MathJax


Scale-free Networks question

 Consider a network with \( N = 10^4\) nodes. If the degree distribution \(p_k\) follows the power law \(p_k \sim k^{\gamma}\), with degree exponent \(\gamma = 3\) and a minimum degree \(k_{\text{min}} = 2\), which the following statements are true?

I. The expected size of the largest hub is \(k_{\text{max}} = 200\).

II. The first moment \(\langle k \rangle\) diverges, meaning the network has no scale.

III. Doubling the number of nodes to \(N = 2 \times 10^4\) would double the expected size of the largest hub to \(k_{\text{max}} = 400\).

IV. The second moment \(\langle k^2 \rangle\) diverges, meaning the network has no characteristic flutuation.


a) I, II and IV

b) I and IV

c) Only I

d) II and III

e) None of the above.


Original idea by: Carlos Trindade

Comentários

  1. Joao Meidanis4 de abril de 2026 às 05:55

    Good question, but I am afraid the statements about and diverging may lead to misunderstandings and dispute. On one hand, obviously for a fixed network the moments do not diverge. On the other hand, some people may argue that for this value of gamma we have divergence when N grows. I prefer to avoid the risk.

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