arXiv subjects

Fei Cao

Publications and source records attributed to Fei Cao.

At least 19 recordsLinked to original sources

arXiv2026

The incremental voter model: mean-field analysis and convergence to equilibrium

We introduce the incremental voter model (IVM), a discrete-opinion multi-agent system where agents undergo step-wise transitions biased by the opinion of a randomly selected persuader. Our incremental voter model comprises a large population of interacting agents, each holding an opinion represented by an element of the discrete set $\{-k,\ldots,0,\ldots,k\}, k \in \mathbb{N}_{+}$. At each update step as time progresses, a pair of distinct agents are selected independently and uniformly at random from the population, and the first agent (viewed as the ``listener'') updates its opinion based on that of the second (viewed as the ``persuader''), adopting a new opinion that differs from its current one by at most one unit. By deriving the mean-field system of nonlinear ordinary differential equations (ODEs) that governs the large-population limit of the agent-based model, we develop a rigorous mathematical framework to study the asymptotic behavior of the opinion distribution in the mean-field limit. These results contribute to a deeper understanding of social influence processes in complex systems, particularly in modeling opinion polarization, and may guide the formulation of more advanced models in future research.

cs.MA↗

arXiv2026

Relocation without preference: A destination-agnostic Schelling-type metapopulation model

In this work, we propose and analyze a novel Schelling-type metapopulation model that examines how random relocations of families between neighborhoods can lead to segregation. The model consists of a large number of houses organized into $N$ neighborhoods with $L$ houses each, without any spatial structure. Houses can be occupied by either a blue or a red family, and families relocate -- to an empty house selected uniformly at random -- at a rate that depends only on the number of families of the other type within the same neighborhood. We study two mean-field regimes: the large $N$ limit with fixed $L$, and the large $L$ limit with fixed $N$. The associated mean-field systems of ODEs are derived, and their long-time behavior is investigated. As is often the case with Schelling-type models, we find a rich interplay between the model parameters and the social structure of the equilibrium distribution, which exhibits segregation in some parameter ranges. Our work demonstrates that segregation patterns can emerge even when the relocation mechanism is destination-agnostic.

physics.soc-ph↗

arXiv2026

Analysis of the long-term behavior of the "Bando--follow-the-leader'' car-following model

In this article, we investigate the long-term behavior of the ``Bando--follow-the-leader'' car-following model, whose well-posedness and stability with respect to delay were analyzed in a recent work \cite{gong2023well}. We first establish the collision-free property of the model with $N+1\in\N_{\geq2}$ vehicles over an infinite time horizon, assuming that the trajectory of the first vehicle is prescribed, by demonstrating the existence of a uniform strictly positive lower bound on the space headway between adjacent vehicles. Furthermore, assuming that the first vehicle travels at a constant velocity and $N\in\N_{\geq1}$ vehicles follow it according to the Bando--follow-the-leader model on a single lane, our main results state that, with certain reasonable constraints imposed on the modeling parameters, all $N$ following vehicles will eventually (i.e., when time goes to infinity) converge to the same headway and velocity with a globally exponential convergence rate. The analytical methods are based on Lyapunov functions and a perturbation argument. Numerical simulations are also provided to illustrate the obtained theoretical convergence guarantees.

math.CA↗

arXiv2026

Wealth exchange under ceiling and flooring constraints: a modified Bennati-Dragulescu-Yakovenko model

We investigate the classical Bennati-Dragulescu-Yakovenko (BDY) dollar exchange model introduced in \cite{dragulescu_statistical_2000} where the effects of wealth ceiling and wealth flooring are explored. In our model, $N$ identical economical agents involved in the BDY game are also subjected to certain policies issued by a (artificial) government, which prevent agents whose wealth exceeds some prescribed threshold value (denoted by $b \in \mathbb N_+$) from receiving money and which prohibit agents whose wealth falls below certain threshold value (denoted by $a \in \mathbb N$) from giving out their money. We derive a mean-field system of coupled nonlinear ordinary differential equations (ODEs) governing the evolution of the distribution of money as the number of agents $N$ tends to infinity and study the large time behavior of the resulting ODE system. The impact of a wealth cap and a wealth floor on economic inequality (measured by the Gini index) will also be explored numerically.

