MO412 - Mateus de Padua Vicente

A research team is investigating the spread of a pathogen in a large contact network using an SIR model. During simulations, they obtained the following three observations: Observation A: When β is increased while μ is kept fixed, the distribution of outbreak sizes shows a sudden transition from mostly small outbreaks to frequent large outbreaks. Observation B : In networks with highly irregular degree distributions, a small fraction of nodes is responsible for the vast majority of early transmissions. Observation C: After the outbreak peaks, the number of infectious nodes always decreases to zero, even when β ≫ μ.  The team proposes the following four hypotheses to explain the observations: A corresponds to the existence of an epidemic threshold related to the transmissibility T = β/(β + μ). B indicates that heterogeneity in the degree distribution affects both the threshold condition and the probability of large outbreaks. C implies that the SIR model can exhibit a sustained...

A group of network scientists is analyzing a large social network to uncover community structures. They are evaluating several theoretical principles and algorithmic strategies to decide which framework better explains the observed patterns. After reviewing the literature, they make the following statements: I. A community structure should balance internal density and external sparsity; however, a subgraph that maximizes internal density alone may fail to capture community boundaries consistent with the overall network topology. II. In hierarchical clustering of networks, bottom-up strategies tend to merge nodes with high topological similarity or shared neighbors, while top-down strategies progressively dismantle inter-community links that maintain global connectivity. III. The distinction between strong and weak communities reflects whether the internal–external link condition must hold for each vertex individually or only on average for the entire subgraph. IV.  Finding all...