Densification and structural transitions in networks that grow by node copying.

Area of Science:

• Network Science
• Statistical Physics
• Complex Systems

Background:

• Understanding the structure and evolution of growing complex networks is crucial in various scientific domains.
• Existing models often fail to capture the nuanced behaviors observed in real-world networks, particularly regarding degree distribution and community structures.

Purpose of the Study:

• To introduce and analyze a novel growing network model, termed the 'copying model'.
• To investigate the impact of a 'copying probability' parameter (p) on network sparsity, degree distribution, and emergent structural properties.
• To explore the transition from normal to anomalous network behaviors in dense regimes.

Main Methods:

• Development of the 'copying model' where new nodes attach to a target node and its neighbors with probability p.
• Mathematical analysis to derive degree distributions and study network properties as a function of p.
• Investigation of clique number scaling and self-averaging properties in different network regimes.

Main Results:

• For p < 1/2, the model generates sparse networks with finite average degrees and power-law degree distributions with a nonuniversal exponent.
• For p >= 1/2, dense networks emerge, exhibiting anomalous behaviors in the number of m-cliques and a lack of self-averaging.
• Linking to second neighbors leads to a near-complete network as the number of nodes increases.

Conclusions:

• The copying model provides a flexible framework for generating diverse network structures.
• The parameter p critically controls the transition between sparse and dense network regimes, influencing structural properties like clique formation.
• The model highlights the potential for anomalous behaviors and absence of self-averaging in dense growing networks.