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Triadic closure in two-mode networks: Redefining the global and local clustering coefficients
Tore Opsahl
TL;DR
Projected one-mode analysis distorts clustering in two-mode networks because some triangles arise from shared affiliations rather than triadic closure. The paper redefines global and local clustering around 4-paths and closed 4-paths, with the global measure requiring closure through a 6-cycle. These coefficients exclude structurally automatic triangles and address the projection-related modeling issues identified for clustering.
Problem
Two-mode networks are often projected because most network measures are defined for one-mode networks, but projected clustering coefficients have distorted baselines and degree associations.
Method
The paper redefines global and local clustering coefficients using 4-paths and closed 4-paths, excluding structures closed by shared affiliations.
Results
The proposed coefficients assess triadic closure without the modeling issues affecting existing clustering coefficients on projected one-mode networks.
Takeaways & Limitations
Two-mode clustering can be evaluated using measures that omit triangles formed by definition in the two-mode structure.
Takeaways & Limitations
The global coefficient requires designating one node set as primary, which can be subjective when that designation is unclear.
Abstract
from arXiv · showhide
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.
Clustering coefficients for one-mode networks
One-mode clustering coefficients quantify triadic closure globally across the network or locally around a focal node. They cannot be applied directly to bipartite two-mode networks because contacts of a node cannot be connected by construction.
- The global clustering coefficient is the fraction of 2-paths closed by a tie between their first and third nodes.It ranges from 0 when no triangles exist to 1 when all 2-paths are closed.
- In classical random networks, the global clustering coefficient equals tie probability, which is the network density.This follows from independent ties.
- The local clustering coefficient measures the fraction of possible ties among a node’s contacts that are present.Its denominator is the number of 2-paths centered on the focal node, and its numerator counts those closed into triangles.
- Direct application of either coefficient to two-mode data is senseless because bipartite structure prevents triangles among a node’s contacts.Two-mode-specific measures are therefore required.
Origins of triangles
Triadic closure is the addition of a tie that closes a 2-path into a triangle. In projected two-mode networks, triangles may also arise mechanically from shared affiliations rather than closure among contacts.
- Triadic closure adds a tie between the endpoints of a 2-path, making the path part of a triangle.Social examples include introducing two contacts or befriending friends’ friends.
- In one-mode networks, clustering research connects closure with strong ties, overlapping social circles, and ties among a person’s contacts.These connections are discussed through Simmel’s and Granovetter’s arguments.
- Burt’s structural-holes perspective emphasizes brokerage opportunities when a person’s contacts are not connected.Disconnected contacts can create opportunities to control information flow.
- Projection can create triangles when three primary nodes share one secondary node, because the shared affiliation links the primary nodes pairwise.A second configuration can also produce a projected triangle, so not every projected triangle represents ordinary triadic closure.
Global clustering coefficient for two-mode networks
The proposed global two-mode coefficient measures closure among three primary nodes using 4-paths and closed 4-paths, excluding triangles created by shared affiliations. It retains boundedness and has a defined random-network expectation.
- Projected one-mode clustering underestimates the clustering baseline of projected random two-mode networks.Projected networks contain more triangles than classical one-mode random networks with a similar triadic-closure tendency.
- The coefficient counts closed 4-paths among all 4-paths, excluding projected 2-paths created when multiple primary nodes share a secondary node.A closed 4-path is part of at least one 6-cycle.
- In the Figure 3 example, the two-mode coefficient is 0.6, whereas the one-mode projection coefficient is 0.75.The projection contains additional 2-paths that were not generated by 4-paths in the original two-mode structure.
- The coefficient ranges from 0 to 1, equals 1 in a fully connected network, and has an expected value determined by the number of secondary nodes and density in classical random two-mode networks.Its numerator is a subset of its positive denominator.
- For weighted networks, the generalized coefficient equals the binary coefficient when all ties have the same value and is approximately equal after random weight reshuffling.Different 4-path weighting rules can produce different values when closed paths have stronger ties than open paths.
