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21 pages, 1330 KB

Open AccessArticle

Multi-Agent Reinforcement Learning Game Model for Market Economic Equilibrium Regulation

by Fang Lin and Ruyue Cao

Mathematics 2026, 14(15), 2777; https://doi.org/10.3390/math14152777 - 4 Aug 2026

Viewed by 423

Market equilibrium regulation constitutes a dynamic decision-making problem on complex networks, where strategic firms, consumers, platforms, and regulators interact under uncertain demand, delayed price information, and networked spillovers. Although existing multi-agent reinforcement learning (MARL) methods succeed at decentralized adaptation, they typically maximize private [...] Read more.

Market equilibrium regulation constitutes a dynamic decision-making problem on complex networks, where strategic firms, consumers, platforms, and regulators interact under uncertain demand, delayed price information, and networked spillovers. Although existing multi-agent reinforcement learning (MARL) methods succeed at decentralized adaptation, they typically maximize private rewards without encoding an explicit equilibrium residual or a rigorous link to market clearing. We introduce an Equilibrium-Residual Mirror Multi-Agent Reinforcement Learning (ERM-MARL) framework for regulating market equilibria. The framework formulates a regulated Markov potential game: agents learn pricing, production, and risk-control policies while a dual regulation layer penalizes violations of market clearing, price volatility, and network risk. An equilibrium-residual shaping mechanism aligns each agent’s policy gradient with a global regulation potential. The analysis establishes existence of equilibrium, uniqueness under strong monotonicity, bounded dual stability, and almost-sure convergence of the stochastic mirror actor–critic recursion. Simulation experiments on networked markets demonstrate faster equilibrium-residual decay, higher welfare, lower price volatility, and greater robustness compared with representative MARL and game-learning baselines. Full article

(This article belongs to the Special Issue Dynamic Analysis and Decision-Making in Complex Networks, 2nd Edition)

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53 pages, 569 KB

Open AccessArticle

Semi-Closed-Form Pricing of Vulnerable Geometric Asian Options Under a Three-Factor Stochastic Volatility Jump-Diffusion Model with Stochastic Interest Rates

by Libin Wang and Ruonan Zhang

Mathematics 2026, 14(15), 2673; https://doi.org/10.3390/math14152673 - 23 Jul 2026

Viewed by 372

Abstract

This paper develops a semi-closed-form pricing framework for vulnerable geometric Asian options under a three-factor stochastic volatility jump-diffusion model with stochastic interest rates. To the best of our knowledge, this is the new framework to simultaneously accommodate common and idiosyncratic volatility, co-jumps, stochastic [...] Read more.

This paper develops a semi-closed-form pricing framework for vulnerable geometric Asian options under a three-factor stochastic volatility jump-diffusion model with stochastic interest rates. To the best of our knowledge, this is the new framework to simultaneously accommodate common and idiosyncratic volatility, co-jumps, stochastic rates, and counterparty default risk in a 2D-FFT setting. The theoretical contribution is threefold: (i) a fully flexible correlation structure among asset returns, volatility factors, and the short rate; (ii) a default intensity jump component correlated with asset jumps; and (iii) a two-dimensional fast Fourier transform (2D-FFT) algorithm that computes prices and all Greeks in a single execution. Numerical results demonstrate speedup factors of 22.6 to 114.2 over Monte Carlo simulation at comparable accuracy (APE below 0.02%), with near-constant time scaling for portfolios of up to 4096 options. Degeneracy tests confirm the framework correctly recovers known closed-form solutions and published benchmarks. The proposed method provides a computationally efficient pricing tool for real-time risk management of credit-risk-embedded derivatives. Full article

(This article belongs to the Special Issue Dynamic Analysis and Decision-Making in Complex Networks, 2nd Edition)

19 pages, 5861 KB

Open AccessArticle

Optimizing the Resilience of the 3E System: A Coupled Supernetwork–ABM Framework for the Yangtze River Delta

by Jiacheng He, Xiaomu Yin, Aonan Zhao and Guochang Fang

Mathematics 2026, 14(14), 2629; https://doi.org/10.3390/math14142629 - 20 Jul 2026

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Abstract

The transition toward carbon neutrality demands not only an understanding of the complex dynamics within energy–economy–environment (3E) systems but also the ability to strategically enhance their resilience against external shocks. Here, we introduce a bidirectional coupled framework that integrates macro-level supernetwork topology with [...] Read more.

