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Dynamical random graphs with memory.

Tatyana S Turova¹

¹Center for Mathematical Sciences, University of Lund, Box 118, Lund 22100, Sweden.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics

|August 22, 2002

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This study introduces a dynamical graph model unifying random growth and random graph models. It identifies critical parameters for phase transitions and analyzes network efficiency.

Area of Science:

Complex Systems
Network Science
Probability Theory

Background:

Markov processes model systems with evolving states.
Branching processes describe population growth.
Mean-field dynamics simplify interactions in large systems.

Purpose of the Study:

To introduce a general class of dynamical graphs.
To unify existing models like random growth and random graphs.
To analyze phase transitions and network efficiency.

Main Methods:

Modeling Markov processes on graphs with evolving vertices and edges.
Utilizing supercritical branching processes for vertex dynamics.
Implementing mean-field edge dynamics with exponential lifetimes and a memory parameter.

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Main Results:

Demonstrated the model bridges random growth (infinite memory) and random graphs (zero memory).
Identified critical parameter values for phase transitions.
Characterized the phase diagram and network properties.

Conclusions:

The proposed dynamical graph model offers a unified framework for studying network evolution.
Understanding memory parameter effects is key to network behavior and efficiency.
The model provides insights into the relationship between different network models.