[Paper Review] Discovering Latent Network Structure in Point Process Data

Scott W. Linderman, Ryan P. Adams|arXiv (Cornell University)|Feb 4, 2014

Point processes and geometric inequalities21 references77 citations

TL;DR

This paper proposes a fully-Bayesian probabilistic model that combines multivariate Hawkes processes with exchangeable random graph priors to discover latent network structures from point process data. By leveraging the Poisson superposition principle for efficient data augmentation, the method enables parallel MCMC inference and successfully uncovers interpretable interaction patterns in financial trading and gang violence data, outperforming baseline models in predictive performance.

ABSTRACT

Networks play a central role in modern data analysis, enabling us to reason about systems by studying the relationships between their parts. Most often in network analysis, the edges are given. However, in many systems it is difficult or impossible to measure the network directly. Examples of latent networks include economic interactions linking financial instruments and patterns of reciprocity in gang violence. In these cases, we are limited to noisy observations of events associated with each node. To enable analysis of these implicit networks, we develop a probabilistic model that combines mutually-exciting point processes with random graph models. We show how the Poisson superposition principle enables an elegant auxiliary variable formulation and a fully-Bayesian, parallel inference algorithm. We evaluate this new model empirically on several datasets.

Motivation & Objective

• To address the challenge of inferring latent network structures when direct observation of edges or vertices is impossible, relying instead on noisy event emissions.
• To model mutual excitation between nodes in point process data as evidence of underlying network interactions.
• To develop a scalable, fully-Bayesian inference framework that accounts for uncertainty in latent network structure.
• To evaluate the model on real-world datasets involving financial trading and urban violence, demonstrating interpretability and predictive power.
• To generalize existing Hawkes process models by incorporating structured random graph priors and nonparametric background rates.

Proposed method

• The model combines multivariate Hawkes processes with exchangeable random graph priors to represent latent network structure.
• It uses the Poisson superposition principle to introduce auxiliary variables that decompose event origins, enabling efficient data augmentation.
• A fully-Bayesian inference scheme is implemented via parallel Markov chain Monte Carlo (MCMC), allowing uncertainty quantification.
• Background rates are modeled using Log-Gaussian Cox Processes (LGCPs) with Gaussian process priors for nonparametric flexibility.
• The framework supports various network priors, including Erdős-Rényi, stochastic block models, and distance-dependent graphs.
• The auxiliary variable formulation enables efficient parallelization of MCMC updates across events and processes.

Experimental results

Research questions

• RQ1 Can we infer latent network structures from indirect, noisy event data where edges and vertices are unobserved?
• RQ2 How can we model mutual excitation between nodes in point process data as evidence of underlying network interactions?
• RQ3 What network priors and background rate models best capture the structure and dynamics of real-world event sequences?
• RQ4 How does the model's predictive performance compare to baselines on financial and crime data?
• RQ5 What conditions ensure the stability of the inferred latent network structure in Hawkes processes?

Key findings

• The model achieved the highest predictive log likelihood on Chicago homicide data using a four-cluster Erdős-Rényi graph prior with interactions, outperforming distance-dependent priors.
• Inferred clusters in the Chicago data correspond to interpretable social structures: 'safe suburbs', 'buffer neighborhoods', and 'gang territories'.
• The blue cluster exhibited strong self-excitation, consistent with bursts of violence during turf wars, and aligned with historical reports of increased gang activity in 1990–1991.
• The model captured a pronounced seasonal pattern in homicide rates, peaking in summer and declining in winter, which was effectively modeled using a LGCP with quadratic and periodic kernels.
• The four-cluster Erdős-Rényi model showed superior predictive performance, suggesting that long-range or local interactions may not be the dominant drivers in this dataset.
• The framework successfully uncovered interpretable, stable network structures in both synthetic and real-world datasets, demonstrating robustness and scalability.