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Random Graphs Question

 

Consider the T network below:

 

Which of the following statements are true?
 
I. T is in supercritical regime because its average degree \(\langle k \rangle = 4.5\) is higher than \(1\).
 
II. T can't be considered a random network because node G has degree \(k = 7\), representing a hub in the network and its degree is far from \(\langle k \rangle = 3.25\).
 
III. T is in connected regime, since its average degree is higher than \(\ln N\)
 
IV. T is a random network with \(N = 8\) nodes and \(p = 0.46\).
 
 
A) I and IV
B) III and IV
C) II and III
D) I and II
E) None of the above
 
 
Original idea by: Carlos Trindade 

Comentários

  1. Joao Meidanis21 de março de 2026 às 09:23

    Dear Carlos, we need to calibrate our understanding of random networks. The categorization into regimes is for large networks. For small networks like the one you show, it does not make much sense. Another issue is to say that a given, specific network is random. We never defined a specific network as being random or not. There is the random network model, that assigns probabilities to all networks of a certain size, depending on a parameter p. When we say that the 10 networks studied in the book do not look like random networks, this should be interpreted as saying that it is very unlikely that they were produced by the process behind the random model specification. In other words, given one on the 10 networks, call it G, if we generate a random network with the same N and any p, a lot of other, very different networks would have a greater probability of having been generated, compared to G. Taking all that into consideration, your question sounds very strange to my ears.

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