@article{OehmeMoewisBoethetal.2024, author = {Oehme, Stephan and Moewis, Philippe and Boeth, Heide and Bartek, Benjamin and von Tycowicz, Christoph and Ehrig, Rainald and Duda, Georg and Jung, Tobias}, title = {Altered knee kinematics after posterior cruciate single-bundle reconstruction - a comprehensive prospective biomechanical in vivo analysis}, volume = {12}, journal = {Frontiers in Bioengineering and Biotechnology}, doi = {10.3389/fbioe.2024.1322136}, year = {2024}, language = {en} } @article{ErnstFalkenhagenWinkelmann2023, author = {Ernst, Ariane and Falkenhagen, Undine and Winkelmann, Stefanie}, title = {Model reduction for calcium-induced vesicle fusion dynamics}, volume = {23}, journal = {Proceedings in Applied Mathematics \& Mechanics}, number = {4}, doi = {10.1002/pamm.202300184}, year = {2023}, abstract = {In this work, we adapt an established model for the Ca2+-induced fusion dynamics of synaptic vesicles and employ a lumping method to reduce its complexity. In the reduced system, sequential Ca2+-binding steps are merged to a single releasable state, while keeping the important dependence of the reaction rates on the local Ca2+ concentration. We examine the feasibility of this model reduction for a representative stimulus train over the physiologically relevant site-channel distances. Our findings show that the approximation error is generally small and exhibits an interesting nonlinear and non-monotonic behavior where it vanishes for very low distances and is insignificant at intermediary distances. Furthermore, we give expressions for the reduced model's reaction rates and suggest that our approach may be used to directly compute effective fusion rates for assessing the validity of a fusion model, thereby circumventing expensive simulations.}, language = {en} } @article{ErbanWinkelmann2025, author = {Erban, Radek and Winkelmann, Stefanie}, title = {Multi-grid reaction-diffusion master equation: applications to morphogen gradient modelling}, volume = {87}, journal = {Bulletin of Mathematical Biology}, arxiv = {http://arxiv.org/abs/arXiv:2405.02117}, doi = {10.1007/s11538-024-01377-y}, pages = {6}, year = {2025}, abstract = {The multi-grid reaction-diffusion master equation (mgRDME) provides a generalization of stochastic compartment-based reaction-diffusion modelling described by the standard reaction-diffusion master equation (RDME). By enabling different resolutions on lattices for biochemical species with different diffusion constants, the mgRDME approach improves both accuracy and efficiency of compartment-based reaction-diffusion simulations. The mgRDME framework is examined through its application to morphogen gradient formation in stochastic reaction-diffusion scenarios, using both an analytically tractable first-order reaction network and a model with a second-order reaction. The results obtained by the mgRDME modelling are compared with the standard RDME model and with the (more detailed) particle-based Brownian dynamics simulations. The dependence of error and numerical cost on the compartment sizes is defined and investigated through a multi-objective optimization problem.}, language = {en} } @article{ArmasMerbisMeylahnetal.2025, author = {Armas, Jay and Merbis, Wout and Meylahn, Janusz M and Rafiee Rad, Soroush and del Razo, Mauricio J}, title = {Risk aversion can promote cooperation}, volume = {6}, journal = {Journal of Physics: Complexity}, number = {1}, arxiv = {http://arxiv.org/abs/2306.05971}, doi = {10.1088/2632-072X/adb234}, year = {2025}, abstract = {Cooperative dynamics are central to our understanding of many phenomena in living and complex systems. However, we lack a universal mechanism to explain the emergence of cooperation. We present a novel framework for modelling social dilemma games with an arbitrary number of players by combining reaction networks, methods from quantum mechanics applied to stochastic complex systems, game theory and stochastic simulations of molecular reactions. Using this framework, we propose a novel and robust mechanism for cooperation based on risk aversion that leads to cooperative behaviour in population games. Rather than individuals seeking to maximise payouts in the long run, individuals seek to obtain a minimum set of resources with a given level of confidence and in a limited time span. We show that this mechanism can lead to the emergence of new equilibria in a range of social dilemma games.