math.PR↗

arXiv2026

The fractal geometry of opinion formation

In this manuscript, we introduce and study a variant of the agent-based opinion dynamics proposed in a recent work [9], within the framework of an interacting multi-agent system, where agents are assumed to interact with each other and update their opinions after each pairwise encounter. Specifically, our opinion model involves a large crowd of $N$ indistinguishable agents, each characterized by an opinion value ranging within the interval $[-1,1]$. At each update time, two agents are picked uniformly at random and the opinion of one agent will either shift by a proportion $\mu \in (0,1]$ towards $+1$, or by a proportion $\lambda \in (0,1]$ towards $-1$, with probabilities depending on the other agent's opinion. We rigorously derive the mean-field limit PDE that governs the large-population limit of the agent-based model and present several quantitative results demonstrating convergence to the unique equilibrium distribution. Remarkably, for a suitable choice of model parameters, the long-term equilibrium opinion profile displays a striking self-similar structure that generalizes the celebrated Bernoulli convolution, a topic extensively studied in the context of fractal geometry [24,51]. These findings also enhance our understanding of the opinion fragmentation phenomenon and may provide valuable insights for the development of more sophisticated models in future research.

math.PR↗

arXiv2025

The Bennati-Dragulescu-Yakovenko model in the continuous setting: PDE derivation and long-time behavior

In this manuscript, we develop and analyze a continuous version of the well-known Bennati-Dragulescu-Yakovenko (BDY) dollar-exchange discrete model. Starting from the conservative BDY exchange mechanism, we rely on kinetic theory for multi-agent systems in order to propose an analogue continuous dynamics, which does not belong to the class of other popular kinetic models for wealth exchange. We employ the quasi-invariant limit procedure to rigorously derive a nonlinear PDE on the half-line, which is a Fokker-Planck equation featuring the boundary value in the drift term. The PDE is supplemented with a nonlinear Robin-type boundary condition encoding conservation of total agents and wealth. We prove existence and uniqueness of the solution, which converges in relative entropy to the unique stationary state that is the Boltzmann-Gibbs (exponential) distribution. We determine the $L^1$ convergence (up to subsequences) of the solution toward this equilibrium: this requires us to make a step forward with respect to established arguments of entropy decay for Fokker-Planck equations. Thus, our results, which bridge the discrete stochastic dynamics with a continuous deterministic evolution equation, provide a novel and influential wealth exchange model in a PDE framework, which opens up many new applicative scenarios and methodological analytical challenges.

math.AP↗

arXiv2025

Mean-field analysis of a random asset exchange model with probabilistic cheaters

We investigate a variant of the standard Bennati-Dragulescu-Yakovenko (BDY) game \cite{dragulescu_statistical_2000} inspired by the very recent work \cite{blom_hallmarks_2024}, where agents involving in a money exchange dynamics are classified into two distinct types which are termed as probabilistic cheaters and honest players, respectively. A probabilistic cheater has a positive probability of declaring to have no money to give to other agents in the system, resulting in a potential financial benefits from being dishonest about his/her financial status. We provide a mean-field description of the agent-based model (in terms of a coupled infinite dimensional system of nonlinear ODEs), in the large population limit where the number of players is sent to infinity, and proves convergence of the coupled mean-field system to its stationary distribution (provided by a mixture of geometric distributions). In particular, the model gives rise to a novel formulation involving a mixture of probability distributions, thereby motivating the introduction of a unusual (generalized) entropy functional tailored to the associated mean-field system.

