The OPTGRAPH Procedure

This example uses Zachary’s Karate Club data (Zachary 1977), which describes social network friendships between 34 members of a karate club at a U.S. university in the 1970s. This is one of the standard publicly available data sets for testing community detection algorithms. It contains 34 nodes and 78 links. The graph is shown in Figure 1.144.

Figure 1.144: Zachary’s Karate Club Graph

The graph can be represented using the following links data set LinkSetIn:

data LinkSetIn;
   input from to weight @@;
   datalines;
 0  9  1  0 10  1  0 14  1  0 15  1  0 16  1  0 19  1  0 20  1  0 21  1
 0 23  1  0 24  1  0 27  1  0 28  1  0 29  1  0 30  1  0 31  1  0 32  1
 0 33  1  2  1  1  3  1  1  3  2  1  4  1  1  4  2  1  4  3  1  5  1  1
 6  1  1  7  1  1  7  5  1  7  6  1  8  1  1  8  2  1  8  3  1  8  4  1
 9  1  1  9  3  1 10  3  1 11  1  1 11  5  1 11  6  1 12  1  1 13  1  1
13  4  1 14  1  1 14  2  1 14  3  1 14  4  1 17  6  1 17  7  1 18  1  1
18  2  1 20  1  1 20  2  1 22  1  1 22  2  1 26 24  1 26 25  1 28  3  1
28 24  1 28 25  1 29  3  1 30 24  1 30 27  1 31  2  1 31  9  1 32  1  1
32 25  1 32 26  1 32 29  1 33  3  1 33  9  1 33 15  1 33 16  1 33 19  1
33 21  1 33 23  1 33 24  1 33 30  1 33 31  1 33 32  1
;

The following statements use the RESOLUTION_LIST= option to represent resolution levels (1, 0.5) in community detection on the Karate Club data. For more information about resolution levels, see the section Resolution List.

proc optgraph
   data_links            = LinkSetIn
   out_nodes             = NodeSetOut
   graph_internal_format = thin;
   community
      resolution_list    = 1.0 0.5
      out_level          = CommLevelOut
      out_community      = CommOut
      out_overlap        = CommOverlapOut
      out_comm_links     = CommLinksOut;
run;

The data set NodeSetOut contains the community identifier of each node. It is shown in Output 1.7.1.

Column community_1 contains the community identifier of each node when the resolution value is 1.0; column community_2 contains the community identifier of each node when the resolution value is 0.5. Different node colors are used to represent different communities in Figure 1.145 and Figure 1.146. As you can see from the figures, four communities at resolution 1.0 are merged to two communities at resolution 0.5.

Figure 1.145: Karate Club Communities (Resolution = 1.0)

Figure 1.146: Karate Club Communities (Resolution = 0.5)

The data set CommLevelOut contains the number of communities and the corresponding modularity values found at each resolution level. It is shown in Output 1.7.2.

The data set CommOut contains the number of nodes contained in each community. It is shown in Output 1.7.3.

The data set CommOverlapOut contains the intensity of each node that belongs to multiple communities. It is shown in Output 1.7.4. Note that only the communities in the last resolution level (the smallest resolution value) are output in this data set. In this example, Node 0 belongs to two communities, with 82.3% of its links connecting to Community 0, and 17.6% of its links connecting to Community 1.

The data set CommLinksOut shows how the communities are interconnected. It is shown in Output 1.7.5. In this example, when the resolution value is 1, the link weight between Communities 0 and 1 is 7, and the link weight between Communities 1 and 2 is 4.