. We develop a rigorous graphon framework for analyzing the flocking dynamics of Cucker-Smale models with evolving weights. First, we rigorously justify this integro-differential model as the graph limit of finite particle systems, establishing both deterministic and probabilistic convergence. For the probabilistic setting, we introduce a hybrid double randomization scheme and patchwise Ho & uml;lder continuity to offer a more encompassing graph-limit framework that includes small-world networks and stochastic block models. Second, we derive a system of dissipative differential inequalities (SDDI) for the timevarying graphon model, which allows us to prove exponential flocking under a priori conditions involving the graphon's connectivity and initial data. Furthermore, by designing a velocity control law based on conservation principles, we demonstrate that the limiting velocity of this heterogeneous system can converge to any point within the relative interior of the convex hull of the initial velocities.

We investigate a Benamou-Brenier type transportation metric for nonnegative measures on a finite reversible Markov chain, which endows the space of measures with a Riemannian structure. Using this geometric framework, we identify a generalized heat equation with source as the gradient flow of the discrete entropy. Moreover, by means of a local Lojasiewicz inequality, we prove exponential convergence of the flow to a unique equilibrium. Our results clarify the role of the Benamou-Brenier formulation in discrete optimal transport for nonnegative measures and provide a coherent geometric interpretation of generalized diffusion equations with source terms.

This paper presents the upper bounds for eigenvalues of the symmetric substochastic matrices, which are related to the measure of irreducibility of matrices. Then three applications in different fields are presented to indicate further applications. In the first one, the bounds for eigenvalues are used to prove that a class of iteration systems could reach a consensus under certain conditions. In the second one, they are used to prove the flocking behaviors of the Cucker-Smale model under rooted leadership. In the last one, a new approach to estimate the upper bounds for eigenvalues of nonnegative matrices is provided.

In this paper, we study a discrete dynamic system designed for asymptotic tracking of target groups under an uncertain environment. We analyze the tracking characteristics of this system in discrete settings by developing discretized energy functionals. Unlike the energy functional of a continuous system, the energy functional of a discrete system may not remain positive, and the estimation of the energy functional is more complicated. Therefore, we introduce cumulative residual energy, which ensures that the energy functional remains positive and non-increasing. Finally, we utilize the convergence of the discrete time step to overcome the velocity estimate difficulty brought by the random effects and infinite length of target particle trajectories, which enables us to establish target tracking for discrete systems. In particular, we also found that, under the controlled step size, the discretized solutions gradually converge to the tracking system of the continuous system. This continuous transition phenomenon has been verified through the simulation presented in this paper. By investigating the behavior of this discrete coupled system, we aim to facilitate practical applications of multi-particle tracking systems with information discontinuity and an uncertain environment.

This paper focuses on the dynamic behavior of an interval-constraint multiagent system. Each agent has a constraint interval that limits its potential consensus values, achieved by encoding a nonsmooth piecewise function into the agent. In addition, the underlying graphs considered are strongly connected. First, a dichotomy of equilibria is identified: either a unique non-consensus equilibrium point or multiple consensus points, depending on whether the intersection of the constraint intervals is empty or not. Then, the set of equilibria is proven to be a global attractor. Structural stability of such a system is also proven based on real-analysis methods, showing that the equilibria have continuous dependence on changes of the constraint intervals. Three running examples are used to illustrate the proposed results. (c) 2025 Elsevier Ltd. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

In this paper, we investigate the geodesic structure and the associated Kantorovich-type duality for a Benamou-Brenier-type transportation metric defined on the space of nonnegative measures over a finite reversible Markov chain. The metric is introduced through a dynamic formulation that combines transport and source costs along solutions of a nonconservative continuity equation, where mass variation is constrained to occur along a fixed strictly positive reference direction. We show that geodesics associated with this metric exhibit a non-locality property: almost every time, they are supported on the whole state space, independently of the choice of endpoints. Moreover, along optimal curves, the source term displays a characteristic temporal profile, with mass creation occurring at early times and subsequent decay as the curve approaches the target measure. As an application of this property, we compare our metric with the shift-transport distance and prove that the latter is always bounded above by our metric. Finally, we establish a Kantorovich-type duality formula in terms of Hamilton-Jacobi subsolutions, which provides a characterization of the metric and highlights the role of the momentum associated with geodesic curves.

We study clustering dynamics for the infinite Cucker-Smale(ICS)model and its connection to the spectrum of the graph Laplacian.For the ICS model,we overcome the challenge of estimating the velocities of particles and derive a system of dissipative differential inequalities(SDDI)in terms of infinite norms.As in the finite ensemble,we show that mono-cluster flocking emerges exponentially fast,and additionally establish a sufficient framework for algebraic multi-cluster flocking.Moreover,for the CS model with a finite system size,we offer a complete spectral characterization of multi-cluster flocking.Specifically,the emergence of n-cluster behavior corresponds to the limit of the n-th eigenvalue of the time-varying Laplacian approaching zero.In contrast,the lower bound of the(n+1)-th eigenvalue remains strictly positive.Furthermore,we extend this framework to the ICS model by characterizing weak n-cluster flocking via spectral asymptotics,where the n-th eigenvalue tends to zero,while both the(n+1)-th eigenvalue and the infimum of the essential spectrum remain strictly positive.Our results bridge spectral analysis and clustering dynamics,providing indirect evidence for the fast emergence of mono-cluster flocking and the slow relaxation of multi-cluster patterns in both finite and infinite particle systems.

