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

2026-356

Regarding the relationships among cliques, strong communities, and weak communities, which of the following statements is correct?

1. Every weak community is a strong community
2. Every clique is a strong community, regardless of the external degrees of its nodes
3. Every \(n\)-clique is a strong community only if the external degree of every node in the clique is smaller than \(n\)
4. Every clique is a weak community, but no clique can be a strong community
5. None of the above

Original idea by: Giuliano Macedo.

2026-355

Analyze the following statements about the Agglomerative (Ravasz) and Divisive (Girvan–Newman) hierarchical clustering algorithms, and determine whether each statement is True (T) or False (F):

 1) The Agglomerative algorithm starts by treating each node in the network as an individual community and repeatedly merges the most similar communities until a stopping condition is reached. This stopping condition occurs when the density inside each community becomes maximal.

2) The Divisive algorithm follows a top-down strategy: it initially treats the entire network as a single community and progressively splits the network into smaller communities.

3) The result of the Agglomerative algorithm depends on the linkage criteria adopted (single, complete, or average linkage).

4) In the Girvan–Newman algorithm, communities are identified by iteratively removing links with high centrality, since these links are more likely to connect different communities.

5) In the Agglomerative hierarchical clustering, once two communities are merged, the algorithm may later separate them again if a stronger similarity pattern is detected.

Choose the correct alternative:

A) F-T-T-F-F

B) T-T-T-F-F

C) F-T-T-T-F

D) F-T-T-T-T

E) None of the above

Original idea by: Gabriela Caspa

2026-354

Consider the following adjacency matrix of an undirected network

$$A=
  \begin{bmatrix}
  0&1&1&0&0&0&0\\
  1&0&1&1&0&0&0\\
  1&1&0&1&1&0&0\\
  0&1&1&0&0&1&0\\
  0&0&1&0&0&1&1\\
  0&0&0&1&1&0&1\\
  0&0&0&0&1&1&0
  \end{bmatrix}$$

Consider the candidate communities 

$$C_1=\{1,2,3,4\}
  \qquad
  C_2=\{5,6,7\}$$

Which statement is correct?

1. \(C_1\) is a clique and \(C_2\) is a strong community
2. \(C_1\) is a strong community but not a clique, while \(C_2\) is a clique
3. Both \(C_1\) and \(C_2\) are strong communities but not cliques
4. \(C_1\) is a weak but not strong community and \(C_2\) is a clique
5. None of the above

Original idea by: Antonio De Cesare Del Nero

2026-353

When analyzing the resilience of complex systems, network science literature establishes a fundamental distinction between static structural robustness and dynamic robustness

Consider Watts' Linear Threshold Model, where a network initialized with all functional nodes undergoes a local shock. In this model, a healthy node \(i\) transitions to a failed state if the fraction \(f_i\) of its inoperable neighbors exceeds a local critical threshold \(\phi\).

Based on the theoretical pillars of Network Robustness and the phase diagrams of this model, select the alternative that correctly describes the relationship between network topology, stability limits, and the propagation of catastrophic cascades:

A) In Watts' cascade model, the occurrence of a global avalanche exhibits a non-monotonic dependence on the network's average connectivity \(\langle k \rangle\). In highly dense networks (high \(\langle k \rangle\)), the system enters a subcritical regime where the failure of a single neighboring node represents a perturbation fraction \(1/k\) that is strictly lower than the critical threshold \(\phi\), thereby locally confining the impact and preventing global cascade propagation.

B) Scale-free networks with a degree exponent \(2 < \gamma \leq 3\) are ultra-robust against random structural failures (\(f_c\) → 1) due to the topological protection provided by hubs. This property automatically guarantees absolute immunity against global dynamic cascades triggered by minor shocks in peripheral nodes, since hubs invariably act as static sinks that absorb the overload and halt the domino effect.

C) The Molloy-Reed criterion (\(\kappa = \langle k^2 \rangle/\langle k \rangle > 2\)), which dictates the existence of a giant connected component in networks under inverse percolation, perfectly defines the dynamic threshold for avalanches. This explains why networks violating this inequality become inherently immune to cascading failures in the branching model, keeping the critical avalanche exponent unaltered.

D) According to the modeling of cascades via branching processes, the exponent α of the avalanche size distribution (\(P(S) \sim S^{-\alpha}\)) is a universal constant fixed at \(\alpha = 3/2\) for any complex network configuration, regardless of the underlying shape of the system's original degree distribution \(p_k\).

 E) None of the above.

Original idea by: Maria Luiza Ramos da Silva 

2026-352

Considering a scale-free network with a degree distribution \(P(k) \sim k^{-\gamma}\) and its critical threshold given by

$$f_c = 1 - \frac{1}{\frac{\langle k^2\rangle}{\langle k\rangle} - 1},$$

select the correct alternative.

1. For \(2 < \gamma < 3\), the second moment of the degree distribution tends to zero, indicating extreme fragility to random failures.
2. For \(\gamma > 3\), the second moment of the degree distribution diverges, making the network completely robust.
3. The value of \(f_c\) is independent of the exponent, depending only on the network size \(N\).
4. For \(2 < \gamma < 3\), the second moment of the degree distribution diverges and \(f_c\) tends to 1, indicating that the network can sustain the random removal of almost all nodes without losing the giant component.
5. None of the above

Original idea by: Julia de Pietro Bigi

2026-351

 Consider the network below:

What is the Pearson degree correlation coefficient (r) for this network using Mark Newman's definition?

A. r = 0, since the average degree of every node’s neighbors equals ⟨k⟩.

B. r = -2/3, indicating the network is assortative.

C. r = +2/3, indicating the network is disassortative.

D. r = -1/2, indicating the network is assortative.

E. None of the above.

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