Spreading Phenomena

Assuming the mean-field approximation (homogeneous mixing) for spreading phenomena, analyze the mathematical consistency of the following statements regarding the temporal and asymptotic solutions of the SI and SIS models:

A) In the initial linear regime ($t \to 0$), the infection growth rate is independent of the removal parameter $\mu$. For both SI and SIS dynamics, the expansion is driven solely by $\beta \langle k \rangle$, resulting in identical initial Lyapunov exponents.

B) The characteristic relaxation time $\tau$ for the SIS model follows the relationship

$$\tau \propto [\mu(R_0 - 1)]^{-1}.$$

This implies that as the system approaches the epidemic threshold ($R_0 \to 1^+$), it experiences critical slowing down, where the time required to reach the steady state diverges.

C) For the SI model, governed by

$$\frac{di}{dt} = \beta \langle k \rangle\, i(1-i),$$

the average degree $\langle k \rangle$ acts as a scaling parameter for the asymptotic amplitude, determining whether the final infection fraction $i(\infty)$ reaches partial ($<1$) or total ($=1$) saturation.

D) The existence of a non-trivial endemic fixed point ($i(\infty) > 0$) in the SIS model requires the recovery rate to exceed the effective transmission rate. Mathematically, the system sustains an epidemic only if the inequality

$$\mu > \beta \langle k \rangle$$

is satisfied.

E) None of the above.

Original idea by: Matteus Vargas Simão da Silva

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