The transition toward carbon neutrality demands not only an understanding of the complex dynamics within energy–economy–environment (3E) systems but also the ability to strategically enhance their resilience against external shocks. Here, we introduce a bidirectional coupled framework that integrates macro-level supernetwork topology with micro-level agent-based modeling (ABM) to diagnose structural vulnerabilities and optimize systemic performance. Applied to 41 cities in the Yangtze River Delta (YRD) from 2010 to 2023, our framework reveals a persistent core–periphery spatial disparity in coupling coordination, underpinned by four distinct network layers—energy flow, economic linkage, environmental impact, and policy synergy—whose densities vary by an order of magnitude. Through coupled evolutionary simulations, we quantify system resilience as a tripartite metric of robustness, adaptability, and recovery, identifying a systemic structural weakness: a robust recovery capacity is offset by substantially lower robustness. To address this, we deploy a genetic algorithm to solve for optimal investment allocation under a budget constraint, demonstrating that a targeted, hub-centric strategy yields a higher marginal resilience gain than uniform distribution. Furthermore, embedding multi-agent reinforcement learning (MARL) and social learning into policy scenario simulations shows that unified environmental standards and a carbon-inclusive mechanism effectively eliminate regulatory arbitrage and accelerate low-carbon behavioral diffusion, improving system-wide robustness by 35% under extreme climate shocks. This work delivers a closed-loop, transferable analytical framework that transforms structural diagnosis into actionable optimization, offering a scientific basis for coordinated regional decarbonization strategies. Full article

(This article belongs to the Special Issue Dynamic Analysis and Decision-Making in Complex Networks, 2nd Edition)

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46 pages, 7426 KB

Open AccessArticle

How Supply-Side Policies Influence High-Quality Technology Diffusion: A Complex Network Simulation Based on Evolutionary Game Theory

by Lin Zhang, Jiakun Wu and Xianwei Liu

Mathematics 2026, 14(14), 2616; https://doi.org/10.3390/math14142616 - 18 Jul 2026

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Abstract

To examine the mechanisms through which supply-side policy instruments influence the early diffusion of high-quality technologies in technology markets, this study constructs a bilateral evolutionary model incorporating finitely rational technology suppliers and demanders based on evolutionary game theory and a bipartite small-world network. [...] Read more.

To examine the mechanisms through which supply-side policy instruments influence the early diffusion of high-quality technologies in technology markets, this study constructs a bilateral evolutionary model incorporating finitely rational technology suppliers and demanders based on evolutionary game theory and a bipartite small-world network. Through multi-agent simulations, the study analyzes the effects of government R&D funding, tax relief and technology transaction subsidies on the diffusion of high-quality technologies and the process of supply-demand strategy adaptation, with the aim of characterizing the role of public fiscal funds in screening for effective technology supply. The study reached the following main conclusions: First, under the baseline scenario, high-quality technologies exhibit a strong tendency toward endogenous diffusion, and supply-side policies primarily serve to accelerate marginal growth in the early stages rather than fundamentally altering the long-term convergence trend. Among these policies, R&D funding, as an ex ante incentive tool, has the most direct impact on early-stage diffusion; however, high-intensity funding leads to a decline in fiscal efficiency; Second, the independent effects of ex post incentive tools like tax relief and technology transaction subsidies are relatively moderate, with the former exhibiting high fiscal efficiency at low to medium intensities, and the latter exerting a moderate incentive effect by increasing the returns on successful transactions, though both suffer from diminishing marginal returns; Third, the supply-demand strategy fit index can be used to help characterize the supply-demand coordination process, but it cannot be directly interpreted as an indicator of technology quality or diffusion quality; Finally, robustness tests indicate that the main conclusions remain stable under perturbations to network topology, initial conditions, and Fermi noise parameters. Overall, the design of supply-side policies should simultaneously consider the timing of policy implementation, market-matching mechanisms, and fiscal cost-effectiveness. Full article

