Power-law relations in random networks with communities

Area of Science:

• Network Science
• Statistical Physics
• Complex Systems

Background:

• Traditional random graph models lack short cycles, limiting their use for networks with community structures.
• Existing models are unsuitable for capturing the intricate community organization observed in real-world networks.

Purpose of the Study:

• To introduce a generalized random graph model that incorporates community structures.
• To enable analytical derivations of network properties like giant components and percolation clusters within the new model.
• To uncover novel power-law relationships in real-world networks analyzed through the proposed model.

Main Methods:

• Development of the hierarchical configuration model (HCM) as a generalization of the configuration model.
• Analytical derivation of network properties, including giant component and percolating cluster sizes.
• Empirical analysis of real-world networks viewed as realizations of the HCM.

Main Results:

• The HCM successfully models networks with community structures while retaining analytical tractability.
• Discovery of two new power-law relations concerning intra-community edges, extra-community edges, and community sizes.
• Establishment of a relationship between the degree distribution exponent (τ) and the community-size distribution exponent (γ), simplifying to τ=γ-1 for dense communities.

Conclusions:

• The hierarchical configuration model provides a powerful framework for studying networks with community structures.
• The identified power-law relations offer new insights into network organization and dynamics.
• The model facilitates a deeper understanding of the interplay between degree distribution and community structure.