Influence network reconstruction from discrete time-series of count data modelled by multidimensional Hawkes processes

TL;DR

This work addresses the challenge of reconstructing influence networks from discrete-time count data by modeling counts with a multidimensional Hawkes/Cox framework. It introduces three complementary inference methods: an ensemble-based EM for small networks with state-space representations and smoothing-based uncertainty quantification, a Majorization-Minimization (MM) approach for batch data with decoupled, parallelizable updates, and an extended Poisson-Kalman Filter (ExPKF) for sequential inference using a second-order posterior approximation. The methods are validated on synthetic data and real-world email datasets, demonstrating accurate recovery of excitation links, robustness to misspecification, and scalable inference for large networks. The results highlight the feasibility of count-data network reconstruction and provide practical tools for applications in sociology, criminology, and communications, while outlining avenues for improving online smoothing and broader applicability.

Abstract

Identifying key influencers from time series data without a known prior network structure is a challenging problem in various applications, from crime analysis to social media. While much work has focused on event-based time series (timestamp) data, fewer methods address count data, where event counts are recorded in fixed intervals. We develop network inference methods for both batched and sequential count data. Here the strong network connection represents the key influences among the nodes. We introduce an ensemble-based algorithm, rooted in the expectation-maximization (EM) framework, and demonstrate its utility to identify node dynamics and connections through a discrete-time Cox or Hawkes process. For the linear multidimensional Hawkes model, we employ a minimization-majorization (MM) approach, allowing for parallelized inference of networks. For sequential inference, we use a second-order approximation of the Bayesian inference problem. Under certain assumptions, a rank-1 update for the covariance matrix reduces computational costs. We validate our methods on synthetic data and real-world datasets, including email communications within European academic communities. Our approach effectively reconstructs underlying networks, accounting for both excitation and diffusion influences. This work advances network reconstruction from count data in real-world scenarios.

Figures (18)

• Figure 1: (Top) Comparison between filtered particles $\lambda_{0:k}^{f(\ell)}$, smoothed particles $\lambda_{0:k}^{s(\ell)}$ and ground truth. The simulated data is also shown in the bar plot. (Bottom, left) The progress of the relative error of the parameter vector $\theta$ over 16 EM iterations. (Bottom, right) The propagation of the relative error for the conditional intensity. Note that we have fixed the number of EM iterations to 16; no stopping criteria have been implemented in this example.
• Figure 2: (Top) Comparison between smoothed particles $\lambda_{0:k}^{s(\ell)}$ and ground truth for the Logistic LGCP. The simulated data is also shown in the bar plot. (Bottom, left) The progress of the relative error of the parameter vector $\theta$ over 35 EM iterations where the change in the relative error of $\theta$ is lower than $10^{-5}$. (Bottom, right) The progress of the relative error for the conditional intensity.
• Figure 3: (Left) Relative error of the parameter vector at each EM-step.(Right) Relative error of the conditional intensity at each EM-step. We stop at 5 iterations when the change in the relative error of $\theta$ is lower than $10^{-5}$.
• Figure 4: From the top to bottom, the plot shows the smoothed path at the final step of the EM for node $1$ to $3$, respectively. For a clear visualisation, only part of the trajectory is shown at the time step $k=100-500$. The bar plot beneath the intensity shows the simulated count data for each node.
• Figure 5: Estimated values of $\alpha^{ij}$ for various data length and the ground truth. The $i-$th row and $j-$ column in the plot indicates $\alpha^{ij}$.