network properties and critical phenomena, especially in large degree regimes.

Contribution

It introduces an analytical method to predict degree correlations and demonstrates that scale-free networks are not self-averaging, affecting their critical behavior.

Findings

Intrinsic correlations arise from structural constraints.

Scale-free networks are not self-averaging.

Correlations significantly influence network critical properties.

Abstract

Very often, when studying topological or dynamical properties of random scale-free networks, it is tacitly assumed that degree-degree correlations are not present. However, simple constraints, such as the absence of multiple edges and self-loops, can give rise to intrinsic correlations in these structures. In the same way that Fermionic correlations in thermodynamic systems are relevant only in the limit of low temperature, the intrinsic correlations in scale-free networks are relevant only when the extreme values for the degrees grow faster than the square-root of the network size. In this situation, these correlations can significantly affect the dependence of the average degree of the nearest neighbors of a given vertice on this vertices's degree. Here, we introduce an analytical approach that is capable to predict the functional form of this property. Moreover, our results indicate that random scale-free networks models are not self-averaging, that is, the second moment of their degree distribution may vary orders of magnitude among different realizations. Finally, we argue that the intrinsic correlations investigated here may have profound impact on the critical properties of random scale-free networks.