3.3.1 Bipartite and Multi-mode Networks

A canonical example of network data comes from the “Southern Women’s Study” (Davis, Gardner, and Gardner 1941). These data represent a classic example of what are known as two-mode network data. Two-mode indicates there are two different classes (or modes) of nodes in the data. In this example, the data represented women, and a series of events each attended. These data were recorded from observations made by a team of five ethnographers over a period of nine months.

Figure 3.4: Example Bipartite Graph. Circles represent actors (labeled w/letters), while squares (numbers) represent events.
Figure 3.4: Example Bipartite Graph. Circles represent actors (labeled w/letters), while squares (numbers) represent events.

As an example of this type of data, see Figure 3.4. In this graph, you can see the participation of six actors (circles, labeled with letters), and their participation in six events (squares/numbers). In the example, B is the most active, and overlaps in at least one event with everyone other than F.

Perhaps most commonly these types of data have been analyzed to examine how the overlaps between members of companies’ boards of directors shape organizational behavior, for example isomorphic tendencies across those linked organizations (Mizruchi 1996), also known as board interlocks. These types of data are readily culled from archival sources (e.g., board interlocks can be extracted from tax filings). In this type of affiliation network, the ties only exist between nodes of different modes - e.g., board members and corporations.

For many years, these types of data were typically analyzed by then transforming the actor-by-event data into actor-actor or event-event networks (Borgatti and Everett 1997).64 More recently, methods have been developed for analyzing the bipartite (non-projected) versions of these graphs directly (Latapy, Magnien, and Del Vecchio 2008). Additional generalizations have applied these concepts to three-mode (tripartite) and higher order networks as well (Melamed, Breiger, and West 2013).

A related idea is the hypergraph. While multi-mode networks allow for ties between nodes of different classes, hypergraphs are a graph generalization that assumes that some edges can connect any number of vertices in the network, instead of network ties existing only between pairs of nodes. The left panel of Figure 3.5 presents one visualization strategy for hypergraphs where each node connected by a particular edge is included within the indicated outlines.65 Similar to the actor-by-event matrix used to store two-mode network data, hyperedges can be stored in an edge-by-vertex matrix, as represented in the right panel of Figure 3.5.

Figure 3.5: Example Hypergraph Visualization and Data Storage. The hypergraph visualization (left) uses outlines to indicate nodes connected by the specified edges. The matrix (right) concisely stores hyperedges with each node connected by the specified edge. NOTE: For simplicity, this visualization drops the ’v’s from vertex labels.
Figure 3.5: Example Hypergraph Visualization and Data Storage. The hypergraph visualization (left) uses outlines to indicate nodes connected by the specified edges. The matrix (right) concisely stores hyperedges with each node connected by the specified edge. NOTE: For simplicity, this visualization drops the ’v’s from vertex labels.

A useful example of hypergraphs comes from email exchanges. While some emails are only between two parties, and could readily be captured with dyadic representations, there are also group emails. A translation of typical network data representations could characterize these group emails as represented by simultaneous connections between each of the pairs of nodes within the group.66 However, in such a representation, you cannot distinguish between a group conversation and multiuple dyadic conversations involving the same members. The hypergraph therefore allows for a single “hyperedge” to connect any number of vertices in the graph.