Matrix product algorithm for stochastic dynamics on networks applied to nonequilibrium Glauber dynamics.

Area of Science:

• Statistical physics
• Computational physics
• Complex systems

Background:

• Stochastic dynamical processes are fundamental to many complex systems, including spin glasses, neural networks, and biological networks.
• Simulating these processes on networks, especially those with cycles, presents significant computational challenges.
• Existing methods like Monte Carlo simulations often struggle with error scaling and accuracy for certain system properties.

Purpose of the Study:

• To introduce and apply an efficient and precise simulation method for stochastic dynamical processes on locally treelike graphs and networks with cycles.
• To leverage techniques from quantum many-body theory for improved computational modeling.
• To enable accurate analysis of complex network dynamics, including those with small expectation values.

Main Methods:

• The study employs a novel approach based on matrix product approximation of edge messages (MPEM).
• This method treats networks with cycles using the cavity method framework.
• Computational cost and accuracy are tunable by adjusting matrix dimensions in MPEM truncations.

Main Results:

• The developed algorithm demonstrates superior error scaling compared to Monte Carlo simulations.
• It accurately simulates both single instances and the thermodynamic limit of stochastic processes.
• The method was successfully applied to analyze nonequilibrium Glauber dynamics in the kinetic Ising model.

Conclusions:

• The MPEM approach provides an efficient and accurate tool for simulating stochastic dynamics on complex networks.
• This method overcomes limitations of traditional simulations, particularly for systems with cycles and for evaluating small expectation values.
• It opens new possibilities for studying decay processes and temporal correlations in diverse technological, biological, and social systems.