Clustering properties of a generalised critical Euclidean network

TL;DR

This paper investigates a generalized model of growing networks with properties like scale-free degree distribution, small diameter, and high clustering, revealing phase boundaries and new universality classes based on connection probabilities.

Contribution

It introduces a generalized network growth model with tunable parameters, identifying conditions for scale-free behavior and discovering a new universality class for certain parameter ranges.

Findings

Network is scale-free for α > -0.5 at β=1 with γ=3.

Scale-free behavior persists for β > 1 when α < -0.5, indicating a new universality class.

The network maintains a small diameter across the scale-free region.

Abstract

Many real-world networks exhibit scale-free feature, have a small diameter and a high clustering tendency. We have studied the properties of a growing network, which has all these features, in which an incoming node is connected to its $i$th predecessor of degree $k_i$ with a link of length $\ell$ using a probability proportional to $k^\beta_i \ell^{\alpha}$. For $\alpha > -0.5$, the network is scale free at $\beta = 1$ with the degree distribution $P(k) \propto k^{-\gamma}$ and $\gamma = 3.0$ as in the Barab\'asi-Albert model ($\alpha =0, \beta =1$). We find a phase boundary in the $\alpha-\beta$ plane along which the network is scale-free. Interestingly, we find scale-free behaviour even for $\beta > 1$ for $\alpha < -0.5$ where the existence of a new universality class is indicated from the behaviour of the degree distribution and the clustering coefficients. The network has a small diameter in the entire scale-free region. The clustering coefficients emulate the behaviour of most real networks for increasing negative values of $\alpha$ on the phase boundary.