Most of the tools of social network analysis deal with structures defined by patterns in a single kind of relationship among actors: friendship, kinship, economic exchange, warfare, etc. Social relations among actors, however, are usually more complex, in that actors are connected in multiple ways simultaneously. In face-to-face groups of persons, the actors may have emotional connections, exchange relations, kinship ties, and other connections all at the same time. Organizations exchange personnel, money, information, and form groups and alliances. Relations among nation-states are characterized by numerous forms of cultural, economic, and political exchange.
Sociologists tend to assume, until proven otherwise, that actors' behavior is strongly shaped by the complex interaction of many simultaneous constraints and opportunities arising from how the individual is embedded in multiple kinds of relationships. The characteristics and behavior of whole populations, as well, may depend on multiple dimensions of integration/cleavage. Solidarity may be established by economic exchange, shared information, kinship, and other ties operating simultaneously.
In this chapter we will look at some of the tools that social network analysts have used to grapple with the complexity of analyzing simultaneous multiple relations among actors. We'll begin by examining some basic data structures for multiplex data, and how they can be visualized. To be useful in analysis, however, the information about multiple relations among a set of actors must somehow be represented in summary form. There are two general approaches: reduction and combination. The "reduction" approach seeks to combine information about multiple relations among the same set of actors into a single relation that indexes the quantity of ties. All of these issues are dealt with in the section on multiplex data basics.
The "combination" approach also seeks to create a single index of the multiplex relations, but attempts to represent the quality of ties. Summarizing the information about multiple kinds of ties among actors as a single qualitative typology is discussed in the section on "role algebra". We won't actually explore the complexities of role algebra analysis, but we will provide a brief introduction to this way of approaching multi-relational complexity.