## 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 R^{2} of 0.9881; The structural hole-measure constraint values can be fitted by 0.75degree^{−1.07} with an R^{2} 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.

**References**

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.

1.John T Scholz | December 23, 2011 at 8:39 pmTore,

any idea of when you might complete the local structural hole procedure for 2-mode networks? John

2.Tore Opsahl | December 23, 2011 at 9:33 pmHi John!

Thank you for your comment. I tried to finish the two-mode section before the holidays. Just this page and the random networks pages left. Hopefully I will try to get this completed soon.

I am not aware of a specific local structural holes measure for two-mode networks. I have an idea for one based on a similar concept as the two-mode local clustering coefficient. At the moment, I am uncertain how well this measure will perform. As such, you might want to use the local clustering coefficient for two-mode networks (the local clustering coefficient is in many ways a simple constraint measure).

What do you think?

Tore