2025-307

Which of the following statements correctly describes the relationship between network structure and the spread of pathogens, ideas, or computer viruses?

A) The structure of the underlying network is irrelevant to the spreading process; the only factor that matters is the inherent contagiousness of the agent (virus or idea). 

B) Biological diseases spread through contact networks, whereas digital viruses and social phenomena spread randomly without following any specific network topology. 

C) The contact network serves as the transmission medium for spreading processes, meaning that the network's topology significantly determines the speed, reach, and pattern of the diffusion. 

D) Spreading phenomena are strictly limited to biological systems (like the SARS outbreak), and network science concepts cannot be effectively applied to digital or social contexts.

E) None of the above.

Original idea by: Augusto Cruz

2025-306

Consider the propagation of a virus modeled by the SIS (Susceptible-Infected-Susceptible) system in two distinct networks:

• A Random Network (Erdős-Rényi).
• A Scale-Free Network.

Both networks have the same number of nodes and the same average degree.

Given that the epidemic threshold is defined by the critical spreading rate, below which the virus dies out exponentially, select the correct statement regarding the behavior of this threshold and immunization strategies:

1. Since both networks have the same average degree , the epidemic threshold will be identical for both, as the average connection density determines the initial spreading speed in any topology.
2. The Random Immunization strategy is equally effective in both types of networks, meaning that the same fraction of nodes have to be immunized in both networks to stop the epidemic.
3. In Scale-Free Networks, with degree exponent \(\gamma\) between 2 and 3, the hubs are less relevant, and the network behaves similarly to a random network.
4. In the Scale-Free Network, the epidemic threshold is given by \(\langle k \rangle / \langle k^2 \rangle\).
5. None of the above.

Original idea by: Giancarlo Maldonado Cárdenas

2025-305

Consider immunization strategies discussed in the context of network epidemics. Which of the following statements is incorrect?

1. Random immunization performs poorly in scale-free networks because it rarely targets highly connected hubs.
2. Targeted immunization raises the epidemic threshold by removing the hubs responsible for rapid spreading.
3. Random immunization always outperforms targeted immunization in networks where \(\langle k^2\rangle\) is finite.
4. Immunizing a randomly selected neighbor of a randomly chosen individual tends to target high-degree nodes due to the friendship paradox.
5. None of the above

Original idea by: Jhonatan Cléto

2025-304

In the context of epidemic spreading on networks, which statement correctly explains why scale-free networks allow epidemics to spread more easily than random networks?

1. Because scale-free networks have a lower average degree \(\langle k\rangle\), making it harder to contain infections.,/li>
2. Because scale-free networks follow homogeneous mixing, allowing each node the same chance to infect others.
3. Because scale-free networks contain hubs with very high degree, causing \(\langle k^2\rangle\) to diverge and making the epidemic threshold \(\lambda_c\) approach zero.
4. Because scale-free networks have stronger community structure, slowing down infection and increasing the epidemic threshold. 
5. None of the above.

Original idea by: Thiago Soares Laitz

2025-294

You are a systems analyst at a major data center, and you are tasked with evaluating the robustness of your facility's network infrastructure against random node failures.

Your analysis of the network topology reveals that the average degree <k> of the nodes is 6.25. Based on your understanding of robustness, what breakdown threshold does the network at the datacenter need to have to display enhanced robustness?

a) 0.85

b) 0.84

c) 0.17

d) 0.16

e) None of the above

Original idea by: Alexandre Petrachini

2025-303

In the context of the classic SI, SIS, and SIR epidemiological models, which of the following statements is incorrect?

A. All three models predict that the epidemic ends only after every individual has been infected at least once.

B. In the Susceptible-Infected (SI) model, each individual in the population is assumed to be either healthy or infected.

C. In the Susceptible-Infected-Susceptible (SIS) model, an individual who becomes infected with the disease can recover and return to the susceptible state, allowing reinfection.

D. In the Susceptible-Infected-Recovered (SIR) model, individuals who have been infected acquire immunity or die from the disease and therefore do not become susceptible again.

E. None of the above

Original idea by: Mylena Roberta

2025-302

Consider the SI model (Susceptible-Infected) under the homogeneous mixing assumption. In this model, the spread of the virus is driven by random contacts between individuals.

You are given the following data for a specific population at time \(t\):

• Total Population (\(N\)): 10,000 individuals

• Current Susceptible individuals (\(S\)): 9,500

• Current Infected individuals (\(I\)): 500

• Average degree (contacts per individual per unit time, \(\langle k\rangle\)): 12

• Transmission probability (\(\beta\)): 0.05

At this specific moment, what is the average rate of new infections?

A) 275

B) 285

C) 570

D) 600

E) None of the above

Original idea by: Alexandre Petrachini