Network Robustness

Using the giant-component breakdown criterion, $\frac{\langle k'^2 \rangle_f}{\langle k' \rangle_f} = 2$, and replacing $f$ by $f_c$ in the expressions for the residual moments, derive the relation that expresses the second moment of the original degree distribution, $\langle k^2 \rangle$, in terms of the original first moment, $\langle k \rangle$, and the critical percolation threshold $f_c$.

The derived relation is:

a) $\langle k^2 \rangle = \langle k \rangle \left( 1 + \frac{1}{1 - f_c} \right)$
b) $\langle k^2 \rangle = \langle k \rangle \frac{2 - f_c}{1 - f_c}$
c) $\langle k^2 \rangle = 2 \langle k \rangle (1 - f_c) + \langle k \rangle f_c$
d) $\langle k^2 \rangle = \frac{\langle k \rangle}{(1 - f_c)^2}$
e) None of the above

Original idea by: Matteus Vargas Simão da Silva