Intrinsic degree-correlations in static model of scale-free networks

TL;DR

This paper analytically investigates degree correlations and clustering in static scale-free networks, revealing crossover behaviors depending on the degree exponent and confirming findings with simulations.

Contribution

It provides an analytical derivation of degree-dependent clustering and neighbor degree functions in static scale-free networks, highlighting differences from growing network models.

Findings

Crossover from degree-independent to degree-dependent behavior for 2<γ<3

Analytical expressions derived for k-dependent functions

Numerical simulations confirm analytical results

Abstract

We calculate the mean neighboring degree function $\bar k_{\rm{nn}}(k)$ and the mean clustering function $C(k)$ of vertices with degree $k$ as a function of $k$ in finite scale-free random networks through the static model. While both are independent of $k$ when the degree exponent $\gamma \geq 3$, they show the crossover behavior for $2 < \gamma < 3$ from $k$-independent behavior for small $k$ to $k$-dependent behavior for large $k$. The $k$-dependent behavior is analytically derived. Such a behavior arises from the prevention of self-loops and multiple edges between each pair of vertices. The analytic results are confirmed by numerical simulations. We also compare our results with those obtained from a growing network model, finding that they behave differently from each other.