Structural Holes in Two-mode Networks
Structural holes-measures are based on density in the local network surrounding focal nodes (Burt, 1992), and form part of a larger set of cohesion metrics (e.g., the clustering coefficients and block modeling). Network cohesion metrics for one-mode networks are often defined around triplets (i.e., three nodes with at least two ties among them) and whether or not these triplets are closed (i.e., they form part of a triangle).
Two-mode networks are often projected onto one-mode networks to be analysed as there are few network metrics specifically designed for them. These projected networks often contain many more triangles than prototypical one-mode networks, and thus overestimates the level of clustering in a network. As such, more triplets are closed in projected two-mode networks than in prototypical one-mode networks.
This overestimation can be seen when plot and regressing metrics with nodes’ two-mode degree. Specifically, when calculating the local clustering coefficient (Watts and Strogatz, 1998) or the structural holes measure constraint (Burt, 1992) on projected two-mode networks, the measures are inversely correlated with nodes’ two-mode degree on a randomly tie reshuffled two-mode network (each node maintains their degree). Below is the average (a) local clustering coefficient and (b) constraint scores for nodes in a random version of the Scientific Collaboration Network (Newman, 2001) for various levels of two-mode degree (the local clustering coefficient values can be fitted by 1.02degree−0.93 with an R2 of 0.9881; The structural hole-measure constraint values can be fitted by 0.75degree−1.07 with an R2 of 0.9879).
As such, it is not advisable to use the one-mode version of the structural holes metric constraint on projected two-mode networks. In these cases until a suitable generalization is proposed, I would recommend that the local clustering coefficient for two-mode networks is used to gauge the cohesion surrounding nodes in a two-mode network.
Burt, R.S., 1992. Structural holes. Harvard University Press, Cambridge, MA.
Newman, M. E. J., 2001. Scientific collaboration networks. II. Shortest paths, weighted networks, and centrality. Physical Review E 64, 016132.
Opsahl, T., 2013. Triadic closure in two-mode networks: Redefining the global and local clustering coefficients. Social Networks 35, doi:10.1016/j.socnet.2011.07.001.
Watts, D. J., Strogatz, S. H., 1998. Collective dynamics of “small-world” networks. Nature 393, 440-442.