Organization of growing random networks

Area of Science:

• Network Science
• Statistical Physics
• Computer Science

Background:

• Growing random networks are fundamental models in understanding complex systems.
• Previous models often assumed simpler attachment rules, limiting applicability to real-world networks.

Purpose of the Study:

• To investigate the organizational development of growing random networks.
• To analyze the impact of attachment probability on network topology and node properties.
• To characterize the degree distribution, age-degree correlations, and component sizes within these networks.

Main Methods:

• Modeling growing random networks with a degree-dependent attachment probability A(k).
• Analyzing the resulting node degree distributions N(k)(t) as a function of time (t) and degree (k).
• Investigating the combined age and degree distributions, and the size distributions of in- and out-components.

Main Results:

• Attachment probability A(k) dictates network structure: slow growth leads to power-law decay, fast growth to a central hub, and linear growth to N(k)(t) ~ tk(-nu).
• Older nodes generally possess higher degrees, and nodes with similar degrees are preferentially connected.
• The in-component (ancestors) shows a robust s(-2) power-law tail, while the out-component (descendants) has a typical size of order ln(t).

Conclusions:

• The attachment probability function A(k) is a critical determinant of growing random network organization.
• The emergence of preferential attachment and age-degree correlations provides insights into network evolution and genealogy.
• The findings offer a framework for understanding the structure and dynamics of diverse real-world complex networks.