Exercise: Suppose you are analyzing a "scale-free" network that follows the Barabási-Albert model. This model generates networks where new nodes tend to preferentially connect to nodes that already have many connections, creating a hub structure. Question: In a "scale-free" network, which of the following statements best describes the degree distribution and its impact on the network's structure? Options: a) All nodes in the network have approximately the same number of connections, leading to a normal degree distribution. b) Nodes with few connections are the most influential in the network, due to the low centralization of hubs. c) The degree distribution follows a power law, meaning most nodes have few connections, while a few nodes have a very high number of connections. d) Preferential attachment ensures that each new node connects to a random node, regardless of how many connections that node already has.
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Question 1: Considering a random graph G(n, p) , which of the following statements correctly describes the properties that emerge near the critical threshold of connectivity as the probability p increases? Options: 1) The emergence of a giant component, where a significant number of nodes connect to form a large connected component. 2) The critical threshold of connectivity occurs approximately when p is equal to log(n)/n. 3) Before the connectivity threshold, the graph is mainly composed of many small and isolated components. 4) After the connectivity threshold, all nodes in the graph become fully connected, forming a complete graph. 5) Upon reaching the critical threshold, the average number of edges in the graph follows a normal distribution centered at n . Correct Answers: a) 1, 2, and 3 are correct. b) 2 and 4 are correct. c) 1 and 5 are correct. d) 3, 4, and 5 are correct. Question 2: Given a random graph G(n, p) in the Erdős–Rényi model, what is the main theoretical reason w