> By Thomas Aynaud and Jean-Loup Guillaume
The modularity is widely used to evaluate the quality of a partition of a graph in communities. Each community contributes to the global modularity according to the formula below, where m is the number of links of the graph, e is the number of links inside a given community C and d is the number of link extremities inside C:
We extracted communities from a co-authoring network obtained from  and we used the Louvain method to compute the communities. On the plot, each level 1 dot corresponds to one community and gives its own contribution to the modularity versus its size. There is a strong correlation between the size and the contribution of a community which is typical of the resolution limit problem: small communities must be merged in order to maximize the modularity. See  for formal details on this assertion.
This is strengthened by the other levels communities. Level 2 communities correspond to all communities obtained when decomposing level 1 communities, i.e., subcommunities of the network. Level 3 are sub-subcommunities, and so on. The more we decompose the network, the smaller the communities are and the smaller their contribution to the global modularity.
Maximizing the modularity is interesting but if one wants to understand the structure of a network, the maximal modularity partition hides a lot of information which can only be unraveled by looking at the complete hierarchy of the community structure.