(This article belongs to the Special Issue Dynamic Analysis and Decision-Making in Complex Networks, 2nd Edition)

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48 pages, 745 KB

Open AccessArticle

Bundle-Constrained Multilayer Flow Networks with Fractional Cuts, Compatibility Gaps, and Decision Extensions

by Ruiliang Li and Xiaoya Su

Mathematics 2026, 14(14), 2567; https://doi.org/10.3390/math14142567 - 16 Jul 2026

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Abstract

Many multilayer service systems require a single unit of demand to use a mutually compatible tuple of layerwise routes. In these systems, layerwise reachability and layerwise maximum-flow values do not determine how much service can be jointly delivered. We introduce bundle-constrained multilayer flow [...] Read more.

Many multilayer service systems require a single unit of demand to use a mutually compatible tuple of layerwise routes. In these systems, layerwise reachability and layerwise maximum-flow values do not determine how much service can be jointly delivered. We introduce bundle-constrained multilayer flow networks, in which physical capacities remain on ordinary directed edges while admissible service units are compatible bundles generated by finite-state compatibility systems. The maximum bundle-flow problem is a finite path-packing linear program over accepting paths in product networks. Its dual is a fractional service-bundle cut problem whose edge and demand-cap weights must intersect every admissible bundle. We prove the resulting flow-cut duality, identify dual weights as capacity supergradients, give a product-network shortest-path oracle for separation and pricing, and establish an unbounded compatibility gap for systems with identical layerwise shadows. These results show that compatibility must be modelled explicitly, since layerwise data alone cannot recover joint service capacity. The same fractional-cut geometry is then applied to affine shocks, minimum restoration costs, and projected adjustment dynamics. Deterministic computations and a synthetic scalability experiment illustrate the compatibility gap, active-cut switching, restoration breakpoints, projected decision regimes, and the column-generation method. Full article

(This article belongs to the Special Issue Dynamic Analysis and Decision-Making in Complex Networks, 2nd Edition)

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19 pages, 1044 KB

Open AccessArticle

Algebraic Topology Modeling and Game Decision Optimization for Multilayer Complex Network Dynamics

by Yandong Yuan

Mathematics 2026, 14(11), 1817; https://doi.org/10.3390/math14111817 - 24 May 2026

Cited by 1 | Viewed by 454

Abstract

Modeling and controlling multilayer complex network dynamics is challenging under coexisting crosslayer interactions, higher-order couplings, and decentralized strategic decisions. Most existing schemes focus on graph-based pairwise structures and overlook topological cavities, mesoscale loops, and layered self-interested actions. This paper presents TopoGame-MND, an algebraic-topological [...] Read more.

Modeling and controlling multilayer complex network dynamics is challenging under coexisting crosslayer interactions, higher-order couplings, and decentralized strategic decisions. Most existing schemes focus on graph-based pairwise structures and overlook topological cavities, mesoscale loops, and layered self-interested actions. This paper presents TopoGame-MND, an algebraic-topological and game-theoretic framework for multilayer network dynamics. We first build a filtration-driven simplicial lifting to unify pairwise and higher-order interactions into a weighted multilayer simplicial complex. A topological state operator using generalized Hodge Laplacians and persistent homology is then constructed to characterize cross-scale diffusion, circulation, and structural inconsistency. A distributed potential-game mechanism is developed with a topology-aware utility, followed by a proximal mirror-best-response algorithm with consensus correction. We prove Nash equilibrium existence and uniqueness, global potential monotone descent, linear convergence, computational complexity, and input-to-state robustness. Simulations on multiplex and interdependent networks validate that TopoGame-MND outperforms baselines in regulation speed, oscillation energy, failure resilience, and robustness, providing a unified way to connect higher-order topology and distributed decision optimization. Full article

