Strongly clustered random graphs via triadic closure: An exactly solvable model.

Area of Science:

• Network Science
• Statistical Physics
• Complex Systems

Background:

• Triadic closure is a key mechanism driving network clustering in real-world systems.
• Understanding motif formation is crucial for characterizing network topology.

Purpose of the Study:

• To introduce and analyze a static triadic closure (STC) model for generating clustered networks.
• To derive analytical expressions for small network motifs based on backbone network properties.
• To investigate the impact of network heterogeneity on transitivity and higher-order motifs.

Main Methods:

• Definition of a static triadic closure (STC) model.
• Derivation of exact expressions for expected motif counts using moments of the backbone degree distribution.
• Analysis of transitions in motif densities based on network heterogeneity.
• Testing approximate relationships between motif densities on real-world network datasets.

Main Results:

• Exact expressions for expected numbers of triangles, 4-loops, diamonds, and 4-cliques derived.
• Demonstrated dependence of transitivity and generalized motifs on backbone network heterogeneity.
• Identified transitions in network structure due to topologically inequivalent triads.
• Validated approximate relationships between motif densities against real-world network data.

Conclusions:

• The static triadic closure (STC) model provides a realistic and analytically tractable mechanism for generating clustered networks.
• Network heterogeneity significantly influences the formation of network motifs and overall transitivity.
• The STC model offers valuable insights into the fundamental processes underlying complex network formation.