From random walks on networks to nonlinear diffusion

Abstract

Mathematical models of motility are often based on random-walk descriptions of discrete individuals that can move according to certain rules. It is usually the case that large masses concentrated in small regions of space have a great impact on the collective movement of the group. For this reason, many models in mathematical biology have incorporated crowding effects and managed to understand their implications. Here, we build on a previously developed framework for random walks on networks to show that in the continuum limit, the underlying stochastic process can be identified with a diffusion partial differential equation. The diffusion coefficient of the emerging equation is in general density-dependent, and can be directly related to the transition probabilities of the random walk. Moreover, the relaxation time of the stochastic process is directly linked to the diffusion coefficient and also to the network structure, as it usually happens in the case of linear diffusion. As a specific example, we study the equivalent of a porous-medium type equation on networks, which shows similar properties to its continuum equivalent. For this equation, self-similar solutions on a lattice and on homogeneous trees can be found, showing finite speed of propagation in contrast to commonly used linear diffusion equations. These findings also provide insights into reaction-diffusion systems with general diffusion operators, which have appeared recently in some applications.

References in corpus (22)

• Synchronization in complex networks
• Random walks and diffusion on networks
• Predator-prey cycles from resonant amplification of demographic stochasticity
• Physical Models of Collective Cell Migration
• Entropy Rate of Diffusion Processes on Complex Networks
• Topology-driven instabilities: the theory of pattern formation on directed networks
• Biased random walks on complex networks: the role of local navigation rules
• Stochastic models in population biology and their deterministic analogs
• Graph partitions and cluster synchronization in networks of oscillators
• Maximal-entropy random walks in complex networks with limited information
• Characteristic times of biased random walks on complex networks
• Diffusion of multiple species with excluded-volume effects
• A stochastic model for wound healing
• Bifurcation Diagram for Compartmentalized Granular Gases
• Hopping in the crowd to unveil network topology
• Nonlocal-interaction equation on graphs: gradient flow structure and continuum limit
• Time and space generalized diffusion equation on graphs/networks
• On the singularity formation and relaxation to equilibrium in 1D Fokker-Planck model with superlinear drift
• Nonlinear random walks optimize the trade-off between cost and prevention in epidemics lockdown measures : the ESIR model
• Self-segregation in heterogeneous metapopulation landscapes
• Reactions, Diffusion and Volume Exclusion in a Heterogeneous System of Interacting Particles
• Chaos in Nonlinear Random Walks with Non-Monotonic Transition Probabilities

Cited by in corpus (2)

• Random walk models for the propagation of signalling molecules in one-dimensional spatial networks and their continuum limit
• A nonlocal-to-local approach to aggregation-diffusion equations