Parsimonious Hawkes Processes for temporal networks modelling

TL;DR

The paper presents MINCH, a parsimonious continuous-time Hawkes framework for temporal networks that explicitly separates highly active influencer nodes from communities. Building on MULCH and CHIP, it models time-ordered interactions with block-structured intensities and kernel mixtures, while isolating hubs and inactive nodes to improve log-likelihood and interpretability. Across MID, Reality Mining, and Enron datasets, MINCH yields systematic improvements in predictive performance and reveals meaningful influencer–community dynamics. The approach offers scalable, interpretable insights into temporal networks and points to future directions including finite-time links and temporal hypergraphs, with potential integration into deeper learning pipelines. $\lambda_{ij}(t)$, $\mu_{ab}$, $\alpha_{ab}^{xy \rightarrow ij}$, $\gamma_{xy \rightarrow ij}(t)$, and kernel mixtures $\sum_q C_{ab}^{q} \beta_q e^{-eta_q t}$ are central to the method.

Abstract

Temporal networks are characterised by interdependent link events between nodes, forming ordered sequences of links that may represent specific information flows in the system. Nevertheless, representing temporal networks using discrete snapshots in time partially cancels the effect of time-ordered links on each other, while continuous time models, such as Poisson or Hawkes processes, can describe the full influence between all the potential pairs of links at all times. In this paper, we introduce a continuous Hawkes temporal network model which accounts both for a community structure of the aggregate network and a strong heterogeneity in the activity of individual nodes, thus accounting for the presence of highly heterogeneous clusters with isolated high-activity influencer nodes, communities and low-activity nodes. Our model improves the prediction performance of previously available continuous time network models, and obtains a systematic increase in log-likelihood. Characterising the direct interaction between influencer nodes and communities, we can provide a more detailed description of the system that can better outline the sequence of activations in the components of the systems represented by temporal networks.

Figures (14)

• Figure 1: MID clusters produced by MINCH
• Figure 2: Event interactions between blocks over time
• Figure 3: MID model parameters for 5 communities (1 : cluster 1, 2: cluster 2, 3 : hub 1, 4 : hub 2, 5 : inactive); from left to right, top to bottom: $\mu$ base intensity, $\alpha_{s}$ self-excitation, $\alpha_{r}$ reciprocal, $\alpha_{c}$ turn continuation, $\alpha_{gr}$ generalized reciprocity, $\alpha_{al}$ allied continuation, $\alpha_{ar}$ allied reciprocity, $\beta_{halfday}$ kernel scaling, $\beta_{2weeks}$ kernel scaling, $\beta_{2months}$ kernel scaling model parameters. Row i and column j in the grids above represent block i to block j parameters.
• Figure 4: MID cluster - hub - inactive interactions. Aggregated events over time between different blocks in the network. Hubs refer to standalone nodes with large activity, and inactive nodes refer to nodes with a very low activity throughout the 21 year period. In particular, we see a large number of events directed from hub 1 to cluster 1, and in both directions between cluster 2 and hub 2.
• Figure 5: Total number of events recorded between blocks