math.PR↗

arXiv2025

Generative diffusion models from a PDE perspective

Diffusion models have become the de facto framework for generating new datasets. The core of these models lies in the ability to reverse a diffusion process in time. The goal of this manuscript is to explain, from a PDE perspective, how this method works and how to derive the PDE governing the reverse dynamics as well as to study its solution analytically. By linking forward and reverse dynamics, we show that the reverse process's distribution has its support contained within the original distribution. Consequently, diffusion methods, in their analytical formulation, do not inherently regularize the original distribution, and thus, there is no generalization principle. This raises a question: where does generalization arise, given that in practice it does occur? Moreover, we derive an explicit solution to the reverse process's SDE under the assumption that the starting point of the forward process is fixed. This provides a new derivation that links two popular approaches to generative diffusion models: stable diffusion (discrete dynamics) and the score-based approach (continuous dynamics). Finally, we explore the case where the original distribution consists of a finite set of data points. In this scenario, the reverse dynamics are explicit (i.e., the loss function has a clear minimizer), and solving the dynamics fails to generate new samples: the dynamics converge to the original samples. In a sense, solving the minimization problem exactly is "too good for its own good" (i.e., an overfitting regime).

math.PR↗

arXiv2024

From Gini index as a Lyapunov functional to convergence in Wasserstein distance

In several recent works on infinite-dimensional systems of ODEs \cite{cao_derivation_2021,cao_explicit_2021,cao_iterative_2024,cao_sticky_2024}, which arise from the mean-field limit of agent-based models in economics and social sciences and model the evolution of probability distributions (on the set of non-negative integers), it is often shown that the Gini index serves as a natural Lyapunov functional along the solution to a given system. Furthermore, the Gini index converges to that of the equilibrium distribution. However, it is not immediately clear whether this convergence at the level of the Gini index implies convergence in the sense of probability distributions or even stronger notions of convergence. In this paper, we prove several results in this direction, highlighting the interplay between the Gini index and other popular metrics, such as the Wasserstein distance and the usual $\ell^p$ distance, which are used to quantify the closeness of probability distributions.

q-fin.MF↗

arXiv2024

Cold plasma with zirconia nanoparticles for lung cancer via TGF-\b{eta} signaling pathway

Despite advancements in lung cancer therapy, the prognosis for advanced or metastatic patients remains poor, yet many patients eventually develop resistance to standard treatments leading to disease progression and poor survival. Here, we described a combination of CAP and nanoparticles (ZrO2 NPs (zirconium oxide nanoparticle) and 3Y-TZP NPs (3% mol Yttria Tetragonal Zirconia Polycrystal Nanoparticle)) for lung cancer therapy. We found that ZrO2 NPs caused obvious damage to the inside of the lung cancer cells. CAP and ZrO2 NPs mainly affected the mitochondria function, leading to a decrease in mitochondrial membrane potential and ATP levels, and causing endoplasmic reticulum stress and cell nucleus internal DNA damage, etc. CAP combined with ZrO2 NPs (CAP@ZrO2) induced lung cancer cell apoptosis by activating the TGF-\b{eta} pathway. CAP@ZrO2 offers a new therapy for the clinical treatment of lung cancer.

physics.bio-ph↗

arXiv2024

Fractal opinions among interacting agents

We investigate an opinion model consisting of a large group of interacting agents, whose opinions are represented as numbers in $[-1,1]$. At each update time, two random agents are selected, and the opinion of the first agent is updated based on the opinion of the second (the ``persuader''). We derive the mean-field kinetic equation describing the large population limit of the model, and we provide several quantitative results establishing convergence to the unique equilibrium distribution. Surprisingly, in some range of the model parameters, the support of the equilibrium distribution exhibits a fractal structure. This provides a new mathematical description for the so-called opinion fragmentation phenomenon.

math.PR↗

arXiv2024

The iterative persuasion-polarization opinion dynamics and its mean-field analysis

In this paper, we introduce the Iterative Persuasion-Polarization (IPP) model to study the dynamics of opinion formation and change within a population. The IPP model integrates mechanisms of persuasion and repulsion, where individuals influence each other through interactions that can either align opinions incrementally or lead to greater divergence. The probability of each interaction type is governed by a parameter $\alpha$, representing the population's receptiveness to persuasion. We investigate how these interaction dynamics shape the long-term distribution of opinions, examining conditions that promote consensus or polarization. By deriving a system of nonlinear and autonomous ordinary differential equations (ODEs), we provide a rigorous mathematical framework for analyzing the distributional behavior of opinions in large populations. Our findings contribute to a deeper understanding of social influence dynamics and their implications in complex social systems.

