Clustering properties of a generalised critical Euclidean network

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 $k^β_i \ell^α$. For $α> -0.5$, the network is scale free at $β= 1$ with the degree distribution $P(k) \propto k^{-γ}$ and $γ= 3.0$ as in the Barabási-Albert model ($α=0, β=1$). We find a phase boundary in the $α-β$ plane along which the network is scale-free. Interestingly, we find scale-free behaviour even for $β> 1$ for $α< -0.5$ 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 $α$ on the phase boundary.