Generation of uncorrelated random scale-free networks

Uncorrelated random scale-free networks are useful null models to check the accuracy and the analytical solutions of dynamical processes defined on complex networks. We propose and analyze a model capable of generating random uncorrelated scale-free networks with no multiple and self-connections. The model is based on the classical configuration model, with an additional restriction on the maximum possible degree of the vertices. We check numerically that the proposed model indeed generates scale-free networks with no two- and three-vertex correlations, as measured by the average degree of the nearest neighbors and the clustering coefficient of the vertices of degree $k$, respectively.

Figure 1

Average nearest-neighbor degree of vertices of degree $k$, ${\overline{k}}_{\mathit{nn}}\left(k\right)$ (a), and average clustering coefficient $\overline{c}\left(k\right)$ (b) for the original CM algorithm with different degree exponents $\gamma$. Network size is $N={10}^5$.

Figure 2

Average nearest-neighbor degree of vertices of degree $k$, ${\overline{k}}_{\mathit{nn}}\left(k\right)$ (a), and average clustering coefficient $\overline{c}\left(k\right)$ (b) for the UCM algorithm with different degree exponents $\gamma$. Network size is $N={10}^5$.

Figure 3

Numerical average clustering coefficient $⟨c⟩$ as a function of the corresponding theoretical value, given by Eq. (5), for the CM (open symbols) and the UCM (solid symbols) algorithms. The different points for each value of $\gamma$ represent different network sizes $N={10}^3$, $3\times {10}^3$, ${10}^4$, $3\times {10}^4$, and ${10}^5$.