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Random Walks and Stochastic Processes in Complex Systems: From Physics to Socio-Economic Phenomena

A special issue of Entropy (ISSN 1099-4300). This special issue belongs to the section "Complexity".

Deadline for manuscript submissions: 28 February 2025 | Viewed by 6928

Special Issue Editors

Dr. Trifce Sandev

E-Mail Website1 Website2
Guest Editor
1. Research Center for Computer Science and Information Technologies, Macedonian Academy of Sciences and Arts, Bul. Krste Misirkov 2, 1000 Skopje, Macedonia
2. Institute of Physics, Faculty of Natural Sciences and Mathematics, Ss. Cyril and Methodius University, Arhimedova 3, 1000 Skopje, Macedonia
3. Institute of Physics & Astronomy, University of Potsdam, D-14776 Potsdam, Germany
Interests: statistical mechanics; mathematical physics; stochastic processes; anomalous diffusion; fractional calculus
Special Issues, Collections and Topics in MDPI journals
Dr. Alexander Iomin

E-Mail Website
Guest Editor
Retired, R&D, Technion, Haifa 32000, Israel
Interests: fractional kinetics; fractional electrostatics; fractional quantum mechanics

Special Issue Information

Dear Colleagues,

Random walks are underlying phenomena of various processes in nature, economics and social behavior. Theoretical investigations of random walks and stochastic processes in complex systems have been of great interest for years. The modeling of random processes in complex systems, including complex networks and graphs, requires an interdisciplinary approach due to the different applications of same/analogous or similar models in various fields, such as physics, biology, computer science, economics and social sciences. 

The vast amount of data obtained experimentally or observed empirically and by means of computer simulations requires new theoretical approaches in order to understand the dynamics of such complex systems, which open new vistas in physics, biology, computer science, engineering and economics. Nowadays, methods of statistical physics have been applied to describe complex social phenomena using stochastic and kinetic differential equations or agent-based models. Analysis of the empirical data from various economic and financial systems has shown that, despite the abundance of proposed models, there is still a lack of models that accurately reproduce and explain the emergence of empirically observable statistical properties. One such example is the famed Ornstein–Uhlenbeck process, which has been used in physics to describe the random motion of a particle in a harmonic potential, but can be also used to model interest and currency exchange rates. Furthermore, geometric Brownian motion, which is an universal model for self-reproducing phenomena, such as population and wealth, can also be used in mathematical finance (in the Black–Scholes model) for asset pricing, but also to model other natural phenomena such as bacterial cell division, fragment sizes in rock crushing processes, as well as being connected to turbulent diffusion governed by inhomogeneous advection–diffusion equations. Moreover, the voter model is a part of the area of sociophysics, and is used, for example, to model opinion dynamics. Different generalizations of the voter model, which can also be related to the diffusion problems in physics, have been introduced and applied to real data, as well.

The purpose of this Special Issue is to reflect the current situation in the application of random walks and stochastic models in various fields of science, such as physics, computer science, economics and social sciences. We kindly invite researchers working in these fields to contribute with original research/review papers dedicated to theoretical modeling and applications.

Dr. Trifce Sandev
Dr. Alexander Iomin
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Entropy is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • random walks
  • stochastic equations
  • kinetic equations
  • diffusion
  • geometric Brownian motion
  • voter model
  • econophysics
  • sociophysics

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Published Papers (6 papers)

