Using the bipartite line graph to visualize 2-mode social networks

Transcription

1 Using the bipartite line graph to visualize 2-mode social networks Malcolm Alexander Griffith University, Qld. Australia Abstract This paper surveys the range of techniques available for the analysis of 2-mode (actor-byevent) datasets and proposes an additional option for visualization and analysis that uses the linegraph of the bipartite adjacency matrix. The nodes of this line-graph are roles or statuses conferred on actors through their participation in events or recognition of their group membership. The line-graph can be structured to have two sets of edges. One set of edges shows the connections between actors created by their common participations or group memberships. The second set of edges show which statuses (memberships) in different groups are connected because the same actor holds both. The paper shows how to generate the line graph using UCINET procedures but also assembles 2-mode data to show details in the line graph not readily available through UCINET procedures. Contact: Dr. Malcolm Alexander, School of Arts, Media and Culture, Griffith University, Nathan, Australia Tel: Fax: Key Words: social network analysis; 2-mode networks; network visualization; bipartite graphs; line graph 1

2 Using the bipartite line graph to visualize 2-mode social networks Malcolm Alexander Introduction: What is 2-mode social network data? 2-mode social network data is generated when a researcher assembles information that links a population of social actors with an array of collective or corporate entities such as committees, boards, or social clubs [Borgatti and Everett, 1997] [Wasserman and Faust, 1994]. The key aspect of 2-mode data is that it does not record direct relations between social actors or direct connections between the collective entities although such direct relations can be derived from this data as 1-mode datasets. Conventionally we label the collective entities as events and speak of an actor-by-event matrix. 2-mode datasets are rectangular (n x k) matrices and the number of actors is nearly always more than the number of events. Conventionally the row headings are the social actors and the column headings the events. An entry in the a ij cell indicates that actor i is associated with event j. The 2-mode data used in this paper is information on the membership of corporate boards of directors in Australia in These are legally regulated positions and information on the companies and the names of the directors was readily available from public sources. The data analysis described in this paper is carried out using UCINET [Borgatti, Everett and Freeman]. The 2-mode data can be assembled for input to UCINET as a dl file. Figure 1 gives the first section of the input file. Figure One: 2-mode data: DL file dl nr=99 nc=10 format=edgelist2 labels embedded data: 125Wallis AMC 126McFarlane AMC 127Alexander AMC 130Cameron AMC 132Halstead AMC 2378Allen AMC 2422Lapthorne AMC 310Mercer ANZ 311Dahlsen ANZ 312Deane ANZ 313Ellis ANZ 314Harper ANZ 316Maitland ANZ This file is read into UCINET using the Data > Import > dl, procedure. What analyses can we do with a 2-mode dataset Data on corporate directorships has been collected for many years to expose the network of interlocking directorates between companies as an inter-corporate network structure [Mintz and Schwartz, 1985]. Recently these datasets have been addressed in a different way. They are mapped to show the network of personal connections among the population of directors built by the bridges between boards created by interlockers. This interpersonal network is then analyzed as an example of small world architecture [Watts. 1999] [Davis, Yoo and Baker, 2001]. These two strategies of analysis utilize the duality of persons and groups originally noted and conceptualized by Breiger [Breiger 1974]. The derivation of the event-by-event matrix from board membership data through matrix multiplication yields the inter-corporate network. The derivation of the actor-by-actor matrix yields the interpersonal network. Within UCINET the procedure Data > Affiliations gives the event-by-event matrix when actioned on columns and the actor-by-actor matrix when actioned on rows. The Netdraw facility then visualizes each network (Figure 2 and 3). 2

3 Figure 2: The Inter-Corporate Network Figure 3: The Interpersonal Network In recent years Borgatti and Everett have developed visualization tools and statistical analyses for 2-mode datasets that include both sets of nodes (actors and events). The Netdraw software produces a visualization of the 2- mode data with all the actors and event nodes. The actor nodes are circles and the event nodes are the squares (Figure 4). The Netdraw visualization works directly from the 2-mode data set in UCINET (Open > UCINET dataset > 2 mode network). Figure 4: 2-Mode Network The inter-corporate and interpersonal views of the 2-mode network highlight certain aspects of the 2-mode network while subordinating less relevant details. The inter-corporate view is the most compressed. It highlights the 3

