Quiz question 4 (21/04/2023) - Calculus

Consider a scale-free network with a fixed constant \(γ = 2.5\) at all times and a number of nodes that increase over time \(t\) according to the relation \(N(t) = e^{10^6 t}\). Which alternative correctly estimates and interprets the variation of the average distance over time (\( \frac{d \left\langle d \right\rangle}{dt} \)):

1. \( e^{10^6 t} \), as the network grows rapidly, \( \frac{d \left\langle d \right\rangle}{dt} \) also increases rapidly over time, causing the network to lose its ultra-small-world property.
2. \( \frac{1}{t \cdot e^{10^6} + ln (t)} \), as the network grows rapidly, \( \frac{d \left\langle d \right\rangle}{dt} \) gets too small, a property of the ultra-small-world present on scale-free networks.
3. \( \frac{e^{ 10^6 t}}{10^6 t} \), as the network grows rapidly, \( \frac{d \left\langle d \right\rangle}{dt} \) also increases rapidly over time, but not too fast, which some time cause the loss of its ultra-small-world property.
4. \( \frac{1}{t (10^6 + ln(t))} \), as the network grows rapidly, \( \frac{d \left\langle d \right\rangle}{dt} \) gets too small, a property of the ultra-small-world present on scale-free networks.
5. none of above

Original idea by: Anderson Nogueira Cotrim