Nonlinear Barab\'asi-Albert Network

TL;DR

This paper explores the nonlinear Barabási-Albert network model, deriving degree distribution formulas, analyzing clustering, assortativity, and path length, revealing phase transitions and structural properties for different nonlinearity parameters.

Contribution

It provides an analytical framework for the degree distribution and structural metrics of nonlinear Barabási-Albert networks across all parameter ranges.

Findings

Degree distribution formula valid for all m and alpha ≤ 1.

Network exhibits phase transition at alpha=1 affecting clustering and path length.

Assortativity varies with alpha, being positive for alpha<1 and negative for alpha>1.

Abstract

In recent years there has been considerable interest in the structure and dynamics of complex networks. One of the most studied networks is the linear Barab\'asi-Albert model. Here we investigate the nonlinear Barab\'asi-Albert growing network. In this model, a new node connects to a vertex of degree $k$ with a probability proportional to $k^{\alpha}$ ($\alpha$ real). Each vertex adds $m$ new edges to the network. We derive an analytic expression for the degree distribution $P(k)$ which is valid for all values of $m$ and $\alpha \le 1$. In the limit $\alpha \to -\infty$ the network is homogeneous. If $\alpha > 1$ there is a gel phase with $m$ super-connected nodes. It is proposed a formula for the clustering coefficient which is in good agreement with numerical simulations. The assortativity coefficient $r$ is determined and it is shown that the nonlinear Barab\'asi-Albert network is assortative (disassortative) if $\alpha < 1$ ($\alpha > 1$) and no assortative only when $\alpha = 1$. In the limit $\alpha \to -\infty$ the assortativity coefficient can be exactly calculated. We find $r=7/13$ when $m=2$. Finally, the minimum average shortest path length $l_{min}$ is numerically evaluated. Increasing the network size, $l_{min}$ diverges for $\alpha \le 1$ and it is equal to 1 when $\alpha > 1$.