}, language = {en} } @article{BernsYildizWinkelmannetal.2025, author = {Berns, Manon and Yildiz, Mirza and Winkelmann, Stefanie and Walter, Alexander}, title = {Independently engaging protein tethers of different length enhance synaptic vesicle trafficking to the plasma membrane}, volume = {603}, journal = {The Journal of Physiology}, number = {20}, doi = {10.1113/JP286651}, pages = {6107 -- 6134}, year = {2025}, abstract = {Synaptic vesicle (SV) trafficking toward the plasma membrane (PM) and subsequent SV maturation are essential for neurotransmitter release. These processes, including SV docking and priming, are coordinated by various proteins, such as SNAREs, Munc13, and Synaptotagmin (Syt), which connect—tether—the SV to the PM. Here, we investigated how tethers of varying lengths mediate SV docking using a simplified mathematical model. The heights of the three tether types—estimated from the structures of the SNARE complex, Munc13, and Syt—defined the SV-to-PM distance ranges for tether formation. Geometric considerations linked SV-to-PM distances to the probability and rate of tether formation. We assumed that SV tethering constrains SV motility and that multiple tethers associate by independent interactions. The model predicted that forming multiple tethers favors shorter SV-to-PM distances. Although tethers acted independently in the model, their geometrical properties often caused sequential assembly, from longer ones (Munc13/Syt), that accelerated SV movement towards the PM, to shorter ones (SNAREs) which stabilized PM-proximal SVs. Modifying tether lengths or numbers affected SV trafficking. The independent implementation of tethering proteins enabled their selective removal to mimic gene knockout situations. This showed that simulated SV-to-PM distance distributions qualitatively aligned with published EM studies upon removal of SNARE and Syt tethers, while Munc13 knockout data were best approximated when assuming additional disruption of SNARE tethers. Thus, while salient features of SV docking can be accounted for by independent tethering alone, our results suggest that functional tether interactions not yet featured in our model are crucial for biological function.}, language = {en} } @article{delRazoDelleSite2025, author = {del Razo, Mauricio and Delle Site, Luigi}, title = {Dynamics of systems with varying number of particles: from Liouville equations to general master equations for open systems}, volume = {18, 001}, journal = {SciPost Physics}, doi = {10.21468/SciPostPhys.18.1.001}, year = {2025}, abstract = {A varying number of particles is one of the most relevant characteristics of systems of interest in nature and technology, ranging from the exchange of energy and matter with the surrounding environment to the change of particle number through internal dynamics such as reactions. The physico-mathematical modeling of these systems is extremely challenging, with the major difficulty being the time dependence of the number of degrees of freedom and the additional constraint that the increment or reduction of the number and species of particles must not violate basic physical laws. Theoretical models, in such a case, represent the key tool for the design of computational strategies for numerical studies that deliver trustful results. In this manuscript, we review complementary physico-mathematical approaches of varying number of particles inspired by rather different specific numerical goals. As a result of the analysis on the underlying common structure of these models, we propose a unifying master equation for general dynamical systems with varying number of particles. This equation embeds all the previous models and can potentially model a much larger range of complex systems, ranging from molecular to social agent-based dynamics.}, language = {en} } @article{delRazoKostre2025, author = {del Razo, Mauricio and Kostr{\´e}, Margarita}, title = {Open reaction-diffusion systems: bridging probabilistic theory and simulations across scales}, volume = {58}, journal = {Journal of Physics A: Mathematical and Theoretical}, number = {14}, doi = {10.1088/1751-8121/adc520}, year = {2025}, abstract = {Reaction-diffusion processes are the foundational model for a diverse range of complex systems, ranging from biochemical reactions to social agent-based phenomena. The underlying dynamics of these systems occur at the individual particle/agent level, and in realistic applications, they often display interaction with their environment through energy or material exchange with a reservoir. This requires intricate mathematical considerations, especially in the case of material exchange since the varying number of particles/agents results in ``on-the-fly'' modification of the system dimension. In this work, we