Spreading Phenomena - Chapter 10

Consider the problem of the propagation of an SIS infection in two types of networks:

1. An aggregated network, constructed by summing all contacts over the course of a week.
2. A temporal network, in which each contact occurs only at the moment when it was recorded.

The figure below shows the fraction of infected individuals I(t) over time in each network.

The structural and epidemiological parameters are:

• Aggregated network:
  • average degree \( \langle k \rangle = 12 \)
  • second moment \( \langle k^2 \rangle = 260 \)
• Temporal network:
  • effective average degree \( \langle k_{\text{eff}} \rangle = 5 \)
  • effective second moment \( \langle k_{\text{eff}}^2 \rangle = 40 \)

Model rates:

• transmission \( \beta = 0.04 \)
• recovery \( \mu = 0.02 \)

Consider the approximate epidemic threshold for general networks:

\[ \lambda_c = \frac{\mu}{\beta}\,\frac{\langle k\rangle}{\langle k^2\rangle}, \qquad\text{where}\qquad \lambda = \frac{\beta}{\mu} \]

Which option correctly explains why the red curve (temporal network) shows much lower propagation than the blue curve (aggregated network)?

A.
The blue curve has a higher threshold because the aggregated connectivity is higher, making it harder for the epidemic to be sustained.

B.
In the temporal network, fewer simultaneous contacts reduce \( \lambda_c \), which facilitates spreading but lowers the final plateau.

C.
The temporal network shows weaker propagation because its second moment \( \langle k^2\rangle \) is much smaller, increasing the threshold \( \lambda_c \) and making sustained transmission more difficult.

D.
The difference between the curves occurs because \( \mu \) is higher in the temporal network — the network structure does not affect the threshold when \( \beta \) and \( \mu \) are fixed.

E.
None of the above.

Original idea by: Luiza Barguil