揭示了从事件数据中恢复潜在网络所需的最短观测时间。
On Observation Time for Recovering Latent Hawkes Networks
- 提出两阶段估计方法，结合截断与分箱筛选及最小二乘优化。
- 证明在稀疏弱耦合条件下，观测时间仅需约 log d 即可精确恢复网络。
- 理论严谨，适合从事点过程建模与网络推断的研究者参考。
工程、社会和自然系统中的动态常由隐藏的网络结构决定，这些结构控制着实体间的相互作用。本文研究从基于事件的观测数据中推断这些网络的问题，此类数据广泛存在于金融、地震学和神经科学等领域。尽管已有大量算法研究，但理论结果稀缺。本文提出一个基本问题：在交互实体数量为 d 时，需要多长的观测时间才能精确恢复底层网络？针对一类具有稀疏、弱相互作用的平稳霍克斯过程，我们证明观测时间量级为 log d 是充分且必要的。上界通过构造两阶段估计器实现：先利用截断和分箱数据进行筛选，再进行最小二乘精修，并应用基于泊松簇表示的集中不等式。下界则结合法诺不等式与雅科布的吉尔萨诺夫公式，针对特定子类网络建立。
原文摘要 · Abstract (English)
Dynamics of interacting systems in engineering, society, and nature often evolve over latent networks that govern which entities can interact. We study the problem of inferring these networks from event-based observations, which arise naturally in finance, seismology, and neuroscience. While there is substantial algorithmic work addressing this important problem, theoretical results are scarce. In this paper we ask the following fundamental question: what is the minimum time that one must observe the dynamics in order to exactly recover the underlying network, as a function of the number $d$ of interacting entities? For a class of stationary Hawkes processes with sparse, weak interactions, we prove that an observation time of order $\log d$ is sufficient and necessary. For the upper bound we construct a two-stage estimator that uses clipped and binned event data for screening, followed by a least-squares refinement, and apply concentration bounds derived from the Poisson cluster representation. For the lower bound we combine Fano's inequality with Jacod's Girsanov formula for point processes on a suitable subclass of networks.
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