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
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?
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
Regarding the relationships among cliques, strong communities, and weak communities, which of the following statements is correct?
Original idea by: Giuliano Macedo.
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
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?
Original idea by: Antonio De Cesare Del Nero
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
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.
Original idea by: Julia de Pietro Bigi