first overview the probabilistic description of reaction-diffusion processes at the particle level, which readily handles varying number of particles. We then extend this model to consistently incorporate interactions with macroscopic material reservoirs. Based on the resulting expressions, we bridge the probabilistic description with macroscopic concentration-based descriptions for linear and nonlinear reaction-diffusion systems, as well as for an archetypal open reaction-diffusion system. Using these mathematical bridges across scales, we finally develop numerical schemes for open reaction-diffusion systems, which we implement in two illustrative examples. This work establishes a methodological workflow to bridge particle-based probabilistic descriptions with macroscopic concentration-based descriptions of reaction-diffusion in open settings, laying the foundations for a multiscale theoretical framework upon which to construct theory and simulation schemes that are consistent across scales.}, language = {en} } @article{MontefuscoHelfmannOkunolaetal.2024, author = {Montefusco, Alberto and Helfmann, Luzie and Okunola, Toluwani and Winkelmann, Stefanie and Sch{\"u}tte, Christof}, title = {Partial mean-field model for neurotransmission dynamics}, volume = {369}, journal = {Mathematical Biosciences}, arxiv = {http://arxiv.org/abs/2307.01737}, doi = {10.1016/j.mbs.2024.109143}, year = {2024}, abstract = {This article addresses reaction networks in which spatial and stochastic effects are of crucial importance. For such systems, particle-based models allow us to describe all microscopic details with high accuracy. However, they suffer from computational inefficiency if particle numbers and density get too large. Alternative coarse-grained-resolution models reduce computational effort tremendously, e.g., by replacing the particle distribution by a continuous concentration field governed by reaction-diffusion PDEs. We demonstrate how models on the different resolution levels can be combined into hybrid models that seamlessly combine the best of both worlds, describing molecular species with large copy numbers by macroscopic equations with spatial resolution while keeping the stochastic-spatial particle-based resolution level for the species with low copy numbers. To this end, we introduce a simple particle-based model for the binding dynamics of ions and vesicles at the heart of the neurotransmission process. Within this framework, we derive a novel hybrid model and present results from numerical experiments which demonstrate that the hybrid model allows for an accurate approximation of the full particle-based model in realistic scenarios.}, language = {en} } @article{LueckeWinkelmannHeitzigetal.2024, author = {L{\"u}cke, Marvin and Winkelmann, Stefanie and Heitzig, Jobst and Molkenthin, Nora and Koltai, P{\´e}ter}, title = {Learning interpretable collective variables for spreading processes on networks}, volume = {109}, journal = {Physical Review E}, number = {2}, arxiv = {http://arxiv.org/abs/2307.03491}, doi = {10.1103/PhysRevE.109.L022301}, pages = {L022301}, year = {2024}, abstract = {Collective variables (CVs) are low-dimensional projections of high-dimensional system states. They are used to gain insights into complex emergent dynamical behaviors of processes on networks. The relation between CVs and network measures is not well understood and its derivation typically requires detailed knowledge of both the dynamical system and the network topology. In this Letter, we present a data-driven method for algorithmically learning and understanding CVs for binary-state spreading processes on networks of arbitrary topology. We demonstrate our method using four example networks: the stochastic block model, a ring-shaped graph, a random regular graph, and a scale-free network generated by the Albert-Barab{\´a}si model. Our results deliver evidence for the existence of low-dimensional CVs even in cases that are not yet understood theoretically.