Question 3 (Scale-free Networks)

Using the continuum formalism, a scale-free network has

\[ p(k)=Ck^{-\gamma}, \qquad k \geq 1, \]

with \(\gamma = 2.5\). Which expression gives the second moment \(\langle k^2 \rangle\) for a finite network with cutoff \(k_{\max}\)?

1. \(\langle k^2 \rangle = \int_1^{k_{\max}} k^{-0.5}\,dk\)
2. \(\langle k^2 \rangle = C\int_1^{k_{\max}} k^{-0.5}\,dk\)
3. \(\langle k^2 \rangle = C\int_1^{k_{\max}} k^{-2.5}\,dk\)
4. \(\langle k^2 \rangle = \int_1^{k_{\max}} k^{2.5}\,dk\)
5. None of the above

Original idea by: Antonio De Cesare Del Nero