Local clustering in scale-free networks with hidden variables

We investigate the presence of triangles in a class of correlated random graphs in which hidden variables determine the pairwise connections between vertices. The class rules out self-loops and multiple edges. We focus on the regime where the hidden variables follow a power law with exponent $\tau \in (2,3)$, so that the degrees have infinite variance. The natural cutoff $h_c$ characterizes the largest degrees in the hidden variable models, and a structural cutoff $h_s$ introduces negative degree correlations (disassortative mixing) due to the infinite-variance degrees. We show that local clustering decreases with the hidden variable (or degree). We also determine how the average clustering coefficient $C$ scales with the network size $N$, as a function of $h_s$ and $h_c$. For scale-free networks with exponent $2<\tau <3$ and the default choices $h_s\sim N^{1/2}$ and $h_c\sim N^{1/(\tau -1)}$ this gives $C\sim N^{2-\tau }lnN$ for the universality class at hand. We characterize the extremely slow decay of $C$ when $\tau \approx 2$ and show that for $\tau =2.1$, say, clustering starts to vanish only for networks as large as $N={10}^9$.

Figure 1

$c\left(h\right)$ for $\tau =2.1,2.5,2.9$ and networks of size $N={10}^6$, using $h_{min}=1$. The markers indicate the average of ${10}^5$ simulations, and the solid lines follow from (17).

Figure 2

$C_{ab}^{max}\left(\tau \right)$ as a function of $\tau$, with $r$ as in Eq. (6), using $h_{min}=1$.

Figure 3

$C_{ab}\left(\tau \right)$ for $\tau =2.1,2.5,2.9$, choices (6) (solid line) and (8) (dashed line) and networks of size $N={10}^k$ for $k=4,5,6$, using $h_{min}=1$. The markers indicate the average of ${10}^5$ simulations, and the solid lines follow from (24) and (F2).

Figure 4

$C_{ab}^{max}\left(\tau \right)$ as a function of $N$ for $\tau$ close to 2, with $r\left(u\right)$ as in (6) and $h_{min}=1$. The marks indicate the value of $N_{\tau ,2}$ as calculated in Table 1.