}, language = {en} } @article{ErnstBankowskiJusyteetal.2025, author = {Ernst, Ariane and Bankowski, Anastasia and Jusyte, Meida and Okunola, Toluwani and Petrov, Tino and Walter, Alexander and Winkelmann, Stefanie}, title = {Parameter Optimization for a Neurotransmission Recovery Model}, volume = {87}, journal = {Bulletin of Mathematical Biology}, publisher = {Bulletin of Mathematical Biology}, doi = {10.1007/s11538-025-01486-2}, year = {2025}, abstract = {We assess the empirical applicability of a simplified model for neurotransmitter release that incorporates maturation, fusion, and recovery of both release sites and vesicles. Model parameters are optimized by fitting the model to experimental data obtained from neuromuscular junction synapses of 3rd-instar Drosophila melanogaster larvae. In particular, the mean-squared error between the local extrema of the simulated total junction current and its experimental counterpart is minimized. We compare three estimation approaches, differing in the choice of optimized parameters and the fusion rate function. Despite the model's minimalistic structure, it demonstrates a compelling ability to replicate experimental data, yielding plausible parameter estimates for five different animals. An additional identifiability analysis based on the profile likelihood reveals practical non-identifiabilities for several parameters, highlighting the need for additional constraints or data to improve estimation accuracy.}, language = {en} } @article{OehmeMoewisBoethetal.2022, author = {Oehme, Stephan and Moewis, Philippe and Boeth, Heide and Bartek, Benjamin and Lippert, Annika and von Tycowicz, Christoph and Ehrig, Rainald and Duda, Georg and Jung, Tobias}, title = {PCL insufficient patients with increased translational and rotational passive knee joint laxity have no increased range of anterior-posterior and rotational tibiofemoral motion during level walking}, volume = {12}, journal = {Scientific Reports}, number = {1}, doi = {10.1038/s41598-022-17328-3}, pages = {1 -- 11}, year = {2022}, language = {en} } @article{FischerHolzhausenRoeblitz2022, author = {Fischer-Holzhausen, Sophie and R{\"o}blitz, Susanna}, title = {Hormonal regulation of ovarian follicle growth in humans: Model-based exploration of cycle variability and parameter sensitivities}, journal = {Journal of Theoretical Biology}, doi = {https://doi.org/10.1016/j.jtbi.2022.111150}, year = {2022}, abstract = {We present a modelling and simulation framework for the dynamics of ovarian follicles and key hormones along the hypothalamic-pituitary-gonadal axis throughout consecutive human menstrual cycles. All simulation results (hormone concentrations and ovarian follicle sizes) are in biological units and can easily be compared to clinical data. The model takes into account variability in follicles' response to stimulating hormones, which introduces variability between cycles. The growth of ovarian follicles in waves is an emergent property in our model simulations and further supports the hypothesis that follicular waves are also present in humans. We use Approximate Bayesian Computation and cluster analysis to construct a population of virtual subjects and to study parameter distributions and sensitivities. The model can be used to compare and optimize treatment protocols for ovarian hyperstimulation, thus potentially forming the integral part of a clinical decision support system in reproductive endocrinology.}, language = {en} } @article{FischerHolzhausenRoeblitz2022, author = {Fischer-Holzhausen, Sophie and R{\"o}blitz, Susanna}, title = {Mathematical modelling of follicular growth and ovarian stimulation}, journal = {Current Opinion in Endocrine and Metabolic Research}, doi = {https://doi.org/10.1016/j.coemr.2022.100385}, year = {2022}, language = {en} } @article{ErnstUngerSchuetteetal.2023, author = {Ernst, Ariane and Unger, Nathalie and Sch{\"u}tte, Christof and Walter, Alexander and Winkelmann, Stefanie}, title = {Rate-limiting recovery processes in neurotransmission under sustained stimulation}, volume = {362}, journal = {Mathematical Biosciences}, arxiv = {http://arxiv.org/abs/2302.01635}, doi = {10.1016/j.mbs.2023.109023}, year = {2023}, abstract = {At chemical synapses, an arriving electric signal induces the fusion of vesicles with the presynaptic membrane, thereby releasing neurotransmitters into the synaptic cleft. After a fusion event, both the release site and the vesicle undergo a recovery process before becoming available for reuse again. Of central interest is the question which of the two restoration steps acts as the limiting factor during neurotrans-mission under high-frequency sustained stimulation. In order to investigate this question, we introduce a novel non-linear reaction network which involves explicit recovery steps for both the vesicles and the release sites, and includes the induced time-dependent output current. The associated reaction dynamics are formulated by means of ordinary differential equations (ODEs), as well as via the associated stochastic jump process. While the stochastic jump model describes a single release site, the average over many release sites is close to the ODE solution and shares its periodic