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 \( N(t) = e^{10^6t} \). Which alternative correctly estimates and interprets the variation of the average distance over time (\( d\langle d \rangle/dt \)):

1. \( e^{10^6t} \), as the network grows rapidly, \( d\langle d \rangle/dt \) also increases rapidly over time, causing the network to lose its ultra-small-world property.
2. \( 1/(te^{10^6} + \ln t) \), as the network grows rapidly, \( d\langle d \rangle/dt \) gets too small, a consequence of the ultra-small-world property of scale-free networks in this regime.
3. \( e^{10^6t}/(10^6t) \), as the network grows rapidly, \( d\langle d \rangle/dt \) 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, \( d\langle d \rangle/dt \) 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