Clustering properties of a generalised critical Euclidean network

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 $\mathrm{\ell }$ using a probability proportional to $k_i^{\beta }{\mathrm{\ell }}^{\alpha }$. For $\alpha >-0.5$, the network is scale free at $\beta =1$ with the degree distribution $P(k) \propto k^{-\gamma}$and$\gamma = 3.0$asintheBarab\acute{a}si-Albertmodel($\alpha =0, \beta =1$)\mathrm{.}Wefindaphaseboundaryinthe$\alpha-\beta$ 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.