CommunityModularity

• To use CommunityModularity, you first need to load the Graph Utilities Package using Needs["GraphUtilities`"].

• A community in a network is a group of vertices such that there is a higher density of edges within the group than between them.

• Given a graph g={V, E}, let the vertex set V be partitioned into k subsets V=V_i such that each subset belongs to one community. The community modularity Q of this partition is defined as

• Q= (e_ii - a_ii²)

• where e_ii is the percentage of number of edges that has both ends in community V_i, and a_i is the percentage of edges that start from community V_i. In other words,

• e_ii = |{(u, v)|uV_i, v V_i, (u, v)E}|/|E|

• and

• a_i = |{(u, v)|uV_i, (u, v)E}|/|E|.

• The community modularity Q is a number less than or equal to 1. A large positive value indicates that the vertex partition gives significant community structure.

• The following option can be given: