Random Walk Approximation for Stochastic Processes on Graphs.

Area of Science:

• Computational Biology
• Mathematical Modeling
• Statistical Physics

Background:

• Stochastic processes on graphs are common in biological systems, like gene networks.
• Exact solutions for these systems are often computationally intractable.
• Existing approximation methods like System Size Expansion (SSE) have limitations.

Purpose of the Study:

• Introduce a novel approximation method, the Random Walk Approximation (RWA).
• Develop a method to approximate the stationary solution of master equations for stochastic processes.
• Provide a simpler, global analytical solution compared to SSE.

Main Methods:

• Developed the Random Walk Approximation (RWA) for master equations.
• Derived theoretically sufficient conditions for RWA validity.
• Estimated the approximation error with respect to particle number.
• Compared RWA against SSE and exact master equation solutions.

Main Results:

• RWA provides a simple analytical approximation for non-linear master equations.
• For linear systems, RWA yields the exact maximum entropy solution.
• RWA demonstrates comparable or superior performance to SSE in tested models.
• RWA shows good performance even when theoretical conditions for validity are not strictly met.

Conclusions:

• RWA is a powerful new tool for approximating stochastic processes in biological networks.
• The method offers advantages in simplicity and applicability over existing techniques.
• RWA provides valuable insights into systems where exact solutions are not feasible.