This paper shows the equivalence class definition of graphons hinders a direct development of dynamics on the graphon space, and hence proposes a state-driven approach to obtain dynamic graphons. The state-driven dynamic graphon model constructs a time-index sequence of the permutation-invariant probability measures on the universal graph space by assigning i.i.d. state random processes to $\mathbbm{N}$ and edge random variables to each of the unordered integer pairs. The model is justified from three perspectives: graph limit definition preservation, genericity, and analysis availability. It preserves the graph limit definition of graphon by applying a bijection between the permutation-invariant probability measures on the universal graph space and the graphon space to obtain the dynamic graphon, where the existence of the bijection is proved. Also, a generalized version of the model is proved to cover the graphon space by an application of the celebrated Aldous-Hoover representation, where generalization is achieved by adding randomness to the edge-generating functions. Finally, analysis of the behavior of the dynamic graphon is shown to be available by making assumptions on the state random processes and the edge random variables.
We consider a particular respondent-driven sampling procedure governed by a graphon. By a specific clumping procedure of the sampled vertices we construct a sequence of sparse graphs. If the sequence of the vertex-sets is stationary then the sequence of sparse graphs converge to the governing graphon in the cut-metric. The tools used are concentration inequality for Markov chains and the Stein-Chen method.
This paper establishes unique solvability of a class of Graphon Mean Field Game equations. The special case of a constant graphon yields the result for the Mean Field Game equations.
We consider a dynamic Erdős-Rényi random graph (ERRG) on $n$ vertices in which each edge switches on at rate $λ$ and switches off at rate $μ$, independently of other edges. The focus is on the analysis of the evolution of the associated empirical graphon in the limit as $n\to\infty$. Our main result is a large deviation principle (LDP) for the sample path of the empirical graphon observed until a fixed time horizon. The rate is $\binom{n}{2}$, the rate function is a specific action integral on the space of graphon trajectories. We apply the LDP to identify (i) the most likely path that startin…
Graph-limit theory focuses on the convergence of sequences of graphs when the number of nodes becomes arbitrarily large. This framework defines a continuous version of graphs allowing for the study of dynamical systems on very large graphs, where classical methods would become computationally intractable. Through an approximation procedure, the standard system of coupled ordinary differential equations is replaced by a nonlocal evolution equation on the unit interval. In this work, we adopt this methodology to explore the continuum limit of random walks, a popular model for diffusion on graphs…
Graphon Particle Systems, Part II: Dynamics of Distributed Stochastic Continuum Optimization
School of Mathematical Sciences, East China Normal University（华东师范大学数学科学学院）；Key Laboratory of Management, Decision and Information Systems, Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院管理决策与信息系统重点实验室）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）；School of Automation, China University of Geosciences（中国地质大学（武汉）自动化学院）
Stochastic heterogeneous SIR model with infection-age dependent infectivity on large random graphs
Rice University（莱斯大学）；Aix–Marseille Université, CNRS, Centrale Marseille, I2M, UMR 7373（艾克斯-马赛大学，法国国家科学研究中心，中央马赛学院，I2M，UMR 7373）；Université Paris-Saclay, Université d’Evry Val d’Essonne, CNRS, LaMME, UMR 8071（巴黎萨克雷大学，埃夫里-瓦勒-德-埃松纳大学，法国国家科学研究中心，LaMME，UMR 8071）
This paper considers stochastic population dynamics driven by Levy noise. The contributions of this paper lie in that (a) Using Khasminskii-Mao theorem, we show that the stochastic differential equation associated with the model has a unique global positive solution; (b) Applying an exponential martingale inequality with jumps, we discuss the asymptotic pathwise estimation of such model.
We prove a law of large numbers for the empirical density of one-dimensional, boundary driven, symmetric exclusion processes with different types of non-reversible dynamics at the boundary. The proofs rely on duality techniques.