We consider the Cucker-Smale type flocking model with pattern formation and velocity regulation. This is a multi-agent model consisting of three terms, taking effects for network coupling, spatial pattern formation control, and pinning control for velocity, respectively. By means of energy function and gradient theory, we show that the relative positions evolve into a target pattern formation, and a target common velocity is achieved by imposing the pinning control on a few agents only. Furthermore, we formulate an optimal control problem of the flocking model by controlling the coupling strength of network, for which the existence of optimal control is analyzed. To minimize the cost function, we utilize the standard gradient descent approach to identify the optimal control. Finally, we carry out numerical simulations to validate the pinning control and optimal control strategy, and investigate the influence of various environments on flocking dynamics.

We derive the discrete forms of the porous medium equation and fast diffusion equation with drift terms on a finite graph as gradient flows of the m-relative entropy, with the global existence of solutions. We show the & Lstrok;ojasiewicz inequality to prove the convergence of solutions and some important functional inequalities. More specifically, the & Lstrok;ojasiewicz inequality holds near the stationary solution with exponent 1 / 2 , which leads to the exactly exponential convergence rate with a finite trajectory length. It can also be applied to show the Talagrand-type inequality and the log-Sobolev-type inequality directly. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

In this paper, we study the propagation of the second spatial-velocity moments for the kinetic Cucker-Smale model with non-compact spatial support. In contrast to compact support, non-compact support leads to a lower bound of zero for the communication weight, which makes the previous approach break down. To address this challenge, we consider two types of initial distributions: exponential decay distributions and polynomial decay distributions. Moreover, our approach uses the infinite-particle mean-field approximation as an intermediary step to analyze the kinetic Cucker-Smale model, with conservation laws of mass and momentum. When initial distributions belong to the aforementioned types of decaying classes and coupling strength exceeds a certain threshold, we show the weak flocking behavior of the kinetic Cucker-Smale model. Specifically, the second velocity moment of the solution centered around the initial average velocity converges to zero, and the second spatial moment around the position of the center of mass remains uniformly bounded in time. The emergence of weak flocking behavior illustrates that even for non-compact support, a certain degree of aggregation can be maintained for the kinetic Cucker-Smale model, as long as the initial distribution exhibits relative concentration.

. We study flocking dynamics for the Motsch-Tadmor (MT) model over infinite graphs. For flocking dynamics, we consider two network topologies, namely the sender network and hierarchical leadership network, and we provide sufficient frameworks leading to asymptotic mono-cluster flocking for these network topologies. In the case of the sender network, the index set is discrete and unbounded. This produces some technical difficulty in estimating relative velocities. We first overcome the challenge of estimating relative velocities by establishing a system of differential inequalities for infinite norms. Then, we derive a mono-cluster flocking estimate using the nonlinear functional approach proposed by Ha and Liu [A simple proof of the Cucker-Smale flocking dynamics and mean-field limit, Commun. Math. Sci. 1 (2009) 297-325]. In particular, for the hierarchical leadership network, we show that the MT model exhibits slow-flocking dynamics, which gives an algebraic relaxation to the flocking state if the initial velocity fluctuations around the leader's velocity are summable.

We consider the Kuramoto-type models for associative-memory networks and its applications in binary pattern retrieval and classification. In this model, the coupling function consists of a Hebbian term and a second-order Fourier term with nonnegative parameter. The theory of the stability/instability has been established in literature for the equilibria corresponding to binary patterns. In this short communication, we investigate the computation method of this model. In practical situations, quick-response is highly desired. However, as the size of the network increases, the high dimension causes heavy computation cost, and even a curse of dimensionality. We provide the discrete-time formulation given by the first-order Euler method, and show that this method is effective in the computation of the continuous-time model. This simplifies the computation and simulations verify that the computation cost is reduced, comparing to the conventional higher-order Runge-Kutta method.

This article investigates the flocking behavior of the Cucker–Smale (CS) model on infinite graphs, considering both standard and cut-off interactions. We introduce the concept of connected infinite graphs with a central vertex group and then derive sufficient conditions for the CS model to produce flocking behavior. For standard interaction, we find that the CS model will exhibit flocking behavior exponentially when the connected infinite graph is equipped with a central vertex group. However, for cut-off interaction, we need the time-varying graph induced by interparticle distance to have a fixed central vertex group and the coupling strength to be above a certain threshold to produce the flocking behavior. Our theoretical analysis shows that if a connected infinite graph has a central vertex group, the second eigenvalue of the corresponding Laplacian is positive, which is crucial for the proof of flocking behavior. The consistent convergence towards flocking may well reveal the advantages and necessities of having a central vertex group in an infinite-particle complex system with sufficient intelligence.

