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2025-007

Question 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 with a degree exponent \(2 Both networks have the same number of nodes \(N\) (where \(N \to \infty\)) and the same average degree \(\langle k \rangle\). Given that the epidemic threshold is defined by the critical spreading rate \(\lambda_c\), below which the virus dies out exponentially, select the correct statement regarding the behavior of this threshold and immunization strategies: A) Since both networks have the same average degree \(\langle k \rangle\), the epidemic threshold \(\lambda_c\) will be identical for both, as the average connection density determines the initial spreading speed in any topology. B) In the Scale-Free Network, the epidemic threshold is given by \(...
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2025-006

Question Consider a network with \( L = 200 \) total edges, composed of multiple communities. Two specific communities, A and B, have total degrees \( k_A = 18 \) and \( k_B = 16 \), with only 2 edges connecting them (\( L_{AB} = 2 \)). Knowing that modularity tends to merge communities when \[ \frac{k_A \cdot k_B}{2L} which of the following statements about detecting these communities is CORRECT? A. Communities A and B will be detected separately because \(\Delta M_{AB} B. The modularity algorithm will inevitably merge A and B due to the resolution limit C. The merger depends exclusively on the number of internal edges in each community D. Such small communities will always have negative modularity E. None of the above Original idea by: Giancarlo Maldonado Cárdenas
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2025-005

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A salesperson wants to plan their route between five cities — A, B, C, D, and E — using the Farthest Insertion algorithm. The initial cycle considered is B–D–E–B, highlighted in dark blue in the figure below. The distances between cities (in arbitrary units) are indicated on the edges of the graph: Map of cities and distances Applying the Farthest Insertion algorithm, determine: The order of insertion of cities outside the initial cycle (A and C). The final cycle resulting after all insertions. A) Order: A, C → Final cycle: B–A–D–C–E–B B) Order: C, A → Final cycle: B–D–C–E–A–B C) Order: C, A → Final cycle: B–C–D–E–A–B D) Order: A, C → Final cycle: B–D–E–C–A–B E) None of the above Original idea by: Giancarlo Maldonado Cárdenas
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2025-004

Consider a simple scale-free network generated through a configuration model with degree exponent  γ = 2.4, minimum degree   kₘᵢₙ = 1, and N = 10⁴ nós. Suppose that the maximum degree is limited by the structural cutoff kₛ ~ N¹ᐟ², and that the empirical degree correlation function follows kₙₙ(k) ~ kᵘ. Based on this scenario, select the correct statement : A) The exponent μ < 0 arises solely from dynamic preferential attachment, since the configuration model preserves assortativity by construction. B) Even though the configuration model is uncorrelated in principle, finite-size effects and the structural cutoff impose an effective dissortativity, resulting in μ < 0. C) Increasing N while maintaining the same γ leads to stronger assortativity, because kₛ grows sublinearly with N. D) Networks with γ < 3 cannot exhibit dissortative mixing, since their hubs dominate the connectivity pattern, forcing μ > 0. E) None of the above. Original idea by: Giancarlo M...
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2025-003

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The figure below shows a power transmission network. Nodes G1 and G2 are generators; C1 and C2 are consumers. The numbers on the edges are the maximum capacities (MW) of the transmission lines.      What is the maximum total flow that can be delivered from the generators (G1, G2) to the consumers (C1, C2)? Alternatives: A) 25 MW B) 30 MW C) 35 MW D) 40 MW E) None of the above Author: Giancarlo Maldonado Cárdenas
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2025-002

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Consider the directed graph below: Applying the Kosaraju-Sharir algorithm, we wish to identify the strongly connected components (SCCs) of the graph. Regarding the steps and the result obtained, analyse the statements below: I. In the first execution of DFS, the completion times of each vertex are used to define the order of exploration in the transposed graph. II. The transposed graph is obtained simply by inverting all the edges of the original graph. III. Each tree in the search forest in the second execution of DFS corresponds to a strongly connected component. IV. The algorithm can identify cycles, but cannot isolate nodes that are not part of any cycle. Which statements are correct? A) I, II, and III only. B) I and IV only. C) II and III only. D) All are correct. E) None of the above. Author: Giancarlo Maldonado Cárdenas
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2025-001

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A group of friends is organizing a secret message run using an oriented and unweighted graph. Each number represents a person, and each area indicates so that the message can be reviewed directly. The graph is shown below: The message starts at friend 1. Considering the execution of the Breadth-First Search (BFS) algorithm starting at 1, and assuming that the adjacency lists are sorted in ascending order, which of the options below correctly shows the order of visitation of the nodes? A. 1 – 2 – 3 – 4 – 5 – 6 B. 1 – 3 – 2 – 5 – 4 – 6 C. 1 – 2 – 3 – 5 – 4 – 6 D. 1 – 3 – 5 – 2 – 4 – 6 E. None of the above Autor: Giancarlo Maldonado Cárdenas
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