The Stochastic Transport Dynamics of a Conserved Quantity on a Complex Network

Area of Science:

• Complex Systems
• Network Science
• Statistical Physics

Background:

• Conserved quantities exhibit emergent stochastic dynamics in various complex systems, including social and biological networks.
• Understanding the transport of these quantities through networks is crucial for analyzing system behavior.

Purpose of the Study:

• To investigate the stochastic dynamics of conserved quantities transported over complex networks.
• To compare a master equation approach with mean-field approximations and stochastic simulations.
• To analyze the impact of different network topologies on quantity distribution.

Main Methods:

• Extension of the Ehrenfest urn model to complex networks.
• Master equation approach for analyzing packet dynamics.
• Mean-field theory for ensemble average evolution.
• Stochastic simulations for validation.
• Analysis of various network topologies (small-world, scale-free, Erdős-Rényi).

Main Results:

• The master equation provides a more comprehensive stochastic description than mean-field theory.
• Mean-field approximations agree well with master equation results in the thermodynamic limit for ensemble averages.
• The standard deviation of node occupation numbers is non-uniform across complex networks.
• A proposed scaling relation accurately constructs asymptotic probability distributions from single-packet network data.

Conclusions:

• The master equation approach offers a detailed probabilistic view of occupation numbers in network nodes.
• Network topology significantly influences the distribution of conserved quantities.
• The developed scaling relation provides an efficient method for determining equilibrium distributions, especially for large packet numbers.