(This article belongs to the Special Issue Dynamic Analysis and Decision-Making in Complex Networks, 2nd Edition)

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24 pages, 1075 KB

Open AccessArticle

The Spatio-Temporal Differentiation and Convergence Characteristics of the Coordinated Development of Digitalization and Greening in China

by Peipei Zhang and Yusen Luo

Mathematics 2026, 14(9), 1574; https://doi.org/10.3390/math14091574 - 6 May 2026

Viewed by 433

Abstract

The synergistic development of digitalization and greening is an important lever for China to accelerate the formation of new quality productive forces. This study adopted the global entropy method and coupling coordination degree model to measure the level of coordinated development between digitalization [...] Read more.

The synergistic development of digitalization and greening is an important lever for China to accelerate the formation of new quality productive forces. This study adopted the global entropy method and coupling coordination degree model to measure the level of coordinated development between digitalization and greening with the panel data of Chinese cities from 2011 to 2022. Spatio-temporal evolution characteristics were explored through kernel density estimation, the Dagum Gini coefficient, and spatial autocorrelation methods. This study further tested the convergence characteristics of coordinated development through a two-way fixed effect model and spatial econometric model. The results show the following: (1) The overall level of coordinated development of digitalization and greening in China is on the rise, with the development level in the eastern region being significantly higher than that in the central and western regions. The degree of differentiation in coordinated development shows a trend of decreasing first and then increasing, mainly due to regional differences. (2) The level of coordinated development between digitalization and greening in China shows a significant positive spatial autocorrelation feature, with a clustering pattern dominated by “low–low” clustering. (3) It is found that the coordinated development of digitalization and greening in China has significant characteristics of σ convergence, spatial β convergence and club convergence. Full article

(This article belongs to the Special Issue Dynamic Analysis and Decision-Making in Complex Networks, 2nd Edition)

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20 pages, 6100 KB

Open AccessArticle

Complex Dynamics of a Supply–Demand–Price Network Model Incorporating a Marginal Feedback Mechanism

by Dingyue Wang, She Han and Mei Sun

Mathematics 2026, 14(8), 1337; https://doi.org/10.3390/math14081337 - 16 Apr 2026

Viewed by 503

Abstract

In this paper, a supply–demand–price network model incorporating a marginal feedback mechanism is proposed to characterize the evolution of market prices. Unlike classical supply–demand models, the marginal effect of excess demand, defined as the rate of change in excess demand, is explicitly introduced [...] Read more.

In this paper, a supply–demand–price network model incorporating a marginal feedback mechanism is proposed to characterize the evolution of market prices. Unlike classical supply–demand models, the marginal effect of excess demand, defined as the rate of change in excess demand, is explicitly introduced into the price adjustment process. As the coefficient of the marginal feedback term varies, the system exhibits rich and complex nonlinear dynamics. In particular, the model gives rise to a centrally symmetric double-wing chaotic attractor, as well as a pair of coexisting single-wing chaotic attractors. The transition routes among different dynamical regimes are systematically analyzed using phase portraits, bifurcation diagrams, and Lyapunov exponents. Furthermore, multistability phenomena are observed, including the coexistence of equilibrium points, limit cycles, and chaotic attractors. The corresponding basins of attraction are illustrated to reveal their intricate and interwoven structures. In addition, the emergence of endogenous chaos is investigated through both theoretical analysis and numerical simulations. Finally, the consistency between the model dynamics and real market data provides empirical evidence supporting the validity and applicability of the proposed framework. Full article

(This article belongs to the Special Issue Dynamic Analysis and Decision-Making in Complex Networks, 2nd Edition)

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