Path statistics, memory, and coarse-graining of continuous-time random walks on networks

Area of Science:

• Complex Systems
• Statistical Physics
• Computational Biology

Background:

• Continuous-time random walks (CTRWs) are fundamental models in physics, chemistry, and biology, often studied on various discrete state spaces.
• Standard CTRW analyses struggle with complex energy landscapes, coarse-grained states, and spatial disorder, which introduce memory effects and non-exponential waiting time distributions.
• Existing methods fail to capture these crucial effects in CTRW analysis, limiting their applicability to more realistic systems.

Purpose of the Study:

• To develop a novel statistical mechanics approach for analyzing first-passage CTRWs on networks with arbitrary complexity.
• To incorporate memory effects and non-exponential waiting time distributions into CTRW analysis on complex energy landscapes.
• To provide a computational tool for calculating higher moments of path statistics and forces along paths.

Main Methods:

• Utilized the statistical mechanics of the nonequilibrium path ensemble to characterize first-passage CTRWs.
• Developed recursion relations, inspired by transfer matrix and exact enumeration techniques, for efficient numerical calculation of path statistics.
• Implemented the algorithm in a Python script named PathMAN (Path Matrix Algorithm for Networks).

Main Results:

• The approach successfully characterizes CTRWs on networks with arbitrary connectivity, energy landscapes, and waiting time distributions.
• Derived exact relations for homogeneous networks, quantifying the accuracy of discrete-time projections of continuous-time processes.
• Demonstrated the importance of non-exponential distributions, memory, and coarse-graining through representative examples.

Conclusions:

• The developed method and PathMAN algorithm offer a robust framework for analyzing complex CTRWs beyond the limitations of existing techniques.
• The findings highlight the critical role of memory and non-exponential waiting times in understanding CTRW behavior in diverse scientific domains.
• PathMAN provides a practical tool for researchers to apply advanced CTRW analysis to their specific models.