2026-352

Considering a scale-free network with a degree distribution \(P(k) \sim k^{-\gamma}\) and its critical threshold given by

$$f_c = 1 - \frac{1}{\frac{\langle k^2\rangle}{\langle k\rangle} - 1},$$

select the correct alternative.

1. For \(2 < \gamma < 3\), the second moment of the degree distribution tends to zero, indicating extreme fragility to random failures.
2. For \(\gamma > 3\), the second moment of the degree distribution diverges, making the network completely robust.
3. The value of \(f_c\) is independent of the exponent, depending only on the network size \(N\).
4. For \(2 < \gamma < 3\), the second moment of the degree distribution diverges and \(f_c\) tends to 1, indicating that the network can sustain the random removal of almost all nodes without losing the giant component.
5. None of the above

Original idea by: Julia de Pietro Bigi