Clustering properties of a generalized critical Euclidean network

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