## References

**Citekey**: @Borgatti2012-vh

Borgatti, S. P. (2012). Social Network Analysis, Two-Mode Concepts in. In R. A. Meyers (Ed.), Computational Complexity (pp. 2912–2924). Springer New York. https://doi.org/10.1007/978-1-4614-1800-9_179

## Notes

A great intro to two-mode SNA. Requires understanding of SNA basics.

## Highlights

In social network analysis, 2-mode data refers to data recording ties between two sets of entities. In this con- text, the term “mode” refers to a class of entities – typically called actors, nodes or vertices – whose members have so- cial ties with other members (in the 1-mode case) or with members of another class (in the 2-mode case). (p. 1)

These kinds of data are often referred to as affilia- tions. (p. 1)

Perhaps the best known example of 2-mode network anal- ysis is contained in the study of class and race by Davis, Gardner and Gardner (henceforth DGG) published in the 1941 book Deep South [6]. (p. 1)

In a sense, what constitutes different modes is up to the researcher. If we collect romantic ties among a group of heterosexual people of both genders, we could construct a 2-mode men-by-women matrix X in which x ij D 1if a romantic tie was observed between man i and woman j, and x ij D 0 otherwise. Or, one could construct a larger 1-mode person-by-person matrix B also consisting of 1s and 0s in which it just happens that 1s only occur in cells where the row and column correspond to persons of different gender. (p. 2)

An edge is simply an unordered pair of nodes ( u, v). (p. 2)

A bipartite graph is a graph in which we can partition all nodes into two sets, V 1 and V 2,suchthatalledgesin- clude a member of V 1 and a member of V 2.Thenumberof nodes in each vertex set is denoted n 1 and n 2, respectively. (p. 2)

Two-Mode Data in SocialNetwork Analysis (p. 2)

2-mode data are common in social network contexts as well. Typical examples include, actor-by-event attendance (as in the DGG data), actor by group mem- bership (such as managers sitting on corporate boards), and actor by trait possession (such as adjective checklist data), and actor by object possession (such as material style of life scales in which inventories are made of household possessions). (p. 3)

Unimodal Approaches to Two-Mode Data (p. 3)

One approach to handling 2-mode data in social network analysis is to convert the data to 1-mode data. This is es- pecially appropriate when the analytical interest focuses primarily on just one of the modes. (p. 3)

In many cases when 2-mode data are collected, the analytical interest is focused on one mode or the other. (p. 3)

However, it can also occur that neither mode domi- nates our analytical focus and the primary interest is in the correspondence of one mode to the other. For example, a university might ask its faculty which courses they prefer to teach. (p. 3)

Unimodal Visualization of Two-Mode Data (p. 4)

Alternatively, one can use a standard graph layout al- gorithm (GLA) to draw the graph induced by dichotomiz- ing the Jaccard similarity matrix (p. 5)

Unimodal Analysis of Two-Mode Data (p. 5)

In general, analysis of 2-mode data transformed into val- ued 1-mode networks proceeds like any other valued net- work. (p. 5)

There are, however,a few consequences that stem from the 2-mode origin of the data. By their very nature, many commonly used measures of similarity and dissimilarity satisfy triangle inequality laws. (p. 5)

Bimodal Visualization of Two-Mode Data (p. 6)

All of the standard ways to visualize networks, such as MDS and GLAs, apply to bipartite graphs. (p. 6)

Bimodal Approaches to Two-Mode Data (p. 6)

In other words, all ties are between vertex sets and none are within-group. (p. 6)

The matrix representation of such a graph can be a rectangular incidence matrix X (as in Fig. 1) or a square bipartite adjacency matrix B with n D n 1 C n 2 rows representing both modes, and an equal number of columns, also representing both modes. (p. 6)

For small datasets, this bimodal visualization is often extremely effective for transmitting a holistic understand- ing of the whole dataset. (p. 7)

Another approach is to treat the bipartite graphs as or- dinary graphs and apply all the standard algorithms and techniques of social network analysis. (p. 7)

This approach works for a small class of methods, but by no means all. For example, calculating transitivity fails be- cause transitive triples are impossible in bipartite graphs (p. 7)

Bimodal Analysis of Two-Mode Data (p. 7)

Given a bimodal perspective, one approach to analyzing 2-mode data is to develop entirely new metrics and algo- rithms designed specifically for 2-mode data. (p. 7)

Finally, a compromise approach is to use the standard metrics and algorithms that apply to general graphs, and then either adjust the outputs via normalization or adjust the baseline expectations for the results [5,9 ] (p. 7)

In other words, taking account of the fact that the observed lack ties between certain nodes (namely, those belonging to different modes) was by de- sign – similar to the concept of structural zeros in log-lin- ear modeling. To date, few techniques of this kind have been developed, the exception being the area of 2-mode centrality measures, which has received significant atten- tion (e. g., [2]). (p. 7)

Bimodal Approaches to Graph Cohesion The simplest and most common measure of network cohesion is den- sity – the number of edges divided by the number of pairs of nodes (using ordered pairs in the case of directed graphs and unordered pairs in the case of undirected graphs). In bipartite graphs, of course, only edges between vertex sets are possible. (p. 8)

Bimodal Approaches to Centrality (p. 8)

Degree centrality, d i, is defined as the number of ties incident upon node i. Degree centrality is typically nor- malized by dividing by the maximum number of ties pos- sible,whichinagraphofn nodes is n 1. (p. 8)

