Generalized Blockmodeling

The goal of blockmodeling is to reduce a large, potentially incoherent network to a smaller comprehensible structure that can be interpreted more readily. In the paper we present some basic ideas and developments in this area. 1 Basic Notions 1.1 Network Let E = {X1, X2, . . . , Xn} be a finite set of units or units. The units are related by binary relations Rt ⊆ E × E, t = 1, . . . , r, r ≥ 1 which determine a network N = (E,R1, R2, . . . , Rr) In the following we restrict our discussion to a single relation R described by a corresponding binary matrix R = [rij]n×n where rij = { 1 XiRXj 0 otherwise In some applications rij can be a nonnegative real number expressing the strength of the relation R between units Xi and Xj. 1.1.1 Example: Student Government In Table 1 and Figure 1 the Student Government network is presented. It consists of communication interactions among twelve members and advisors of the Student Government at the University in Ljubljana (Tina Hlebec, 1993). The results of the measurement are not real interactions among actors but cognition about communication interactions. Data were collected with face to face interviews. Interviews were conducted in May 1992. Communication flow among actors was identified by the following question: Of the members and advisors of the Student Government, whom do you (most often) talk with? The content of the communication flow was limited to the matters of the Student Government. The time frame was also defined: the question was referred to the six months period. One respondent refused to cooperate in the experiment. As he was not considered in the analysis, the network consists of eleven actors. Table 1: Student Government Matrix m p m m m m m m a a a 1 2 3 4 5 6 7 8 9 10 11 minister 1 1 0 1 1 0 0 1 0 0 0 0 0 p.minister 2 0 0 0 0 0 0 0 1 0 0 0 minister 2 3 1 1 0 1 0 1 1 1 0 0 0 minister 3 4 0 0 0 0 0 0 1 1 0 0 0 minister 4 5 0 1 0 1 0 1 1 1 0 0 0 minister 5 6 0 1 0 1 1 0 1 1 0 0 0 minister 6 7 0 0 0 1 0 0 0 1 1 0 1 minister 7 8 0 1 0 1 0 0 1 0 0 0 1 adviser 1 9 0 0 0 1 0 0 1 1 0 0 1 adviser 2 10 1 0 1 1 1 0 0 0 0 0 0 adviser 3 11 0 0 0 0 0 1 0 1 1 0 0