Triadic Closure in Configuration Models with Unbounded Degree Fluctuations.

Area of Science:

• Network Science
• Statistical Physics
• Complex Systems

Background:

• The configuration model is a null model for analyzing scale-free networks with power-law degree distributions.
• Understanding local clustering is crucial for characterizing network structure and function.
• Real-world networks often exhibit modular or hierarchical organization.

Purpose of the Study:

• To investigate the behavior of local clustering coefficient c(k) in the configuration model.
• To determine how local clustering scales with node degree (k) and graph size (n).
• To connect the observed clustering patterns to network modularity and triangle counting in random graphs.

Main Methods:

• Utilizing the configuration model to generate random graphs with specified degree distributions.
• Deriving analytical results for the local clustering coefficient c(k).
• Analyzing the scaling behavior of c(k) with respect to k, n, and the power-law exponent.

Main Results:

• Local clustering c(k) progressively decreases as node degree k increases.
• c(k) also decreases with increasing graph size n.
• For large graphs, c(k) follows a power law, c(k) ~ k^(-gamma), where gamma is the degree distribution's exponent.

Conclusions:

• The observed fall-off in local clustering indicates modular or hierarchical structures within scale-free networks.
• Results align with findings from hidden-variable models.
• A precise method for counting triangles in the configuration model, considering multi-edges and degree specifications, is established.