On Observation Time for Recovering Latent Hawkes Networks

We study the problem of recovering the parameters of a multivariate Ornstein-Uhlenbeck (OU) process from steady-state observational and interventional data. In many applications, such as large-scale gene perturbation experiments, only stationary "snapshot" measurements are available, making standard stochastic differential equation estimation methods that rely on time-series trajectories inapplicable. We first establish an identifiability result: one intervention per strongly connected component (SCC) of the drift graph suffices to recover all OU process parameters generically up to a global scaling factor. This holds provided that the SCC condensation graph is connected with a single root and certain spectral nondegeneracy assumptions hold. We propose a recursive learning algorithm that orders SCCs topologically and, for each component, isolates its marginal dynamics and solves a linear system derived from the steady-state moment equations, leveraging parameters recovered for upstream components. Building on this theoretical foundation, we propose a regularized least-squares estimator that jointly minimizes residuals of the steady-state mean and covariance equations across observational and interventional data. Experimental results validate our theoretical findings in recovering parameters of the underlying OU process.

Many important outcomes unfold as dynamic cascades, including product adoption, disease spread, financial distress, and information diffusion. A central challenge is to recover the hidden influence network behind these cascades. Existing methods typically assume a specific diffusion model, and their performance degrades substantially when that assumption is misspecified. We propose CascadeNet, a Jacobian-based machine learning framework for network recovery that does not require specifying a diffusion mechanism. The key idea is that the underlying influence structure can be characterized by the Jacobian of the one-step transition function. CascadeNet first constructs a flexible estimator of the transition function, and further applies Neyman-orthogonal debiasing via the Riesz representer, so that the debiased Jacobian is $\sqrt{n}$-consistent and asymptotically normal, enabling formal inference on the network structure. We validate CascadeNet in both a simulation exercise and a real-world empirical application. In simulations, where the data-generating process is known, CascadeNet achieves the highest network recovery accuracy across nine common data-generating processes. In an empirical application to COVID-19 transmission across Spain's 52 provinces, CascadeNet recovers transmission networks that are significantly correlated with the true inter-province mobility network, whereas networks recovered by baseline methods show no significant alignment with the ground truth.

In this paper, we propose deep learning based NeuroMem-FHP framework for estimating the parameters of the fractional Hawkes process (FHP), a self-exciting point process that captures long-range dependence through a fractional Mittag-Leffler excitation kernel. Two neural architectures, namely a Long Short-Term Memory (LSTM) network and a Transformer, are developed to estimate the model parameters $(μ,γ,α,β)$ directly from sequences of inter-arrival times without requiring computationally intensive likelihood optimization. Experiments on synthetic data that both neural models significantly outperform the classical Maximum Likelihood Estimation (MLE) method, with the Transformer achieving the highest estimation accuracy (MSE = $0.1634$), followed by the LSTM (MSE = $0.1752$), compared to MLE (MSE = $2.8032$). An ablation study further examines the effects of key hyperparameters on model performance. The proposed framework is also on two real-world high-frequency datasets, namely AAPL NBBO transaction data and Montgomery County 911 emergency call records. Using a predictive validation approach, event sequences simulated from the estimated parameters closely reproduce the empirical distribution, tail behavior, and temporal dependence structure of the observed data. These results demonstrate that Transformer-based parameter estimation provides an accurate and efficient alternative to conventional estimation techniques for FHP and offers a promising framework for modeling event-driven systems with long-memory dynamics.