Local clustering coefficient for two-mode networks
The proposed local two-mode coefficient replaces projection-based 2-paths with 4-paths centered on a focal node. This removes the automatic degree-related clustering pattern while preserving key coefficient properties.
- The projected local clustering coefficient is inversely related to a node’s two-mode degree.A node linked to one secondary node shared with at least two others automatically receives a coefficient of 1.
- In Figure 4, projected local clustering follows 1.02degree^-0.93 with R² = 0.9881, while constraint follows 0.75degree^-1.07 with R² = 0.9879.The averages come from 10 random reshufflings at each degree level.
- The redefined local coefficient uses 4-paths centered on the focal node, with endpoints in the focal node’s node set.It counts as closed those paths whose endpoints share an additional common node outside the path, forming part of a 6-cycle.
- For each node, the proposed coefficient ranges from 0 to 1 and equals 1 when all centered 4-paths are closed.With randomly placed ties, its expected value matches the proposed global coefficient’s expectation.
- The weighted extension uses the same 4-path values as the global coefficient and retains the binary coefficient’s stated properties.It is designed to remain roughly equal to the binary coefficient under random weight reshuffling.
Empirical test
Applications across four two-mode datasets show that the proposed coefficients can distinguish structural triadic closure from triangles created automatically by projection. The redefined measures generally produce weaker, more calibrated clustering signals than projected one-mode coefficients, while the weighted forum analysis suggests stronger 4-paths are more likely to close.
- Davis Southern Women: In Davis’s Southern Women data, the observed one-mode coefficient was 0.93, while the observed two-mode coefficient was 0.77; neither was extreme relative to its simulated null distribution.81% of simulated networks had lower one-mode coefficients, and 44% had lower two-mode coefficients.
- Scientific collaboration: In Newman’s collaboration network, the projected coefficient was 0.36 versus 0.28 for the proposed two-mode coefficient, with both exceeding their simulated null distributions.The 97.5 percentiles were 0.0004 for the projected coefficient and 0.0006 for the proposed coefficient.
- Interlocking directorates: In the Norwegian directorate network, clustering was 0.68 in projection but 0.0114 in the two-mode network, indicating a weaker triadic-closure effect than projection suggested.The corresponding simulated 97.5 percentiles were 0.0042 and 0.0055.
- Weighted networks: In the weighted online-community network, the weighted coefficient exceeded the binary coefficient, suggesting stronger 4-paths were more likely to be closed than weaker ones.The analysis used student-to-group ties weighted by the number of characters posted.
- Local coefficients: The difference between two-mode and projected local coefficients was larger for women attending fewer events, with correlation -0.69 and p-value less than 0.001.This result corroborates that projection bias is greater for nodes attending fewer events.
- Local coefficients: The Davis local example shows how projection can make all contacts appear tied when eleven of Flora’s twelve contacts attended the same event, whereas the redefined coefficient remained below 1.Helen, the twelfth contact, connected to the others through different events.
Conclusion
The paper argues that projection can bias clustering measures because shared affiliations automatically create cliques and triangles. It therefore redefines global and local coefficients to assess triadic closure without those structurally predetermined triangles, while noting important scope limitations and future extensions.
- Problem: Projected two-mode networks can contain larger cliques and automatically formed triangles, biasing measures based on clustering or ties among contacts.When non-projected nodes have degree greater than 2, triangles are automatically formed in the one-mode projection.
- Contribution: The paper redefines both global and local clustering coefficients for two-mode networks.The redefinitions exclude structures closed by definition, such as three nodes connected to a common node.
- Contribution: The redefined coefficients assess triadic closure without the modeling issues affecting existing coefficients on projected one-mode networks.They focus on network structures that are not closed by definition.
- Limitations: The global coefficient requires one node set to be designated as responsible for tie generation, and primary nodes must begin and end the 4-paths.This is especially problematic when the primary node set is ambiguous, as in interlocking directorates.
- Future scope: The proposed coefficients are a first step toward redefining other one-mode measures, including structural-holes measures such as constraint.The paper notes that constraint is inversely associated with two-mode degree in projections of random two-mode networks.