2021-041

Considering the following network and its partition in green and pink nodes:

The modularity \( M \) of a partition can be computed using:

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

where \( n_c \) is the number of sets in the partition and, for each set \( S \), we denote by \( L_c \) the number of links connecting two nodes in the set \( S \), and by \( k_c \) the sum of all degrees of nodes in \( S \). What is the value of \( M \) for the partition depicted above (round to 2 decimal places)?

1. \( M = 0.45 \)
2. \( M = 0.12 \)
3. \( M = 0 \)
4. \( M = 0.5 \)
5. None of the above

Original idea by: Tiago Almeida