Matching is an important prerequisite for point clouds registration, which is to establish a reliable correspondence between two point clouds. This paper aims to improve recent theoretical and algorithmic results on discrete optimal transport (DOT), since it lacks robustness for the point clouds matching problems with large-scale affine or even nonlinear transformation. We first consider the importance of the used prior probability for accurate matching and give some theoretical analysis. Then, to solve the point clouds matching problems with complex deformation and noise, we propose an improved DOT model, which introduces an orthogonal matrix and a diagonal matrix into the classical DOT model. To enhance its capability of dealing with cases with outliers, we further bring forward a relaxed and regularized DOT model. Meantime, we propose two algorithms to solve the brought forward two models. Finally, extensive experiments on some real datasets are designed in the presence of reflection, large-scale rotation, stretch, noise, and outliers. Some state-of-the-art methods, including CPD, APM, RANSAC, TPS-ICP, TPS-RPM, RPMNet, and classical DOT methods, are to be discussed and compared. For different levels of degradation, the numerical results demonstrate that the proposed methods perform more favorably and robustly than the other methods.

Based on some new elementary estimates for the space–time derivatives of the heat kernel, we use a bootstrapping approach to establish quantitative estimates on the optimal decay rates for the Lq(Rd) (1≤q≤∞, d∈N) norm of the space–time derivatives of solutions to the (modified) Patlak-Keller–Segel equations with initial data in L1(Rd), which implies the joint space–time analyticity of solutions. When the L1(Rd) norm of the initial datum is small, the upper bound for the decay estimates is global in time, which yields a lower bound on the growth rate of the radius of space–time analyticity in time. As a byproduct, the space analyticity is obtained for any initial data in L1(Rd). The decay estimates and space–time analyticity are also established for solutions bounded in both space and time variables. The results can be extended to a more general class of equations.

We show the optimal decay rate estimates of the space-time derivatives of solutions to the fractional Navier-Stokes equations, which yields the joint space-time analyticity. Consequently, the lower bounds on the growth rate (in time) of radius of space analyticity, time analyticity, and joint spacetime analyticity of solutions are obtained. The proofs only involve real variable methods.

Given a set of standard binary patterns and a defective pattern, the binary pattern retrieval task is to find the closest pattern to the defective one among these standard patterns. The associative-memory network of Kuramoto oscillators consisting of a Hebbian coupling term and a second-order Fourier term can be applied to this task. When the memorized patterns stored in the Hebbian coupling are mutually orthogonal, recent studies show that the network is capable of distinguishing the memorized patterns from most other patterns. However, the orthogonality usually fails in real situations. In this paper, we present a unified approach for the application of this model in pattern retrieval problems with any general set of standard patterns. By subgrouping the standard patterns and employing an orthogonal lift of each subgroup, this approach makes use of the theory in the case of mutually orthogonal memorized patterns. In particular, the error-free retrieval can be guaranteed, which requires that the retrieved pattern must coincide with one of the standard patterns. As illustrative simulations, pattern retrieval tests for partly sheltered Arabic number symbols are presented.

In this paper, we study a one dimensional nonlinear equation with diffusion −ν(−∂xx)α2 for 0 ≤ α ≤ 2 and ν > 0. We use a viscous-splitting algorithm to obtain global nonnegative weak solutions in space L1(R)∩H1/2(R) when 0 ≤ α ≤ 2. For the subcritical case 1 < α ≤ 2 and critical case α = 1, we obtain the global existence and uniqueness of nonnegative spatial analytic solutions. We use a fractional bootstrapping method to improve the regularity of mild solutions in the Bessel potential spaces for the subcritical case 1 < α ≤ 2. Then, we show that the solutions are spatial analytic and can be extended globally. For the critical case α = 1, if the initial data ρ0 satisfies −ν < inf ρ0 < 0, we use the method of characteristics for complex Burgers equation to obtain a unique spatial analytic solution to our target equation in some bounded time interval. If ρ0 ≥ 0, the solution exists globally and converges to steady state.

In this paper, we focus on the critical exponent for the Cucker–Smale model in Rd(d≥1) under group-hierarchical multi-leadership (GHML) topology. The GHML is an asymmetric topology with a group hierarchical structure and multiple leaders. The exponent β in communication weight function measures the decay rate with respect to the distance of particles. In literature, for d≥2, the critical exponent for unconditional flocking is proven to be 1/2 only for symmetric topologies or hierarchical leadership. For general digraphs, the exponent below which the unconditional flocking occurs depends on the digraph and is less than 1/2. In this paper, we prove that the critical exponent is 1/2.

本文研究两类无限图上Cucker-Smale模型的集群行为.第一类图为全连通的无限图,本文得到在初值关于位置无界的情形下,仍然会出现对应的集群行为（速度的一致性）,称之为编队行为.第二类图为局部连接有限的无限图.首先研究Laplace算子谱的下确界大于0的局部有限无限图,得到Cucker-Smale模型发生群集行为的充分条件.其次研究具有Poincaré不等式的局部有限无限图,借助图的几何性质,得到时变图的衰减性,并将其应用在Cucker-Smale模型上,得到Cucker-Smale模型的非线性稳定性（小初值的群集行为）.