Structure detection methods - Complex Networks CSYS/MATH 303, Spring, 2011

We now turn to applications of our methods to real-world network data. Our first such example is taken from one of the classic studies in social network analysis. Over the course of two years in the early 1970s, Wayne Zachary observed social interactions between the members of a karate club at an American university "36#. He constructed networks of ties between members of the club based on their social interactions both within the club and outside it. By chance, a dispute arose during the course of his study between the club’s administrator and its principal karate teacher over whether to raise club fees, and as a result the club eventually split in two, forming two smaller clubs, centered around the administrator and the teacher. In Fig. 8, we show a consensus network structure extracted from Zachary’s observations before the split. Feeding this network into our algorithms, we find the results shown in Fig. 9. In the leftmost two panels, we show the dendrograms generated by the shortest-path and random-walk versions of our algorithm, along with the modularity measures for the same. As we see, both algorithms give reasonably high values for the modularity when the network is split into two communities—around 0.4 in each case—indicating that there is a strong natural division at this level. What is more, the divisions in question correspond almost perfectly to the actual divisions in the club revealed by which group each club member joined after the club split up. $The shapes of the vertices representing the two factions are the same as those of Fig. 8.! Only one vertex, vertex 3, is misclassified by the shortest-path version of the method, and none are misclassified by the random-walk version—the latter gets a perfect score on this test. $On the other hand, the two-community split fails to produce a local maximum in the modularity for the random-walk method, unlike the shortest-path method, for which there is a local maximum precisely at this point.! In the last panel of Fig. 9, we show the dendrogram and modularity for an algorithm based on shortest-path betweenness but without the crucial recalculation step discussed in Sec. II. As the figure shows, without this step, the algorithm fails to find the division of the network into the two known groups. Furthermore, the modularity does not reach nearly such high values as in the first two panels, indicating that the divisions suggested are much poorer than in the cases with the recalculation.