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Spreading Phenomena

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 wher...
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Network Robustness

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 fi of its inoperable neighbors exceeds a local critical threshold φ (fi > φ). 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) Scale-free networks with a degree exponent 2 < γ ≤ 3 are ultra-robust against random structural failures (fc → 1) due to the topological protection provided by hubs. This property automatically guarantees absolute immunity against global dynamic cascades triggered by minor shocks in peri...
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Barabási-Albert

In the study of complex networks, the Barabási-Albert model describes how a node's connectivity evolves over time using the following equation: ki(t) = m * (t / ti)^0.5 In this formula, ti represents the time node i was introduced to the network, and t is the current time. Based on this dynamics, what fundamental conclusion can be drawn regarding the formation of major "hubs"? A) A node's growth rate is purely linear and depends exclusively on the number of new connections (m) added at each step, regardless of the entry time. B) There is a "first-mover advantage," meaning that older nodes are more likely to become hubs because they have been exposed to the preferential attachment mechanism for a longer period. C) A node's degree tends to decrease as the network grows (as t increases toward infinity), allowing newer nodes to quickly reach the same level of connectivity as veteran nodes. D) The resulting degree distribution is independent of the nodes...
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Network Flow

In the Push-Relabel algorithm for finding the maximum flow in a network, the state of an intermediate node u is defined by its excess flow e(u) and its height h(u). Consider this snapshot for node u: Excess flow: e(u) = 8 Current heights: h(u) = 4; h(v) = 4; h(w) = 5 Residual capacities: c(u, v) = 5; c(u, w) = 10 What is the next operation for node u? A) Push to node v, sending 5 units of flow. B) Relabel node u, new height becomes h(u) = 5. C) Push to node w, sending 8 units of flow. D) Relabel node u, new height becomes h(u) = 6. Original idea by: Maria Luiza Ramos da Silva
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DFS vs BFS

You are analyzing a directed graph that represents a complex network of dependencies between software modules. You need to ensure the system is free of circular dependencies and find the minimum number of steps to reach a specific module from the source. Based on the properties of Breadth-First Search (BFS) and Depth-First Search (DFS) presented in the course, which statement correctly identifies the best approach? A) BFS should be used to find the shortest path (minimum steps) because it provides parent links with a distance guarantee, while DFS is required to detect circular dependencies by identifying "backward edges." B) DFS should be used to find the shortest path because its "finishing times" represent the most efficient order, while BFS is the only tool capable of "edge classification" to find cycles. C) In this directed graph, both BFS and DFS will naturally identify "forward" and "cross" edges, making them equally effective ...
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