2026-368

Consider the following partitions over the same graph:

Which alternative lists the partitions in ascending order of modularity ?

A) PA, PB, PC, PD

B) PD, PA, PB, PC

C) PB, PA, PD, PC

D) PA, PD, PB, PC

E) None of the above

Original idea by: Fernando de Facio Rossetti

2026-367

Consider the context of a disease spreading through a global network. Which of the following statements is false?

a) If a disease has different fatality rates depending on specific factors, such as COVID-19 being more dangerous for older people, then the best vaccination strategy for minimizing the number of deaths would be to prioritize high-risk groups. However, if the objective is to eradicate the disease, the best strategy would be to prioritize nodes with the highest number of links, since these nodes contribute more to disease transmission.

b) A disease with a very high fatality rate tends to be less likely to spread globally and become a pandemic, especially if infected individuals die before transmitting the disease to many others. In network terms, death removes a node from the transmission network, reducing the number of possible future infections.

c) If a vaccine exists and 90% of the population is vaccinated at random, the remaining 10% of unvaccinated individuals will not be affected at all by the existence of the vaccine.

d) Early identification of a disease, together with measures such as quarantine and social isolation, is one of the first and most important steps to prevent a local outbreak from becoming a pandemic.

e) None of the above is false.

Original idea by: Henrique Campos Padula

2026-366

The network below shows the contacts in a small population, modelled with the SI epidemic model.

Given a transmission rate β = 0.1, what is the characteristic time τ of the epidemic?

1. τ = 4
2. τ = 5
3. τ ≈ 2.86
4. τ = 0.5
5. None of the above

Original idea by: Gustavo P. C. P. da Luz

2026-365

How do assortative degree correlations typically influence the initial spread of a pathogen compared to neutral or disassortative networks?

1. They have no noticiable impact on epidemic dynamics, since the network scale regime influences the initial spread more than degree correlations.
2. They slow down the spread because hubs are disconnected from susceptible nodes.
3. They increase the epidemic threshold, making outbreaks less likely.
4. They accelerate the spread because highly connected nodes are more likely to be linked to other highly connected nodes.
5. None of the above.

Original idea by: Giuliano Macedo.

2026-364

A zombie infection spreads according to the SI model under the homogeneous mixing assumption. Researchers observe that faster zombies encounter healthy individuals more frequently, so the infection rate is proportional to the zombie walking speed:

$$\beta =c{v}_z,$$

where $v_z$ is the average zombie speed and $c$ is a proportionality constant.

A mutation doubles their average speed. The contact network remains unchanged, with

$$⟨k⟩=8$$

and

$$⟨k^2⟩=100.$$

According to the SI model on networks, by what factor does the characteristic time change after the mutation?

A) It is reduced by half.

B) It doubles.

C) It increases by a factor of four.

D) It remains unchanged.

E) None of the above.

Original idea by: George Gigilas Junior

2026-363

Analyze the following statements about the impact of degree correlations on spreading phenomena in networks, and determine whether each statement is True (T) or False (F):

1. In the SIS model applied to scale-free networks whose second moment diverges, the epidemic threshold remains zero regardless its degree correlation (assortative, disassortative, or neutral).
2. Assortative correlation accelerates pathogen propagation because hubs, which are typically among the first nodes to become infected, preferentially connect to other hubs, facilitating rapid transmission through the most connected parts of the network.
3. In scale-free networks, a weakly infectious pathogen will always die out despite hubs existence, unlike random networks where the homogeneous connectivity allows the pathogen to spread steadily across the population.
4. In networks with finite second moment, disassortative correlations raise the epidemic threshold λ_c, making it harder for a pathogen to sustain itself in the network, while assortative correlations produce the opposite effect by lowering λ_c.

Choose the correct alternative:

A) T-T-T-F

B) F-T-F-T

C) T-T-F-T

D) T-F-T-T

E) None of the above

Original idea by: Gabriela Caspa

2026-362

Consider an epidemic spreading on a large scale-free network whose degree distribution follows

$$P(k) \sim k^{-\gamma}$$

with \(2 < \gamma < 3\). According to Network Epidemics theory field, which of the following statements is correct?

A) The epidemic spreads more slowly than in a random network because hubs act as bottlenecks.

