Diffusion on hierarchical systems of weakly-coupled networks

2013/03/31 by Grzegorz Siudem, Janusz A. Hołyst
Mathematics · Neuroscience · Physics and Astronomy · #Adiabatic process #Balance equation #Complex Network Analysis Techniques #Computer science #Diffusion #Entropy (arrow of time) #Entropy production #Functional Brain Connectivity Studies #Markov chain #Markov model #Markov process #Mathematics #Network topology #Opinion Dynamics and Social Influence #Partition (number theory) #Physics #Statistical physics #Thermodynamics #nlin.CD #physics.soc-ph

paper · pdf · doi:10.1016/j.physa.2018.08.078

published as Grzegorz Siudem, Janusz A. Hołyst, Diffusion on hierarchical systems of weakly-coupled networks, Physica A: Statistical Mechanics and its Applications, Volume 513, 2019 · 13 pages, 7 figures

openalex publication_date 2018/08/14 · arxiv created 2018/12/13 · arxiv updated 2018/12/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We analyse diffusion dynamics on weakly-coupled networks (interconnected networks) by means of separation of time scales. Using an adiabatic approximation we reduced the system dynamics to a Markov chain with aggregated variables and derived a transport equation that is analogous to Fick's First Law and includes a driving force. Entropy production is a sum of microscopic entropy transport, which results from the particle's migration between networks of different topologies and macroscopic entropy production of the Markov chain. Equilibrium particles partition between different sub-networks depends only on internal sub-network parameters. Our framework, confirmed by numerical simulations, is also useful for considering diffusion in nested systems corresponding to hierarchical networks with several different time scales thus it can serve to uncover hidden hierarchy levels from observations of diffusion processes.

Citations

• Diffusion on complex networks: a way to probe their large-scale topological structures
• Catastrophic cascade of failures in interdependent networks
• Community structure in social and biological networks
• Entropy rate of diffusion processes on complex networks
• Emergence of scaling in random networks
• Ferromagnetic phase transition in Barabási–Albert networks
• Mean-field theory for scale-free random networks
• Uncovering the overlapping community structure of complex networks in nature and society
• Maps of random walks on complex networks reveal community structure
• Pair approximation for theq-voter model with independence on complex networks