Aline - MO412 A

A group of explorers is mapping the connections between caves on a mysterious island. Each cave is a node, and the tunnels connecting them are edges. Studies indicate that the cave network follows a scale-free distribution with N = 20000 caves. The minimum number of tunnels from a cave (k min ) is 3, and the network follows a power-law with exponent \(\gamma = 2.6\). The most valuable treasure is hidden in the most connected cave ("biggest hub"). The explorers want to estimate how many tunnels this cave has to plan their expedition. Based on this information, which statement is correct? A) The biggest hub has approximately \(k_{\max} \approx 3000\) tunnels. Since the network follows the small-world property, explorers will need many steps to reach most caves, as the average distance between two caves grows as \(\ln N\). B) The biggest hub has about \(k_{\max} \approx 500\) tunnels, and the average distance between two caves grows as \(\ln \ln N\), showing the ultra...

Consider a connected Erdős-Rényi random graph \( G(n, p) \) with \( n = 100 \) nodes and edge probability \( p = 0.1 \). The average degree is \( \langle k \rangle = 10 \). Which of the following statements is correct? The degree distribution is concentrated around the mean, the average distance scales linearly with \( n \), and the clustering coefficient is \( 0.05 \). The degree distribution is approximately Poisson, the average distance is 2, and the clustering coefficient is roughly equal to \( p \). The degree distribution is uniform, the average distance is independent of \( n \), and the clustering coefficient grows with \( n \). The degree distribution is Poisson, the average distance is 3, and the clustering coefficient is zero. None of the above Original idea by: Aline Azevedo