Partitioning Signed Two-Mode Networks

Abstract

Structural balance theory forms the foundation for a generalized blockmodel method useful for delineating the structure of signed social one-mode networks for social actors (for example, people or nations). Heider's unit formation relation was dropped. We re-examine structural balance by formulating Heider's unit formation relations as signed two-mode data. Just as generalized blockmodeling has been extended to analyze two-mode unsigned data, we extend it to analyze signed two-mode network data and provide a formalization of the extension. The blockmodel structure for signed two-mode networks has positive and negative blocks, defined in terms of different partitions of rows and columns. These signed blocks can be located anywhere in the block model. We provide a motivating example and then use the new blockmodel type to delineate the voting patterns of the Supreme Court justices for all of their nonunanimous decisions for the 2006–07 term. Interpretations are presented together with a statement of further problems meriting attention for partitioning signed two-mode data.

Keywords:

• generalized blockmodeling
• signed networks
• signed two-mode networks
• structural balance

We appreciate greatly the support provided by the Slovenian Research Agency under a grant, Contract Number 1000-07-780005. Comments from two anonymous reviewers helped us improve our presentation.

Notes

¹In contrast, for poq-triples there is only one kind of social entity—people—and the ties are social relations (and perceptions of them). The asymmetry in terms of types—actors and social objects—was discarded as part of the generalization of Heider's ideas in order to focus attention on social relations.

²Of course, it depends on the importance of x. Depending on the circumstances, it would seem that agreeing or not about the merits of U2 as a band is less consequential than disagreeing or not over the morality of an unprovoked invasion of another country.

³For Cartwright and Harary, a network is balanced if it has two plus-sets, one of which can be empty, with this characteristic sign pattern. For Davis, a graph was clusterable into two or more plus-set with this sign pattern. We prefer to use the term k-balance where k is the number of plus-sets.

⁴We represent edges as a pair of reciprocated arcs as a way of capturing the notion of unordered pairs (two element sets). Formally, balance theoretic partitioning can proceed in terms of arcs, edges or both.

⁵When justices recuse themselves from particular cases, there are null unit formation ties.

⁶The same formalization holds, in principle, when arcs are used rather than edges.

⁷See Doreian et al. (2005, pp. 185–186), for the meaning of this term for one-mode data. The idea extends natural to two-mode partitions and consists of all partitions corresponding to the set of all possible exact partitions given a type of equivalence.

⁸Viewing inconsistencies in this fashion is an application of the line indices of imbalance proposed by Harary, Norman, and Cartwright (Harary et al.1965, pp. 348–352). An alternative approach using proportions of balanced cycles does not address the partitioning problem. Further, counting cycles is computationally too complex (Hummon and Fararo, 1995) to be applied here.

⁹This is the optimal value of the criterion function for these specific values of k ₁ and k ₂. We use “best” in this sense throughout the proof. For given values of k ₁ and k ₂, the value of P(C = C _{1(k 1)}, C _{2(k 2)}) is optimal but need not be globally optimal.

¹⁰Doreian et al. (2004; 2005) used only the “important” decisions for their partitions of the Supreme Court in the years they considered. The implications of using the full set of decisions are discussed below.

¹¹Even so, different justices can have different rationales for supporting a unanimous decision.

¹²Henceforth we will use the label conservative in place of so-called conservative and liberal in place of so-called liberal.

¹³While Justice Thomas recused himself from the case, the null cell does not contribute to P(C).

¹⁴This is the number of inconsistencies and the value of the criterion function, P(C) = α  + (1− α) 𝒫 is 0.5 with α = 0.5.

¹⁵It is a future topic of research to consider the details of the cases to discern if there are coherent ideological commonalities leading to systematic differences in voting behavior.

¹⁶These are not the two one-mode networks that can be constructed from this two-mode network.