2022-185

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

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

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