structure. The reason for this can be traced back to the insight that recovery dynamics of vesicles and release sites are statistically almost independent. A sensitivity analysis on the recovery rates based on the ODE formulation reveals that neither the vesicle nor the release site recovery step can be identified as the essential rate-limiting step but that the rate- limiting feature changes over the course of stimulation. Under sustained stimulation the dynamics given by the ODEs exhibit transient dynamics leading from an initial depression of the postsynaptic response to an asymptotic periodic orbit, while the individual trajectories of the stochastic jump model lack the oscillatory behavior an asymptotic periodicity of the ODE-solution.}, language = {de} } @article{StraubeWinkelmannHoefling2023, author = {Straube, Arthur and Winkelmann, Stefanie and H{\"o}fling, Felix}, title = {Accurate reduced models for the pH oscillations in the urea-urease reaction confined to giant lipid vesicles}, volume = {127}, journal = {The Journal of Physical Chemistry B}, number = {13}, arxiv = {http://arxiv.org/abs/arXiv:2212.14503}, doi = {10.1021/acs.jpcb.2c09092}, pages = {2955 -- 2967}, year = {2023}, abstract = {This theoretical study concerns a pH oscillator based on the urea-urease reaction confined to giant lipid vesicles. Under suitable conditions, differential transport of urea and hydrogen ion across the unilamellar vesicle membrane periodically resets the pH clock that switches the system from acid to basic, resulting in self-sustained oscillations. We analyse the structure of the phase flow and of the limit cycle, which controls the dynamics for giant vesicles and dominates the pronouncedly stochastic oscillations in small vesicles of submicrometer size. To this end, we derive reduced models, which are amenable to analytic treatments that are complemented by numerical solutions, and obtain the period and amplitude of the oscillations as well as the parameter domain, where oscillatory behavior persists. We show that the accuracy of these predictions is highly sensitive to the employed reduction scheme. In particular, we suggest an accurate two-variable model and show its equivalence to a three-variable model that admits an interpretation in terms of a chemical reaction network. The faithful modeling of a single pH oscillator appears crucial for rationalizing experiments and understanding communication of vesicles and synchronization of rhythms.}, language = {en} } @article{delRazoWinkelmannKleinetal.2023, author = {del Razo, Mauricio and Winkelmann, Stefanie and Klein, Rupert and H{\"o}fling, Felix}, title = {Chemical diffusion master equation: formulations of reaction-diffusion processes on the molecular level}, volume = {64}, journal = {Journal of Mathematical Physics}, number = {1}, arxiv = {http://arxiv.org/abs/2210.02268}, doi = {10.1063/5.0129620}, year = {2023}, abstract = {The chemical diffusion master equation (CDME) describes the probabilistic dynamics of reaction--diffusion systems at the molecular level [del Razo et al., Lett. Math. Phys. 112:49, 2022]; it can be considered the master equation for reaction--diffusion processes. The CDME consists of an infinite ordered family of Fokker--Planck equations, where each level of the ordered family corresponds to a certain number of particles and each particle represents a molecule. The equations at each level describe the spatial diffusion of the corresponding set of particles, and they are coupled to each other via reaction operators --linear operators representing chemical reactions. These operators change the number of particles in the system, and thus transport probability between different levels in the family. In this work, we present three approaches to formulate the CDME and show the relations between them. We further deduce the non-trivial combinatorial factors contained in the reaction operators, and we elucidate the relation to the original formulation of the CDME, which is based on creation and annihilation operators acting on many-particle probability density functions. Finally we discuss applications to multiscale simulations of biochemical systems among other future prospects.