However, in any (non-trivial) bipartite graph, no node can be connected to all others, since this would mean within- mode ties. Instead, for a node in V 1, the maximum num- ber of ties it could have is n 2,whereasforanodeinV 2,the maximum number of ties is n 1. Hence a natural adjust- ment is to provide two separate normalization formulas as follows: (p. 8)

For measures of cohesion that are not expressed as fractions of maximum possible values, such as average geodesic path length, the need to renormalize is not as great, since the raw values are directly interpretable. (p. 8)

Closeness centrality is ordinarily defined as the sum of geodesic distances from a node to all others (p. 8)

In the bipartite case, the maximum number of nodes that a node can be distance 1 from is the number of nodes in the other class. For other nodes in its own class, the clos- est it can be is two links away. (p. 9)

Cohesive Subgroups Detecting cohesive subgroups is somewhat more difficult in bipartite graphs than in ordi- nary graphs. (p. 9)

However, in bipartite graphs, cliques are impossible, since in any connected triple, two nodes must be non-adjacent. (p. 9)

Betweenness centrality refers to the sum of shares of short- est paths that pass through a given node. (p. 9)

However, other classical notions of cohesive sub- groups make more sense for bipartite graphs. In particular, relaxations of the clique concept based on distance, such as n-cliques, n-clans and n-clubs have good interpreta- tions in bipartite graphs and work well in practice. In fact, 2-cliques defined on the bipartite graph have the property of bipartite completeness, meaning that, within the bipar- tite subgraph induced by the 2-clique, every node of one mode has a tie with every node of the other mode, which is to say that all possible ties are present. In this sense, for bipartite graphs, 2-cliques capture the underlying idea of a clique better than cliques do. This notion has been for- malized in the definition of a biclique,whichisdefined as a maximal complete bipartite subgraph. (p. 9)

This maximum is appropriate for bipar- tite graphs only when one mode has just one node; oth- erwise we must take account of the sizes of each vertex set. (p. 9)

These are structural equivalence and regular equivalence. Generalizations to 2-mode (and higher) data were developed by Borgatti [3] and Borgatti and Everett [4] and Everett and Borgatti [7]. (p. 10)

Structural Equivalence In the best definition of structural equivalence [8], two nodes u and v are said to be struc- turally equivalent if there exists a graph automorphism that would be an identity except that u andv are mapped to each other. In other words, given a diagram of the graph, you could swap nodes u and v (and no other nodes) with- out changing the structure of the network one iota. (p. 10)

The success of bicliques as 2-mode analogues of cliques suggests the possibility of creating 2-mode analogues for other cohesive subgroup concepts that we previously dis- missed as unusable in the 2-mode context. (p. 10)

As a result, one approach to identifying structurally equivalent nodes is to compute a similarity measure among rows and columns of the ad- jacency matrix defining the graph. (p. 10)

This approach requires no modification for the bi- partite case. (p. 10)

Bimodal Approaches to Positions and Roles (p. 10)

Another approach to structural equivalence is known as blockmodeling. Instead of (or as a result of) measur- ing the extent of structural equivalence between nodes, we partition the nodes into classes such that nodes in the same class are structurally equivalent (or nearly so). Given such a partition of nodes, we can reorder and partition the corresponding rows and columns of the adjacency ma- trix. This in turn induces to partition of matrix cells into matrix blocks. (p. 11)

A more elegant (and computationally efficient) ap- proach is to work from the 2-mode incidence matrix X. In this case, we redefine a blockmodel to refer to a coordi- nated pair of partitions, one for the rows and one for the columns. We then redefine structural equivalence as fol- lows. (p. 11)

to the extent that they have ties (lack of ties) to corre- sponding others who are themselves regularly equivalent to each other. (p. 11)

In empirical work, of course, we do not expect to ob- tain perfect 1-blocks and 0-blocks. Instead we seek parti- tions that minimize the number of errors (where we de- fine an error as either a 1 inside a 0-block or a 0 inside a1-block). (p. 11)

Regular Equivalence Two nodes are said to be regularly equivalent (i. e., play the same structural role) to each other (p. 11)

Conclusion In this chapter we have considered methods of visualizing and analyzing 2-mode network data. An examination of the chapter suggests that four strategies are employed for handling 2-mode data. First, there is converting the data into two separate 1-mode datasets,and analyzing each sep- arately. Second, there is using existing 1-mode methods on a bipartite representation of the 2-mode data, and es- sentially ignoring its special features. Third, there is using 1-mode methods but either adjusting the interpretation or applying some kind of normalization to adjust the results. Finally, there is developing new methods designed specifi- cally for the 2-mode case. (p. 12)

Although it would be a mistake to think of 2-mode data as an advance over 1-mode data, it is important to note that there are many cases were extending network analysis methodology to more than 2 modes is desirable. For exam- ple, we might analyze membership in organizations over time, yielding a 3-way, 3-mode data matrix of relations that is person by organization by period. Similarly, we might be interested in modeling the intellectual landscape of an academic field, representing publications as k-mode bundles of authors, journals, years, institutional affiliations and topics. (p. 12)

both examples are fascinating. So excited. (p. 12)

- Bonacich P (1991) Simultaneous group and individual central- ities. Soc Netw 13:155–168 (p. 12)