[Paper Review] The Multivariate Community Hawkes Model for Dependent Relational Events in Continuous-time Networks

Hadeel Soliman, Lingfei Zhao+3 more|arXiv (Cornell University)|May 2, 2022

Point processes and geometric inequalities5 citations

TL;DR

The paper proposes the Multivariate Community Hawkes (MULCH) model, a flexible continuous-time network model that uses structured multivariate Hawkes processes to capture dependent relational events across node pairs via community-based excitation matrices. It significantly outperforms existing models in predicting future events and generating realistic temporal motifs, including triangles and reciprocity, by incorporating sociologically inspired excitations like participation shifts.

ABSTRACT

The stochastic block model (SBM) is one of the most widely used generative models for network data. Many continuous-time dynamic network models are built upon the same assumption as the SBM: edges or events between all pairs of nodes are conditionally independent given the block or community memberships, which prevents them from reproducing higher-order motifs such as triangles that are commonly observed in real networks. We propose the multivariate community Hawkes (MULCH) model, an extremely flexible community-based model for continuous-time networks that introduces dependence between node pairs using structured multivariate Hawkes processes. We fit the model using a spectral clustering and likelihood-based local refinement procedure. We find that our proposed MULCH model is far more accurate than existing models both for predictive and generative tasks.

Motivation & Objective

• To address the limitation of existing continuous-time network models that assume conditional independence between node pairs, preventing accurate modeling of higher-order network motifs.
• To develop a tractable yet flexible model that captures complex dependencies such as reciprocity, participation shifts, and alliance dynamics in dynamic networks.
• To improve predictive performance and generative fidelity for timestamped relational event data, especially for motifs involving multiple events within short time windows.
• To enable model-based exploratory analysis of real-world networks, such as military disputes, by revealing latent community behaviors and response patterns.

Proposed method

• The MULCH model combines a stochastic block model (SBM) with a multivariate Hawkes process, using a structured excitation matrix α to induce dependence between node pairs based on community memberships.
• It employs a spectral clustering algorithm to initialize community assignments, followed by likelihood-based local refinement to optimize model parameters.
• The model incorporates multiple types of excitation: self-excitation, reciprocal excitation, allied continuation, turn continuation, generalized reciprocity, and allied reciprocity, inspired by sociological theories of interaction.
• The conditional intensity function for each node pair (i,j) is modeled as λ_ij(t) = μ_ij + Σ_k α_ij^k * γ_k(t - t_s), where excitation α_ij^k depends on the community memberships of nodes i and j.
• The model is fitted using a two-stage procedure: spectral clustering for initial community detection and iterative optimization via likelihood maximization.
• The excitation matrix α is constrained by block structure, allowing efficient computation while preserving rich dependence patterns across node pairs.

Experimental results

Research questions

• RQ1 Can a continuous-time network model effectively capture higher-order temporal motifs such as triangles and reciprocity, which are common in real-world networks?
• RQ2 How can dependence between node pairs be modeled in a tractable way without assuming conditional independence?
• RQ3 To what extent does incorporating sociologically motivated excitation types (e.g., participation shifts) improve model performance on real data?
• RQ4 Can the MULCH model outperform existing models in both predictive accuracy and generative fidelity for timestamped relational event data?
• RQ5 What insights into collective behavior can be uncovered by analyzing excitation patterns in real-world networks such as military disputes?

Key findings

• The MULCH model significantly outperforms existing models—CHIP and BHM—in predicting future events on the Reality and Enron datasets, with higher log-likelihood scores on held-out data.
• MULCH successfully replicates the relative frequencies of 3-node temporal motifs, including reciprocated and triangle-like patterns, matching real-world data more closely than models with only self-excitation or random assignment.
• In the military disputes case study, the model identified NATO members (block 1) and states like Yugoslavia and Libya (block 2) as forming a high-excitation pair for allied continuation, correctly reflecting historical conflicts such as the 1999 NATO bombing of Yugoslavia.
• The model revealed strong self-excitation (α ≈ 0.9) and reciprocity (α ≈ 0.7), with weaker but meaningful excitations for turn continuation (α ≈ 0.03) and allied continuation (α ≈ 0.04), indicating coordinated state responses.
• The excitation pattern for block pair (2,1) showed high turn continuation, reflecting rapid retaliatory actions by Yugoslavia against NATO states, consistent with historical narratives.
• The checkerboard pattern of motif counts generated by MULCH closely matches actual data, indicating that the model captures the complex, non-uniform distribution of temporal motifs observed in real networks.