Article: Triadic closure in two-mode networks: Redefining the global and local clustering coefficients

December 21, 2011 at 9:00 am 5 comments

A paper called “Triadic closure in two-mode networks: Redefining the global and local clustering coefficients” that I have authored will be published in the special issue of Social Networks on two-mode networks (2013). Unfortunately, the copyright agreement prevents me from uploading a pdf of the published paper to this blog. However, if you have access to Social Networks, you can download the paper directly. Otherwise, a preprint with the exact same text is available (arXiv:1006.0887).

Abstract

As the vast majority of network measures are defined for one-mode networks, two-mode networks often have to be projected onto one-mode networks to be analyzed. A number of issues arise in this transformation process, especially when analyzing ties among nodes’ contacts. For example, the values attained by the global and local clustering coefficients on projected random two-mode networks deviate from the expected values in corresponding classical one-mode networks. Moreover, both the local clustering coefficient and constraint (structural holes) are inversely associated to nodes’ two-mode degree. To overcome these issues, this paper proposes redefinitions of the clustering coefficients for two-mode networks.

Motivation

The clustering coefficients for one-mode networks are a measure of cohesion or group formation. These measures are 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. These networks often contain many more triangles than prototypical networks, and thus overestimates the level of clustering in a network. Methodological issues exist at a local level as well. 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.

Reinforcement: 4-cycles and 3-pathsAs a result, a host of clustering measures for two-mode networks has been developed. For example, Robin and Alexander (2004) defined a coefficient as the number of four-cycles divided by the number of three-paths. Four-cycles in two-mode networks are the smallest possible cycle (like triangles are the smallest possible cycle in one-mode networks). However, this measure is distinctly different from the idea of triadic closure as the measure only include two primary nodes. In fact, a four-cycle is an indication of reinforcement or agreement between two-nodes and not cohesion or group formation.

The paper proposes redefinitions of the global and local clustering coefficients for two-mode networks. The measures are defined around 4-paths or triplets of primary nodes in two-mode networks. Specifically, the global coefficient is defined as the number of 4-paths that are closed divided by the total number, while the local is similar but focused on 4-paths centred on the focal node. For more details, see the paper (Social Networks; arXiv) or the tnet documentation (tnet » Two-mode Networks » Clustering).

Want to test it with your data?

The clustering_tm and clustering_local_tm-functions in tnet allows you to calculate the global and local clustering coefficients for two-mode networks (both binary and weighted) on your own dataset.

# Load tnet
library(tnet)

# Load a sample network (Figure 3A of the paper)
net <- rbind(
c(1,1),
c(1,2),
c(2,1),
c(2,3),
c(3,2),
c(3,3),
c(4,3))

# Calculate global clustering coefficient
clustering_tm(net)

# Calculate local clustering coefficient
clustering_local_tm(net)

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.

Robins, G., Alexander, M., 2004. Small worlds among interlocking Directors: Network structure and distance in bipartite graphs. Computational and Mathematical Organization Theory 10 (1), 69–94.

Watts, D. J., Strogatz, S. H., 1998. Collective dynamics of “small-world” networks. Nature 393, 440-442.

If you use any of the information in this post, please cite: 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.

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5 Comments Add your own

  • 1. Bob  |  January 29, 2013 at 11:39 am

    This is great, have you implemented a local 4 cycle clustering coefficient? i.e. local version of Robins and Alexander’s work?

    Reply
    • 2. Tore Opsahl  |  January 29, 2013 at 11:51 am

      Hi Bob!

      I have not implemented a local version of Robin and Alexander’s (2004) coefficient. The global one is implemented in the reinforcement_tm-function. Without looking up the paper again, is a local coefficient implemented?

      As mentioned on Clustering in Two-mode Networks (http://toreopsahl.com/tnet/two-mode-networks/clustering/), Robin and Alexander’s coefficient is similar to reinforcement of a one-mode tie between two primary nodes. Could you project a binary two-mode network and then study the link weights to answer your question?

      Best,
      Tore

      Reply
      • 3. Bob  |  January 29, 2013 at 12:16 pm

        Ps I think they just have a global 4 cycle clustering coef in that paper

      • 4. Tore Opsahl  |  January 29, 2013 at 1:24 pm

        Hi Bob,

        I am not sure how a local version of the measure can be implemented, but could the unit of analysis be the number of three paths where the focal node is either node 2 or 3 (in other words, in the middle). The numerator would be the number of these paths that are “closed” (part of a 4-cycle), and the denominator could be the total number of these paths. What do you think?

        Tore

      • 5. Bob  |  January 29, 2013 at 2:02 pm

        Hi Tore, This is exactly the sort of implementation I had hoped existed, using open and closed 3 paths on nodes of the focal set with degree >2.

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