physics.soc-ph↗

arXiv2024

Quantitative convergence guarantees for the mean-field dispersion process

We study the discrete Fokker-Planck equation associated with the mean-field dynamics of a particle system called the dispersion process. For different regimes of the average number of particles per site (denoted by $\mu > 0$), we establish various quantitative long-time convergence guarantees toward the global equilibrium (depending on the sign of $\mu - 1$), which is also confirmed by numerical simulations. The main novelty/contribution of this manuscript lies in the careful and tricky analysis of a nonlinear Volterra-type integral equation satisfied by a key auxiliary function.

math.PR↗

arXiv2024

Sticky dispersion on the complete graph: a kinetic approach

We study a variant of the dispersion process on the complete graph introduced in the recent work [17] under the mean-field framework. We adopt a kinetic perspective (as opposed to the probabilistic approach taken in [17] and many other related works) thanks to the reinterpretation of the model in terms of a novel econophysics model. Various analytical and quantitative results regarding the large time behaviour of the mean-field dynamics are obtained and supporting numerical illustrations are provided.

math.PR↗

arXiv2024

On the equivalence between Fourier-based and Wasserstein distances for probability measures on $\mathbb N$

In this manuscript we investigate the equivalence of Fourier-based metrics on discrete state spaces with the well-known Wasserstein distances. While the use of Fourier-based metrics in continuous state spaces is ubiquitous since its introduction by Giuseppe Toscani and his colleagues [9, 14, 16] in the study of kinetic-type partial differential equations, the introduction of its discrete analog is recent [2] and seems to be far less studied. In this work, various relations between Fourier-based metrics and Wasserstein distances are shown to hold when the state space is the set of non-negative integers $\mathbb N$. Lastly, we also describe potential applications of such equivalence of metrics in models from econophysics which motivate the present work.

math.PR↗

arXiv2023

A biased dollar exchange model involving bank and debt with discontinuous equilibrium

In this work, we investigate a biased dollar exchange model with collective debt limit, in which agents picked at random (with a rate depending on the amount of dollars they have) give at random time a dollar to another agent being picked uniformly at random, as long as they have at least one dollar in their pockets or they can borrow a dollar from a central bank if the bank is not empty. This dynamics enjoys a mean-field type interaction and partially extends the recent work \cite{cao_uncovering_2022} on a related model. We perform a formal mean-field analysis as the number of agents grows to infinity and as a by-product we discover a two-phase (ODE) dynamics behind the underlying stochastic $N$-agents dynamics. Numerical experiments on the two-phase (ODE) dynamics are also conducted where we observe the convergence towards its unique equilibrium in the large time limit.

math.PR↗

arXiv2022

From the binomial reshuffling model to Poisson distribution of money

We present a novel reshuffling exchange model and investigate its long time behavior. In this model, two individuals are picked randomly, and their wealth $X_i$ and $X_j$ are redistributed by flipping a sequence of fair coins leading to a binomial distribution denoted $B \circ (X_i+X_j)$. This dynamics can be considered as a natural variant of the so-called uniform reshuffling model in econophysics [2,14]. As the number of individuals goes to infinity, we derive its mean-field limit, which links the stochastic dynamics to a deterministic infinite system of ordinary differential equations. The main result of this work is then to prove (using a coupling argument) that the distribution of wealth converges to the Poisson distribution in the $2$-Wasserstein metric. Numerical simulations illustrate the main result and suggest that the polynomial convergence decay might be further improved.

math.PR↗

arXiv2022

Uniform propagation of chaos for a dollar exchange econophysics model

We study the poor-biased model for money exchange introduced in [2]: agents are being randomly picked at a rate proportional to their current wealth, and then the selected agent gives a dollar to another agent picked uniformly at random. Simulations of a stochastic system of finitely many agents as well as a rigorous analysis carried out in [2,16] suggest that, when both the number of agents and time become large enough, the distribution of money among the agents converges to a Poisson distribution. In this manuscript, we establish a uniform-in-time propagation of chaos result as the number of agents goes to infinity, which justifies the validity of the mean-field deterministic infinite system of ordinary differential equations as an approximation of the underlying stochastic agent-based dynamics.

math.PR↗