2022-185

The generalized modularity \( M \) of a network with \( L \) links partitioned into \( n_c \) communities can be calculated as:

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

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. Consider the following statements about \( M \):

1. Higher values of \( M \) correspond to better community structures.
2. \( M = 0 \) when the entire network is taken as a single community.
3. \( M \) cannot be negative.
4. \( M \) cannot exceed one.

What is correct to assert:

1. Only I is true
2. I, II, and III are true
3. I, II, and IV are true
4. I and II are true, III and IV are false
5. None of the above

Original idea by: Marcelo Silva