Finite connected components in infinite directed and multiplex networks with arbitrary degree distributions

Area of Science:

• Network Science
• Statistical Physics
• Complex Systems

Background:

• Understanding the structure of complex networks is crucial in various scientific fields.
• Existing models often simplify network topology, limiting applicability to real-world systems.
• Characterizing connected components in generalized network models remains an active research area.

Purpose of the Study:

• To derive exact expressions for the size distributions of weakly and multilayer connected components.
• To analyze these distributions in two generalized network models: directed and multiplex networks.
• To investigate the critical behavior and exponents of these size distributions.

Main Methods:

• Development of exact analytical expressions for component size distributions.
• Application of the configuration model generalizations for directed and multiplex networks.
• Asymptotic analysis to determine critical exponents under specific conditions.

Main Results:

• Derived computable, polynomial-time expressions for size distributions.
• Identified specific critical exponents (-3/2) for two-layer connected components in multiplex networks.
• Revealed two critical exponents (-1/2 and -3/2) for weakly connected components in directed networks.

Conclusions:

• The derived expressions provide a precise framework for analyzing network components.
• The findings offer quantitative insights into the phase transitions and critical phenomena in these network types.
• This work advances the theoretical understanding of complex network structures and their properties.