Two-mode analysis and representation

Most network analytic techniques require one-mode adjacency matrices as inputs for processing. Therefore, the conversion of two-mode networks to one-mode networks prior to the analysis is often a necessary step. However, it is important to note that there are

reductions of data with the conversion. As Faust (1997) put it, “In going from the affiliation relation to either the actor co-membership relation or the event overlap relation, one loses information about the patterns of affiliation between actors and events” (p. 189). In a derived one-mode actor network, for example, relations are defined in terms of the number of groups or events two actors have in common in their affiliations. Information on which groups or events are involved is lost in the process of conversion. Considering that groups or events themselves can have significantly different structural features, the loss of information can not be ignored depending on the substantive research problems. In fact, increasingly researchers are concerned that “important structural features of the relations between the elements of one mode can only be completely understood if one simultaneously considers the way in which these same elements form relations among the elements of the other mode” (Field et al., 2006, p.100). What is desirable in studying an affiliation network is to look at all three possible patterns of relations: actor-actor, event-event, and actor-event.

Faust (1997) points out that most affiliation network studies measuring centrality of actors or events overlooked the duality of the data. She argued that centrality measures developed for one-mode networks might not be appropriate for studying affiliation networks and a different conceptualization that takes into account “the relationship between centrality of actors and the centrality of events to which they belong, or the relationship between the centrality of events and the centrality of their members” (p.165) is necessary. She discussed five existing centrality measures – degree, eigenvector, closeness, betweenness, and flow betweenness – and the application and interpretation of those measures for affiliation network data.

an affiliation network as a bipartite graph was suggested by Wilson (1982). A graph is bipartite if its nodes can be partitioned into two mutually exclusive subsets, every tie links nodes from different subsets, and no tie is within a subset. Since an affiliation relation always links an actor and an event, a bipartite graph can be constructed such that a set of actors and a set of events comprise two different subsets in the graph. The number of nodes in this graph is the sum of the number of actors and the number of events (Faust 1997; Borgatti & Everett, 1997). A bipartite graph not only allows a representation of an affiliation network without losing any information but also enables direct extensions of existing network methods based on graph theoretic concepts, since it is a graph (Everett & Borgatti, 2005).

Using a bipartite graph approach, Borgatti and Everett (1997) discuss analyses of two-mode data with an emphasis on how to apply traditional network analytic concepts and techniques such as density, centrality, and subgroup analysis to affiliation networks and what concerns arise in the applications. For example, the maximum number of possible links between nodes, which is used as a standard denominator for normalizing the observed value when calculating density of a network, is different in two-mode data because there can be no links within a set, but only between sets. Everett and Borgatti (2005) further develop the discussion of two-mod data analysis and apply graph centrality measures directly to two- mode data with normalizations.

Another possible representation of an affiliation network is a Galois lattice (Wasserman & Faust, 1994). According to Freeman and White (1993), Galois lattices provide a better way to visualize an affiliation network, because whereas a bipartite graph shows only ties between different subsets, a Galois lattice can reveal relationships among actors or among events as well as between actors and events. As Mische and Pattison (2000)

put it, “Galois lattice analysis makes possible a simultaneous graphical representation of both the ‘between set’ and ‘within set’ relations implied by a two-mode data array” (p. 170). However, it is not useful for a large dataset because the picture quickly gets too complicated to see any pattern as the number of elements to be included increases.

2.3.5 Conclusion

We have reviewed theories and methods of social network analysis. Social network analysis has been used in a variety of substantive areas. While social network analysis was originally concerned with a relatively small dataset in sociological or anthropological studies, network concepts and measures such as connectivity have been successfully applied to very large datasets, including studies of internet link typology.

In many research areas, the social context of a problem at hand is of interest. However, social connections or structures are not always observable or salient in many datasets, especially if the data are generated or collected without the explicit intention to establish such connections. Affiliation networks can be a very useful analytical tool in such circumstances. They are also particularly useful in representing a situation where actors do not necessarily have a social tie but are involved in the same kinds of events or with the same artifacts. In fact, it is possible to induce social networks from any dataset that can be represented as a bipartite graph. For example, Perugini et al. (2004) noted the possibility of building an affiliation network for collaborative filtering.

Social bookmarking data can be represented as tripartite graph, because there are three distinct kinds of entities in a system: user, tag, and resource. Because of the complexity of computation required to process a tripartite graph, a tripartite graph is often transformed to

a bipartite or a unipartite graph. In the case of social bookmarking data, there can be three bipartite graphs: user-resource, user-tag, resource-tag. Mika (2005) used two bipartite graphs, user-tag and resource-tag, to induce networks of tags based on co-occurrences, in an attempt to derive semantic relationships among tags based on tagging data.

As reviewed above, once a one-mode network is induced from a bipartite graph (two-mode data), a full range of network analytic concepts and methods can be applied to the network. This means, for example, that we can find cohesive subgroups in a network of users, either based on their common items (possibly representing shared interests) or on their common usage of tags. Even though we have no data on direct social ties among users of a social bookmarking system, affiliation network analysis presents a theoretically and methodologically sound way to investigate the social dimensions of social bookmarking.

2.4 Social bookmarking studies