Number of Loops of Size $h$ in Growing Scale-Free Networks

The hierarchical structure of scale-free networks has been investigated focusing on the scaling of the number $N_h(t)$ of loops of size $h$ as a function of the system size. In particular, we have found the analytic expression for the scaling of $N_h(t)$ in the Barabási-Albert (BA) scale-free network. We have performed numerical simulations on the scaling law for $N_h(t)$ in the BA network and in other growing scale-free networks, such as the bosonic network and the aging nodes network. We show that in the bosonic network and in the aging node network the phase transitions in the topology of the network are accompained by a change in the scaling of the number of loops with the system size.

Figure 1

Scaling of $N_h(t)$ as a function in a BA network for $t$ up to $t={10}^4$. The inset shows the values of the measured exponent $\psi$ defined in (16) for $h=3,4,5$.

Figure 2

Scaling of the number of loops $N_h(t)$ of size $h$ with the system size $t$ in the FGR and in the BE phase of a bosonic network. The plotted data have been obtained for a network with $p(ϵ)=2ϵ$ and $ϵ\in (0,1)$ (which has $T_c=0.7$ [19]) at $T=1.5$ (FGR phase) and at $T=0.5$ (BE phase). In the insets we report the value of the exponents $\psi$ (2.6, 3.9, 5.2) and $\xi$ (0.78, 1.28, 156) found in the simulations for $h=3,4,5$, respectively.

Figure 3

The order parameter $k_{\max }/mt$ for a network with aging of the nodes, size $t={10}^4$ and $m=2$. We distinguish among three regions of the phase space: $\alpha >1$, $\alpha \in (-1,1)$, and $\alpha <-1$. In the insets we report the typical behavior of $N_h(t)$ as a function of $t$ for $h=3,4,5$ in the three regions.