}, language = {en} } @article{LueckeHeitzigKoltaietal.2023, author = {L{\"u}cke, Marvin and Heitzig, Jobst and Koltai, P{\´e}ter and Molkethin, Nora and Winkelmann, Stefanie}, title = {Large population limits of Markov processes on random networks}, volume = {166}, journal = {Stochastic Processes and their Applications}, arxiv = {http://arxiv.org/abs/2210.02934}, doi = {10.1016/j.spa.2023.09.007}, year = {2023}, abstract = {We consider time-continuous Markovian discrete-state dynamics on random networks of interacting agents and study the large population limit. The dynamics are projected onto low-dimensional collective variables given by the shares of each discrete state in the system, or in certain subsystems, and general conditions for the convergence of the collective variable dynamics to a mean-field ordinary differential equation are proved. We discuss the convergence to this mean-field limit for a continuous-time noisy version of the so-called "voter model" on Erdős-R{\´e}nyi random graphs, on the stochastic block model, as well as on random regular graphs. Moreover, a heterogeneous population of agents is studied. For each of these types of interaction networks, we specify the convergence conditions in dependency on the corresponding model parameters.}, language = {en} } @article{SteudleWinkelmannFuerstetal.2024, author = {Steudle, Gesine and Winkelmann, Stefanie and F{\"u}rst, Steffen and Wolf, Sarah}, title = {Understanding Memory Mechanisms in Socio-Technical Systems: the Case of an Agent-based Mobility Model}, volume = {27}, journal = {Advances in Complex Systems}, doi = {10.1142/S0219525924400034}, year = {2024}, abstract = {This paper explores memory mechanisms in complex socio-technical systems, using a mobility demand model as an example case. We simplified a large-scale agent-based mobility model into a Markov process and discover that the mobility decision process is non-Markovian. This is due to its dependence on the system's history, including social structure and local infrastructure, which evolve based on prior mobility decisions. To make the process Markovian, we extend the state space by incorporating two history-dependent components. Although our model is a very much reduced version of the original one, it remains too complex for the application of usual analytic methods. Instead, we employ simulations to examine the functionalities of the two history-dependent components. We think that the structure of the analyzed stochastic process is exemplary for many socio-technical, -economic, -ecological systems. Additionally, it exhibits analogies with the framework of extended evolution, which has previously been used to study cultural evolution.}, language = {en} } @article{EngelOliconMendezWehlitzetal.2024, author = {Engel, Maximilian and Olic{\´o}n-M{\´e}ndez, Guillermo and Wehlitz, Nathalie and Winkelmann, Stefanie}, title = {Synchronization and random attractors in reaction jump processes}, journal = {Journal of Dynamics and Differential Equations}, arxiv = {http://arxiv.org/abs/2207.00602}, doi = {10.1007/s10884-023-10345-4}, year = {2024}, abstract = {This work explores a synchronization-like phenomenon induced by common noise for continuous-time Markov jump processes given by chemical reaction networks. Based on Gillespie's stochastic simulation algorithm, a corresponding random dynamical system is formulated in a two-step procedure, at first for the states of the embedded discrete-time Markov chain and then for the augmented Markov chain including random jump times. We uncover a time-shifted synchronization in the sense that—after some initial waiting time—one trajectory exactly replicates another one with a certain time delay. Whether or not such a synchronization behavior occurs depends on the combination of the initial states. We prove this partial time-shifted synchronization for the special setting of a birth-death process by analyzing the corresponding two-point motion of the embedded Markov chain and determine the structure of the associated random attractor. In this context, we also provide general results on existence and form of random attractors for discrete-time, discrete-space random dynamical systems.}, language = {en} } @article{delRazoLammaMerbis2026, author = {del Razo, Mauricio and Lamma, Tommaso and Merbis, Wout}, title = {Field theories and quantum methods for stochastic reaction-diffusion systems}, volume = {98}, journal = {Rev. Mod. Phys.