Deep graph kernel point processes

TL;DR

This paper presents a novel point process model for discrete event data over graphs, where the event interaction occurs within a latent graph structure, and significantly extends the existing deep spatio-temporal kernel for point process data.

Abstract

Point process models are widely used for continuous asynchronous event data, where each data point includes time and additional information called "marks", which can be locations, nodes, or event types. This paper presents a novel point process model for discrete event data over graphs, where the event interaction occurs within a latent graph structure. Our model builds upon Hawkes's classic influence kernel-based formulation in the original self-exciting point processes work to capture the influence of historical events on future events' occurrence. The key idea is to represent the influence kernel by Graph Neural Networks (GNN) to capture the underlying graph structure while harvesting the strong representation power of GNNs. Compared with prior works focusing on directly modeling the conditional intensity function using neural networks, our kernel presentation herds the repeated event influence patterns more effectively by combining statistical and deep models, achieving better model estimation/learning efficiency and superior predictive performance. Our work significantly extends the existing deep spatio-temporal kernel for point process data, which is inapplicable to our setting due to the fundamental difference in the nature of the observation space being Euclidean rather than a graph. We present comprehensive experiments on synthetic and real-world data to show the superior performance of the proposed approach against the state-of-the-art in predicting future events and uncovering the relational structure among data.

Figures (14)

• Figure 1: Graph kernel, inter-event dependence, and conditional intensity recovery for the 16-node synthetic data set with 2-hop graph influence. The first column reflects the ground truth, while the subsequent columns reflect the results obtained by GraDK (with three architectures), SAHP-G, and DNSK, respectively.
• Figure 2: Dynamic influence among nodes learned by GraDK on synthetic data generated by the 50-node graph with non-separable diffusion kernel. Both true and learned kernels are evaluated at three-time lags of $0.3, 0.8$, and $1.5$.
• Figure 3: Learned graph kernels for the theft data set evaluated at $t'=0$ and $t-t'=1$; our proposed method can capture complex inter-node dependence compared with prior work DNSK using a spatio-temporal kernel.
• Figure 4: Dynamic graph influence learned by GraDK on the Sepsis data. The learned kernel is evaluated at four time lags $2, 4, 6$, and $8$. The node radii are proportional to the background intensity on each node, and the edge widths are proportional to the influence magnitude. Blue and red edges represent the excitation and inhibition effects, respectively. The history time $t'$ is set to be $0$ for all four panels.
• Figure B1: An example of modeling events on an 8-node graph using graph filter bases in L3Net: (a) The latent graph structure. Blue and red nodes represent the $1$st and $2$nd order neighbors of $v_0$, denoted by $N_{v_0}^{(1)}$ and $N_{v_0}^{(2)}$, respectively. (b) Three graph filter bases $B^{(0)}$, $B^{(1)}$, and $B^{(2)}$ capture the dependencies between events. Hollow circles represent events observed on each node. Colored lines indicate the potential influence of the earliest type-$v_0$ event on future events captured by different bases.