B) In the SIS model, the epidemic threshold \(\lambda_c\) tends to zero as the network size becomes very large (\(N \to \infty\)).

C) The characteristic spreading time increases with the second moment \(\langle k^2 \rangle\).

D) The early-time behavior of the SI model is linear in time rather than exponential.

E) None of the above.

Original idea by: Carlos Trindade

2026-361

Given a network with a defined community division, which has positive modularity, choose the correct alternative:

a) Applying a degree-preserving exchange where two inter-community edges are removed and two intra-community edges are created will increase the modularity.

b) Merging two different communities into one will increase the modularity.

c) Removing an edge between nodes in different communities will increase the modularity.

d) Adding an intra-community edge to the network will necessarily increase the modularity.

e) None of the above.

Original idea by: João Pedro Carolino Morais

2026-360

Sophia is collecting stickers for the 2026 World Cup album. To complete her collection, she plans to leave home (V0), visit all the sticker exchange points (V1, V2, V3, V4) in the city of Campinas exactly once, and then return home. The figure below shows a weighted adjacency matrix of an undirected graph.

	V0	V1	V2	V3	V4
V0	0	14	16	0	0
V1	14	0	11	12	14
V2	16	11	0	14	17
V3	0	12	14	0	15
V4	0	14	17	15	0

• each vertex (V) represents a sticker exchange point; 
• each link represents the possibility of traveling between two points; 
• the weight of each edge represents the amount Sophia will spend in R$ when taking that route; 
• there is only a connection between two vertices when the weight shown in the matrix is greater than zero. 

Considering this information, help Sophia choose a route where she visits all the sticker exchange points exactly once, returns home, and has the lowest possible total cost.

What is the minimum-cost cycle for Sophia? 

a) V0 -> V1 -> V2 -> V3 -> V4 -> V1 -> V0.

b) V0 -> V1 -> V3 -> V4 -> V2 -> V0.

c) V0 -> V2 -> V1 -> V3 -> V4 -> V1 -> V0.

d) V0 -> V2 -> V3 -> V4 -> V1 -> V0.

e) None of the above.

Original idea by: Tássia Martins

2026-359

A research team is modeling the spread of a highly contagious pathogen in a densely packed, fully connected community where the homogeneous mixing assumption holds. 

Because they are modeling the disease beyond its initial early stages, they cannot use the simple exponential approximation. Instead, they use the exact SI model equation:

i(t)=i0exp(βt)1−i0+i0exp(βt)

At the start of the observation (t=0), exactly 10% of the population is infected (i0=0.1). After a certain number of days t, the infection's exponential growth factor reaches eβt=9.

What is the fraction of infected individuals i(t) in the population at this time?

a) 81.8%

b) 33.3%

c) 10.0%

d) 50.0%

e) None of the above.

Original idea by: Matheus de Oliveira Saldanha

2026-358

Consider a Euclidian TSP with 5 cities located in a 2D plane at the following coordinates:

A (1, 2); B (5, 3); C (6, 7); D (2, 8); E (4, 5).

A delivery starts at city A and uses Nearest Neighbor Heuristic to construct a complete tour. Which alternative represents the total distance (cost) of the resulting tour, rounded to two decimals?

a) 13.31

b) 17.16

c) 15.52

d) 19.39

e) None of the above

Original idea by: Matheus Rufino

2026-357

When modeling the spread of a pathogen or an information cascade, the choice between the SI, SIS, and SIR compartmental frameworks depends entirely on the biological or behavioral traits of the agents involved. Under the homogeneous mixing assumption, each of these three classical models leads to a fundamentally different outcome in its final regime.

Which of the following statements correctly identifies the long-term behavior (final regime) of the SIR model and explains how it differs from the SI and SIS models?

A) In the SIR model, the fraction of infected individuals eventually drops to zero because infected individuals transition into a removed state where they develop permanent immunity or die.

B) The SIR model is the only framework where the entire population eventually ends up infected at the same time, unlike the SIS model which always maintains a mix of healthy and sick individuals.

C) In the final regime of the SIR model, the system reaches a steady endemic state where a fixed, non-zero fraction of the population remains actively infected forever.

D) The SIR model differs because it completely lacks an initial exponential growth regime, making its spread linear and predictable from day one.

E) None of the above.

Original idea by: Maria Luiza Ramos da Silva