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 why the binomial distribution is used to model the number of edges present in the graph?

Options:

a) Because the binomial distribution is the only discrete distribution that can model independent processes with a constant probability.

b) Because the binomial distribution can approximate any discrete distribution when n is large and p is small.

c) Because the binomial distribution describes the number of successes in a series of independent trials, where each edge in the graph has a fixed probability of being present or not.

d) Because the binomial distribution guarantees that the graph will always be connected when n tends to infinity.

Comentários

  1. Joao Meidanis2 de setembro de 2024 às 15:13

    Sorry, we can only consider one question per student. We looked at question 1. It is a nice question, but, it has a few problems. In the introductory text, it is said that we will consider the critical threshold of connectivity, but some of the statements refer to the critical threshold (around = 1). Also statement 4 says that 'all nodes become fully connected', but being fully connected is a graph property, not a node property. Discarded.

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