Ergodic Network Stochastic Differential Equations
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遍历网络随机微分方程
- 研究问题
- 如何从高频离散观测估计网络节点间相互作用的随机微分方程，并分别处理网络结构已知和未知的高维情形？
- 方法与思路
- 作者以节点自身动态、邻居反馈和布朗运动驱动的随机项建立网络模型；已知图下研究识别，未知图下采用自适应 Lasso。
- 材料陈述的贡献
- 摘要称建立已知网络下的参数识别条件，并为特定模型子类设计未知网络的迭代估计程序，辅以多种图结构模拟和应用。
- 阅读关注点
- 应核对遍历性和参数识别条件、高频采样与网络规模的关系，以及未知网络算法适用的子类和有向边的因果解释依据。
- 阅读边界
- 这是 arXiv 预印本，同行评审或期刊接收未核实；摘要未说明识别条件及应用数据，有向网络本身不足以确立因果关系。
原文摘要
We propose a novel framework for Network Stochastic Differential Equations (N-SDE), where each node in a network is governed by an SDE influenced by interactions with its neighbors. The evolution of each node is driven by the interplay of three key components: the node's intrinsic dynamics (\emph{momentum effect}), feedback from neighboring nodes (\emph{network effect}), and a \emph{stochastic volatility} term modeled by Brownian motion. Our primary objective is to estimate the parameters of the N-SDE system from high-frequency discrete-time observations. The motivation behind this model lies in its ability to analyze very high-dimensional time series by leveraging the inherent sparsity of the underlying network graph. We consider two distinct scenarios: \textit{i) known network structure}: the graph is fully specified, and we establish conditions under which the parameters can be identified, considering the linear growth of the parameter space with the number of edges. \textit{ii) unknown network structure}: the graph must be inferred from the data. For this, we develop an iterative procedure using adaptive Lasso, tailored to a specific subclass of N-SDE models. In this work, we assume the network graph is oriented, paving the way for novel applications of SDEs in causal inference, enabling the study of cause-effect relationships in dynamic systems. Through extensive simulation studies, we demonstrate the performance of our estimators across various graph topologies in high-dimensional settings. We also showcase the framework's applicability to real-world datasets, highlighting its potential for advancing the analysis of complex networked systems.