2023-209

Consider a scale-free network with a fixed $\gamma =2.5$ at all times and a number of nodes that increases over time according to the relation $N(t)=e^{{10}^{6}t}$. Which alternative correctly estimates and interprets the variation of the average distance over time ($d⟨d⟩/dt$):

1. $e^{{10}^{6}t}$, as the network grows rapidly, $d⟨d⟩/dt$ also increases rapidly over time, causing the network to lose its ultra-small-world property.
2. $1/(t{e}^{{10}^6}+\ln t)$, as the network grows rapidly, $d⟨d⟩/dt$ gets too small, a consequence of the ultra-small-world property of scale-free networks in this regime.
3. $e^{{10}^{6}t}/({10}^{6}t)$, as the network grows rapidly, $d⟨d⟩/dt$ also increases rapidly over time, but not too fast, which may cause the loss of its ultra-small-world property in some cases.
4. $1/(t(e^{{10}^6}+\ln t))$, as the network grows rapidly, $d⟨d⟩/dt$ gets too small, a consequence of the ultra-small-world property of scale-free networks in this regime.
5. None of above

Original idea by: Anderson Nogueira Cotrim