Finding globally optimal macrostructure in multiple relation, mixed-mode social networks

From the outset, computational sociologists have stressed leveraging multiple relations when blockmodeling social networks. Despite this emphasis, the majority of published research over the past 40 years has focused on solving blockmodels for a single relation. When multiple relations exist, a reductionist approach is often employed, where the relations are stacked or aggregated into a single matrix, allowing the researcher to apply single relation, often heuristic, blockmodeling techniques. Accordingly, in this article, we develop an exact procedure for the exploratory blockmodeling of multiple relation, mixed-mode networks. In particular, given (a) $N_1$ actors, (b) $N_2$ events, (c) an $\left(N_1\times N_1\right)$ binary one-mode network depicting the ties between actors, and (d) an $\left(N_1\times N_2\right)$ binary two-mode network representing the ties between actors and events, we use integer programming to find globally optimal $\left(P_1\times P_1|P_1\times P_2\right)$ image matrices and partitions, where $P_1$ and $P_2$ represent the number of actor and event positions, respectively. Given the problem’s computational complexity, we also develop an algorithm to generate a minimal set of non-isomorphic image matrices, as well as a complementary, easily accessible heuristic using the network analysis software Pajek. We illustrate these concepts using a simple, hypothetical example, and we apply our techniques to a terrorist network.