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Research

25 pages, 2010 KiB  
Article
Congestion Transition on Random Walks on Graphs
by Lorenzo Di Meco, Mirko Degli Esposti, Federico Bellisardi and Armando Bazzani
Entropy 2024, 26(8), 632; https://doi.org/10.3390/e26080632 - 26 Jul 2024
Viewed by 679
Abstract
The formation of congestion on an urban road network is a key issue for the development of sustainable mobility in future smart cities. In this work, we propose a reductionist approach by studying the stationary states of a simple transport model using a [...] Read more.
The formation of congestion on an urban road network is a key issue for the development of sustainable mobility in future smart cities. In this work, we propose a reductionist approach by studying the stationary states of a simple transport model using a random process on a graph, where each node represents a location and the link weights give the transition rates to move from one node to another, representing the mobility demand. Each node has a maximum flow rate and a maximum load capacity, and we assume that the average incoming flow equals the outgoing flow. In the approximation of the single-step process, we are able to analytically characterize the traffic load distribution on the single nodes using a local maximum entropy principle. Our results explain how congested nodes emerge as the total traffic load increases, analogous to a percolation transition where the appearance of a congested node is an independent random event. However, using numerical simulations, we show that in the more realistic case of synchronous dynamics for the nodes, entropic forces introduce correlations among the node states and favor the clustering of empty and congested nodes. Our aim is to highlight the universal properties of congestion formation and, in particular, to understand the role of traffic load fluctuations as a possible precursor of congestion in a transport network. Full article
(This article belongs to the Special Issue Random Walks and Stochastic Processes in Complex Systems: From Physics to Socio-Economic Phenomena)
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9 pages, 608 KiB  
Article
Modelling Heterogeneous Anomalous Dynamics of Radiation-Induced Double-Strand Breaks in DNA during Non-Homologous End-Joining Pathway
by Nickolay Korabel, John W. Warmenhoven, Nicholas T. Henthorn, Samuel Ingram, Sergei Fedotov, Charlotte J. Heaven, Karen J. Kirkby, Michael J. Taylor and Michael J. Merchant
Entropy 2024, 26(6), 502; https://doi.org/10.3390/e26060502 - 8 Jun 2024
Viewed by 985
Abstract
The process of end-joining during nonhomologous repair of DNA double-strand breaks (DSBs) after radiation damage is considered. Experimental evidence has revealed that the dynamics of DSB ends exhibit subdiffusive motion rather than simple diffusion with rare directional movement. Traditional models often overlook the [...] Read more.
The process of end-joining during nonhomologous repair of DNA double-strand breaks (DSBs) after radiation damage is considered. Experimental evidence has revealed that the dynamics of DSB ends exhibit subdiffusive motion rather than simple diffusion with rare directional movement. Traditional models often overlook the rare long-range directed motion. To address this limitation, we present a heterogeneous anomalous diffusion model consisting of subdiffusive fractional Brownian motion interchanged with short periods of long-range movement. Our model sheds light on the underlying mechanisms of heterogeneous diffusion in DSB repair and could be used to quantify the DSB dynamics on a time scale inaccessible to single particle tracking analysis. The model predicts that the long-range movement of DSB ends is responsible for the misrepair of DSBs in the form of dicentric chromosome lesions. Full article
(This article belongs to the Special Issue Random Walks and Stochastic Processes in Complex Systems: From Physics to Socio-Economic Phenomena)
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12 pages, 1160 KiB  
Article
Entropy Production in Reaction–Diffusion Systems Confined in Narrow Channels
by Guillermo Chacón-Acosta and Mayra Núñez-López
Entropy 2024, 26(6), 463; https://doi.org/10.3390/e26060463 - 29 May 2024
Viewed by 739
Abstract
This work analyzes the effect of wall geometry when a reaction–diffusion system is confined to a narrow channel. In particular, we study the entropy production density in the reversible Gray–Scott system. Using an effective diffusion equation that considers modifications by the channel characteristics, [...] Read more.
This work analyzes the effect of wall geometry when a reaction–diffusion system is confined to a narrow channel. In particular, we study the entropy production density in the reversible Gray–Scott system. Using an effective diffusion equation that considers modifications by the channel characteristics, we find that the entropy density changes its value but not its qualitative behavior, which helps explore the structure-formation space. Full article
(This article belongs to the Special Issue Random Walks and Stochastic Processes in Complex Systems: From Physics to Socio-Economic Phenomena)
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30 pages, 3660 KiB  
Article
Stochastic Compartment Model with Mortality and Its Application to Epidemic Spreading in Complex Networks
by Téo Granger, Thomas M. Michelitsch, Michael Bestehorn, Alejandro P. Riascos and Bernard A. Collet
Entropy 2024, 26(5), 362; https://doi.org/10.3390/e26050362 - 25 Apr 2024
Cited by 1 | Viewed by 1337
Abstract
We study epidemic spreading in complex networks by a multiple random walker approach. Each walker performs an independent simple Markovian random walk on a complex undirected (ergodic) random graph where we focus on the Barabási–Albert (BA), Erdös–Rényi (ER), and Watts–Strogatz (WS) types. Both [...] Read more.
