MathJax


Communities Question

 In the context of community detection, the generalized modularity \(M\) of a network with \(L\) links and \(n_c\) communities is given by:

\[ M = \sum_{c=1}^{n_c}{\left[ \frac{L_c}{L} - \left( \frac{k_c}{2L} \right)^2 \right]}\]

where \(L_c\) is the total number of links within the community \(C_c\) and \(k_c\) is the total degree of the nodes in this community. Considering the following statements:

I. Modularity has a resolution limit, as it can't detect communities smaller than a factor proportional to \(\sqrt{2L}\).

II. A community \(C_c\) contributes positively to \(M\) only if the fraction of internal links \(\frac{L_c}{L}\) exceeds the expected fraction \(\left(\frac{k_c}{2L}\right)^2\) under a null model that preserves the degree sequence.

III. The modularity \(M\) is bounded in the interval \([0,1]\), where \(M = 1\) corresponds to a perfectly modular network.

IV. The partition with the maximum modularity \(M\) offers an optimal community structure.

It is correct to say that:

A) Only I and II are correct.

B) I, III and IV are correct.

C) I, II and IV are correct.

D) Only I and IV are correct.

E) None of the above


Original idea by: Carlos Trindade

Comentários

  1. Joao Meidanis23 de maio de 2026 às 06:23

    Nice question, but I'm afraid statement IV is not right or wrong. It depends on the acceptance of a hypothesis.

    ResponderExcluir

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