4 inter-corporate connections but leaves aside all details about the social actors on the boards and all details about the persons making the links. Thus, for instance, we cannot know whether the four corporations that are fully connected are linked through one director sitting on all four boards or linked through a larger number of independent connections. On the other hand, the interpersonal visualization retains too much detail. Directors sitting on each board form a cluster. The clustering appears only through the specification of all the binary ties between directors. Directors who are on two or more boards are set in between these clusters. The interpersonal network serves to give a sense of the clustering and connection in this particular network however there is a surfeit of detail. The full 2-mode visualization provided by Netdraw provides a clear view of both the personal and intercorporate network and their connections. The corporate nodes centre the sites of clustering that are the basis of the personal networks. The pattern of bridging created by the interlocking directorates stands out and the interlocking directors are placed midway between the corporate clusters to which they belong. The bipartite line-graph: a further visualization The Netdraw visualization provides a comprehensive view of the 2-node network. However, Borgatti and Everett [Borgatti and Everett, 1997] also developed 2-mode versions of network measures of density, centrality (degree, closeness, betweenness and eigenvector) centralization and subgroups. These statistics involve an important move. They are based on an initial conversion of the 2-mode network data to combine both sets of nodes (actors and events) into one vector. The rectangular 2-mode data is read into a combined adjacency matrix that is a square, 2- mode adjacency matrix showing all the original edges between actors and events and, in the symmetrical rectangle, the same edges running between events and actors. This procedure is available in UCINET as the Transform > Bipartite procedure. (Note the option Make result symmetric must be set to yes.) The line-graph option in UCINET can be applied to this symmetrised, bipartite output. It produces an additional view of the 2-mode network (Figure 5). This view is similar to the personal network view in that the boards appear simply as close clusters of individuals. However, the links between boards created by the interlocking directors now appear as edges between nodes within each cluster. Thus, rather than placing the node between the two clusters, as occurs in the personal view and the 2-mode view, the interlocking director has a node for each board seat they hold. The cluster that is the whole board is then positioned close to the boards (clusters) with which they have the most links. Figure 5: The line-graph of the symmetrised bipartite, 2-mode dataset This linegraph view of the 2-mode network retains the clarity of the Netdraw 2-mode visualization but incorporates the details of how the interlocking occurs which comes from the personal network view. Thus the interlocking directors who sit on only two boards form a single edge between two clusters. Where there are two interlocking directors there are two, parallel edges. Where a director holds three board seats there is a triangle between the clusters. As currently implemented in UCINET the labels for the nodes of the line graph are simply the conjunction of the system generated numbers created when the original data is read into UCINET. This makes it very tedious to 4

5 interpret the line-graph view and it is very difficult to assign attribute data to the nodes and edges of the line-graph. This is unfortunate since it is useful for interpretation of the line-graph to distinguish between edges that occur because they connect seats on the same board and those that occur because the same person is holding seats on different boards. A second aid to interpretation is to color the nodes (board seats) according to whether their occupant holds one, two, three etc board seats to highlight how interlocks are being made. It is possible to achieve both these features by creating the nodes of the line-graph at the data management level (using Excel or Access), creating two sets of relations and associating actor attributes with nodes. Figure 6 shows the same data with these refinements: Interlocking or bridging ties are black while intra-board ties are gray; board seats held by interlockers are gray if they hold just two seats and black if they hold three. We have also added circles to show board clusters and added some labels. Figure 6: Line-graph view with tie differentiation and node attribute details This line-graph view allows us to see the complexity of the interlocking at the heart of this network. Three directors (Gough, Goode and Jackson) each hold three positions. They all sat on the board of Pac Dun which is thus a central board. Pairs of the threesome sat on the boards of three companies close to Pac Dun (CSR, ANZ and BHP). By contrast, the ties to the Comm Bank board that also bring it close to Pac Dun, are made independently by two other directors. Other boards were not so closely linked to the centre of the network. The line-graph visualization of the 2-modes network thus shares the advantages of Netdraw s 2-mode view but allows for clearer visualizations of complex interlocks and overlaps. Conclusion The bipartite line-graph visualization of 2-mode networks presented in this paper is an additional option for researchers working with and visualizing 2-mode data. Although the sample dataset used in this paper is fairly simple it does suggest how this additional visualization offers advantages to researchers dealing with complex 2- mode networks. Visualizations are only exploratory tools however. Using the line-graph also suggests that we can conceptualize 2-mode datasets (and perhaps higher mode data) in a new way. With the line-graph procedure, the edges of the initial, rectangular 2-mode dataset are recognized as distinct social entities. The nodes of the line-graph are neither the social actors nor the collective entities (events, boards, groups etc) of the original data collection. In this dataset they are the seats on the company boards. More generally they can be the statuses, roles, posts, positions, jobs or (university) chairs, recognized by organizations or some other collective or corporate body but given a social reality only when filled by an individual social actor. These entities can be seen as the building blocks of the larger structures visible in the other views of the 2-mode data. Starting with these entities will involve new ways of exploring these larger networks and structures. 5

6 References [Borgatti and Everett, 1997] Borgatti, S. P. and M. G. Everett (1997). "Network analysis of 2-mode data." Social Networks 19: [Borgatti, Everett and Freeman, 2002] Borgatti, S. P., M. G. Everett, L. C. Freeman, (2002), Ucinet 6 for Windows: Software for social network analysis. Harvard, Analytic Technologies. [Breiger 1974] Breiger, R. L., (1974), "The Duality of Persons and Groups " Social Forces 53(2) (December): [Davis, Yoo and Baker, 2001] Davis, G. F., M. Yoo, Wayne E. Baker, (2001), The Small World of the Corporate Elite, Social Networks Sunbelt XXI Conference, New Orleans. [Mintz and Schwartz, 1985] Mintz, B. and M. Schwartz, (1985), The Power Structure of American Business Chicago, University of Chicago Press. [Wasserman and Faust, 1994] Wasserman, S. and K. Faust, (1994), Social Network Analysis: Methods and Applications. New York, Cambridge University Press. [Watts. 1999] Watts, D. J., (1999), Small worlds: the dynamics of networks between order and randomness, Princeton, N.J., Princeton University Press. 6