}, arxiv = {http://arxiv.org/abs/2409.13377}, doi = {10.1103/9qlw-gyd7}, year = {2026}, abstract = {Complex systems are composed of many particles or agents that move and interact with one another. In most real-world applications, these systems involve a varying number of particles/agents that change due to interactions with the environment or their internal dynamics. The underlying mathematical framework to model these systems must incorporate the spatial transport of particles/agents and their interactions, as well as changes to their copy numbers, all of which can be formulated in terms of stochastic reaction-diffusion processes. However, the standard probabilistic representation of these processes can be overly complex because of the combinatorial aspects arising due to the non-linear interactions and varying particle numbers. In this manuscript, we review the main field theory representations of stochastic reaction-diffusion systems, which handle these issues "under-the-hood''. First, we focus on bringing techniques familiar to theoretical physicists —such as second quantization, Fock space, path integrals and quantum field theory— back into the classical domain of reaction-diffusion systems. We demonstrate how various field theory representations, which have evolved historically, can all be unified under a single basis-independent representation. We then extend existing quantum-based methods and notation to work directly on the level of the unifying representation, and we illustrate how they can be used to consistently obtain previous known results in a more straightforward manner, such as numerical discretizations and relations between model parameters at multiple scales. Throughout the work, we contextualize how these representations mirror well-known models of chemical physics depending on their spatial resolution, as well as the corresponding macroscopic (large copy number) limits. The framework presented here may find applications in a diverse set of scientific fields, including physical chemistry, theoretical ecology, epidemiology, game theory and socio-economical models of complex systems, specifically in the modeling and multi-scale simulation of complex systems with varying numbers of particles/agents. The presentation is done in a self-contained educational and unifying manner such that it can be followed by researchers across several fields.}, language = {en} } @article{LueckeWinkelmannKoltai2026, author = {L{\"u}cke, Marvin and Winkelmann, Stefanie and Koltai, Peter}, title = {Accurate mean-field equation for voter model dynamics on scale-free networks}, volume = {113}, journal = {Physical Review E}, arxiv = {http://arxiv.org/abs/2509.13485}, doi = {10.1103/vkpx-5cvt}, pages = {034311}, year = {2026}, abstract = {Understanding the emergent macroscopic behavior of dynamical systems on networks is a crucial but challenging task. One of the simplest and most effective methods to construct a reduced macroscopic model is given by mean-field theory. The resulting approximations perform well on dense and homogeneous networks but poorly on scale-free networks, which, however, are more realistic in many applications. In this paper, we introduce a modified version of the mean-field approximation for voter model dynamics on scale-free networks. The two main deviations from classical theory are that we use degree-weighted shares as coarse variables and that we introduce a correlation factor that can be interpreted as slowing down dynamics induced by interactions. We observe that the correlation factor is only a property of the network and not of the state or of parameters of the process. This approach achieves a significantly smaller approximation error than standard methods without increasing dimensionality.}, language = {en} } @article{WehlitzPavliotisSchuetteetal.2026, author = {Wehlitz, Nathalie and Pavliotis, Grigorios and Sch{\"u}tte, Christof and Winkelmann, Stefanie}, title = {Data-driven Reduction of Transfer Operators for Particle Clustering Dynamics}, arxiv = {http://arxiv.org/abs/2601.02932}, year = {2026}, abstract = {We develop an operator-based framework to coarse-grain interacting particle systems that exhibit clustering dynamics. Starting from the particle-based transfer operator, we first construct a sequence of reduced representations: the operator is projected onto concentrations and then further reduced by representing the concentration dynamics on a geometric low-dimensional manifold and an adapted finite-state discretization. The resulting coarse-grained transfer operator is finally estimated from dynamical simulation data by inferring the transition probabilities between the Markov states. Applied to systems with multichromatic and Morse interaction potentials, the reduced model reproduces key features of the clustering process, including transitions between cluster configurations and the emergence of metastable states. Spectral analysis and transition-path analysis of the estimated operator reveal implied time scales and dominant transition pathways, providing an interpretable and efficient description of particle-clustering dynamics.}, language = {en} }