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 can depend on the linkage criteria adopted, such as 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

2026-350

Epidemiologists are studying the spreading of a disease through an interaction network. After collecting data, they find that the network's degree distribution follows a power law \(p_k \sim k^{-\gamma}\) with exponent \(gamma = 2.5\), suggesting that a small number of highly connected individuals ("superspreaders") play a dominant role in transmission. The network has a minimum degree \( k_{min} = 2\), meaning every user follows at least two others, and an average degree \( \langle k \rangle = 8\). Structural analysis reveals a cutoff \(k_s = 64\), beyond which the simple-graph constraint begins to limit how hubs can connect to one another. Based on this information, which of the following alternatives is correct?

1. \(k_{max} = 128\), and the network presents structural assortativity because \(k_s < k_{max}\).
2. \(k_{max} = 64\), and the network does not present structural disassortativity because \(\gamma > 2\).
3. \(k_{max} = 128\), and the network presents structural disassortativity because \(k_s < k_{max}\).
4. \(k_{max} = 64\), and the network does not present structural disassortativity because \(k_s \geq k_{max}\).
5. None of the above

Original idea by: Carlos Trindade