From random walks on networks to nonlinear diffusion.

Area of Science:

• Mathematical Biology
• Network Science
• Stochastic Processes

Background:

• Mathematical models of motility often use random walks, with crowding effects significantly impacting collective movement.
• Previous frameworks for random walks on networks provide a basis for analyzing complex group dynamics.

Purpose of the Study:

• To establish a connection between random walks on networks and diffusion partial differential equations in the continuum limit.
• To investigate the properties of density-dependent diffusion coefficients and relaxation times in these network models.

Main Methods:

• Developed a framework for random walks on networks.
• Analyzed the continuum limit to derive partial differential equations.
• Studied a porous-medium-type equation on networks as a specific example.

Main Results:

• The underlying stochastic process of random walks on networks can be identified with a diffusion partial differential equation.
• The diffusion coefficient is generally density-dependent and linked to transition probabilities.
• Relaxation time depends on the diffusion coefficient and network structure; self-similar solutions exhibit finite propagation speed.

Conclusions:

• This work bridges discrete random walk models on networks with continuous diffusion equations.
• The findings offer insights into reaction-diffusion systems with complex diffusion operators.
• The derived models capture emergent behaviors like finite propagation speed, relevant for biological systems.