We study epidemic spreading in complex networks by a multiple random walker approach. Each walker performs an independent simple Markovian random walk on a complex undirected (ergodic) random graph where we focus on the Barabási–Albert (BA), Erdös–Rényi (ER), and Watts–Strogatz (WS) types. Both walkers and nodes can be either susceptible (S) or infected and infectious (I), representing their state of health. Susceptible nodes may be infected by visits of infected walkers, and susceptible walkers may be infected by visiting infected nodes. No direct transmission of the disease among walkers (or among nodes) is possible. This model mimics a large class of diseases such as Dengue and Malaria with the transmission of the disease via vectors (mosquitoes). Infected walkers may die during the time span of their infection, introducing an additional compartment D of dead walkers. Contrary to the walkers, there is no mortality of infected nodes. Infected nodes always recover from their infection after a random finite time span. This assumption is based on the observation that infectious vectors (mosquitoes) are not ill and do not die from the infection. The infectious time spans of nodes and walkers, and the survival times of infected walkers, are represented by independent random variables. We derive stochastic evolution equations for the mean-field compartmental populations with the mortality of walkers and delayed transitions among the compartments. From linear stability analysis, we derive the basic reproduction numbers RM,R0 with and without mortality, respectively, and prove that RM<R0. For RM,R0>1, the healthy state is unstable, whereas for zero mortality, a stable endemic equilibrium exists (independent of the initial conditions), which we obtained explicitly. We observed that the solutions of the random walk simulations in the considered networks agree well with the mean-field solutions for strongly connected graph topologies, whereas less well for weakly connected structures and for diseases with high mortality. Our model has applications beyond epidemic dynamics, for instance in the kinetics of chemical reactions, the propagation of contaminants, wood fires, and others. Full article
(This article belongs to the Special Issue Random Walks and Stochastic Processes in Complex Systems: From Physics to Socio-Economic Phenomena)
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15 pages, 2484 KiB  
Article
Non-Markovian Diffusion and Adsorption–Desorption Dynamics: Analytical and Numerical Results
by Derik W. Gryczak, Ervin K. Lenzi, Michely P. Rosseto, Luiz R. Evangelista and Rafael S. Zola
Entropy 2024, 26(4), 294; https://doi.org/10.3390/e26040294 - 27 Mar 2024
Viewed by 950
Abstract
The interplay of diffusion with phenomena like stochastic adsorption–desorption, absorption, and reaction–diffusion is essential for life and manifests in diverse natural contexts. Many factors must be considered, including geometry, dimensionality, and the interplay of diffusion across bulk and surfaces. To address this complexity, [...] Read more.
The interplay of diffusion with phenomena like stochastic adsorption–desorption, absorption, and reaction–diffusion is essential for life and manifests in diverse natural contexts. Many factors must be considered, including geometry, dimensionality, and the interplay of diffusion across bulk and surfaces. To address this complexity, we investigate the diffusion process in heterogeneous media, focusing on non-Markovian diffusion. This process is limited by a surface interaction with the bulk, described by a specific boundary condition relevant to systems such as living cells and biomaterials. The surface can adsorb and desorb particles, and the adsorbed particles may undergo lateral diffusion before returning to the bulk. Different behaviors of the system are identified through analytical and numerical approaches. Full article
(This article belongs to the Special Issue Random Walks and Stochastic Processes in Complex Systems: From Physics to Socio-Economic Phenomena)
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14 pages, 407 KiB  
Article
Random Walks on Comb-like Structures under Stochastic Resetting
by Axel Masó-Puigdellosas, Trifce Sandev and Vicenç Méndez
Entropy 2023, 25(11), 1529; https://doi.org/10.3390/e25111529 - 9 Nov 2023
Cited by 2 | Viewed by 1326
Abstract
We study the long-time dynamics of the mean squared displacement of a random walker moving on a comb structure under the effect of stochastic resetting. We consider that the walker’s motion along the backbone is diffusive and it performs short jumps separated by [...] Read more.
We study the long-time dynamics of the mean squared displacement of a random walker moving on a comb structure under the effect of stochastic resetting. We consider that the walker’s motion along the backbone is diffusive and it performs short jumps separated by random resting periods along fingers. We take into account two different types of resetting acting separately: global resetting from any point in the comb to the initial position and resetting from a finger to the corresponding backbone. We analyze the interplay between the waiting process and Markovian and non-Markovian resetting processes on the overall mean squared displacement. The Markovian resetting from the fingers is found to induce normal diffusion, thereby minimizing the trapping effect of fingers. In contrast, for non-Markovian local resetting, an interesting crossover with three different regimes emerges, with two of them subdiffusive and one of them diffusive. Thus, an interesting interplay between the exponents characterizing the waiting time distributions of the subdiffusive random walk and resetting takes place. As for global resetting, its effect is even more drastic as it precludes normal diffusion. Specifically, such a resetting can induce a constant asymptotic mean squared displacement in the Markovian case or two distinct regimes of subdiffusive motion in the non-Markovian case. Full article
(This article belongs to the Special Issue Random Walks and Stochastic Processes in Complex Systems: From Physics to Socio-Economic Phenomena)
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