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A General Marked Point Process Framework For Self-Exciting Network Evolution

This paper proposes a novel, well-posed framework based on path-dependent nonlinear marked Hawkes processes to model the continuous-time evolution of self-exciting networks, enabling flexible joint inference on update timing and network structure through a demonstrated application to social network data.

Original authors: Duncan A Clark, Conor J. Kresin, Charlotte M. Jones-Todd

Published 2026-03-20

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CC BY 4.0

Original authors: Duncan A Clark, Conor J. Kresin, Charlotte M. Jones-Todd

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are watching a bustling city grow from a single empty lot into a sprawling metropolis. You want to understand why new buildings appear, when they appear, and how they connect to the existing city.

Most traditional models treat this like a static map: they look at the finished city and try to guess the rules. Others look at the timeline of construction but ignore the shape of the buildings.

This paper introduces a new, powerful way to watch the city grow in real-time. The authors call it HawkesNet. Think of it as a "smart camera" that doesn't just record when a building is built, but also what kind of building it is and how it changes the neighborhood.

Here is the breakdown of their idea using simple analogies:

1. The Core Idea: The "Social Butterfly" Effect

In the real world, people (or nodes in a network) often act in bursts. If you post a funny video, your friends might comment immediately. Those comments might trigger more comments from other friends. This is called self-excitation.

The Old Way: Imagine a model that says, "On average, 5 people comment per hour." It treats every minute as independent.
The HawkesNet Way: This model understands that the moment you post, the "temperature" of the room spikes. The next comment is much more likely to happen in the next 5 minutes than 5 hours later. It captures that bursty nature of human interaction.

2. The Big Innovation: "The Marked Point Process"

This is the fancy math term, but here is the simple version:

Imagine a timeline of events.

The Point: The time something happens (e.g., "At 2:00 PM, a new person joined the party").
The Mark: The content of what happened (e.g., "They brought 3 new friends and started a conversation with the host").

Most models struggle to predict both the time and the content together. They usually say, "We know when it happens, but the content is random," or vice versa.

HawkesNet treats the time and the content as a single, inseparable package.

Analogy: Think of a snowball rolling down a hill.
- The time it rolls is the "Point."
- The size and shape of the snowball (how much snow it picked up) is the "Mark."
- The paper argues that you can't predict the size of the snowball without knowing how fast it's rolling, and you can't predict the speed without knowing how much snow is on it. They feed into each other.

3. The "Dynamic Mark Space" (The Shape-Shifting Puzzle)

This is the most clever part of the paper.

In a normal puzzle, the pieces are fixed. In a network, the "pieces" (who can connect to whom) change as the network grows.

Scenario: You are at a conference. You can only introduce two people if they are both there.
The Problem: If you try to model this with a fixed set of rules, you get stuck.
The Solution: HawkesNet uses a Dynamic Mark Space. Imagine the puzzle board itself is alive. As new people arrive, the board expands, and new "valid moves" (edges) appear instantly. The model knows that you can't connect Person A to Person B if Person B hasn't arrived yet. It adapts its rules in real-time based on the current state of the network.

4. Two Ways to Build the Network

The authors show two specific ways this model can work, like two different architects:

The "Popular Kid" Approach (Preferential Attachment):
New people tend to connect to the most popular people already at the party. If someone has 100 friends, they are 100 times more likely to get a new friend than someone with 1 friend. This explains why some networks become "scale-free" (a few super-hubs, many small nodes).
The "Social Logic" Approach (Change Statistics):
This is more like human psychology. People connect based on specific patterns:
- "I like connecting to people who share a mutual friend" (Triadic closure).
- "I like connecting to people who are active right now."
- "I avoid connecting to people who are too far away in the network."
  The model learns these "social rules" by looking at how the network changes when a new edge is added.

5. Why This Matters (The "Conference" Test)

To prove it works, the authors tested HawkesNet on real data from the ACM Hypertext 2009 conference. They tracked who talked to whom, second by second.

The Result: The model successfully predicted that people tend to form small groups (triangles) and that activity spikes when someone new enters a conversation.
The Takeaway: It showed that the timing of an interaction and the type of interaction are deeply linked. You can't understand the network by looking at the timeline alone, or the structure alone. You need both.

Summary

HawkesNet is a new statistical tool that treats a growing network like a living, breathing organism. It understands that:

Events trigger more events (Self-excitation).
The "what" and the "when" are connected (Non-separable).
The rules of connection change as the network grows (Dynamic Mark Space).

It allows researchers to finally ask and answer questions like: "Does the network grow because of random chance, or because of specific social rules that evolve over time?" And it does so with a mathematical framework that is stable, predictable, and ready for real-world data.

1. Problem Statement

Dynamic networks (e.g., social interactions, protein interactions, co-authorship) evolve over continuous time with complex dependencies between when updates occur (timing) and what updates occur (topology).

Limitations of Existing Models:
- Discrete-time models (e.g., SAOM, TERGM, STERGM) treat time as discrete steps and often assume the node set is fixed, failing to model node arrivals or the continuous nature of real-world interactions.
- Standard Point Processes: While Hawkes processes model self-excitation (where past events increase the likelihood of future events), classical formulations often assume separability. This means the timing of events and the nature of the event (the "mark") are conditionally independent given the history.
- The Core Issue: In network growth, timing and topology are non-separable. The current network structure influences the rate of future updates (e.g., a highly connected node attracts more edges), and the timing of updates influences the network structure (e.g., bursts of activity create specific motifs). Existing models either ignore this feedback loop or rely on restrictive Markovian assumptions that discard long-term history.

2. Methodology

The authors propose HawkesNet, a novel framework modeling network evolution as a Mark Path Dependent Marked Point Process (MPP).

A. Theoretical Framework

Marked Point Process (MPP): The network is viewed as a sequence of events $(t_i, m_i)$ where $t_i$ is the time of an update and $m_i$ is the "mark" representing the specific change (new nodes and/or edges).
Dynamic Mark Space: Unlike standard MPPs with a fixed mark space, HawkesNet utilizes a dynamic mark space ( $\mathcal{M}_t$ ). The set of admissible updates depends on the current state of the network $\mathcal{G}_{t-}$ . For example, an edge can only be added between nodes that exist (or are added simultaneously).
Non-Separable Intensity: The conditional intensity $\lambda(t, m | \mathcal{H}_t)$ $λ (t, m ∣ H_{t})$ is decomposed into:
$\lambda(t, m | \mathcal{H}_t) = \lambda_g(t | \mathcal{H}_t) \times q(m | t, \mathcal{H}_t)$
- $\lambda_g$ : The ground intensity (rate of any update occurring).
- $q$ : The mark PMF (probability distribution of which specific update occurs).
- Crucial Innovation: Both components depend on the full marked history $\mathcal{H}_t$ . This allows for topology-timing feedback: the network structure affects the rate of updates, and the timing affects the structural evolution.

B. Mark Path Dependence

The framework introduces Mark Path Dependent Hawkes Processes. In classical linear Hawkes processes, the influence of a past event is fixed at its birth. In HawkesNet, the influence of a past update (e.g., an edge) can change over time as the network evolves (e.g., that edge becomes part of a triangle or a high-degree cluster). The excitation kernel $g$ depends on the current graph state, not just the time elapsed since the event.

C. Specific Model Specifications

The paper proposes two induced models for the mark distribution $q$ :

Preferential Attachment (BA-style): New nodes attach to existing nodes with probability proportional to their time-decayed degree.
Change Statistic (CS-style): Inspired by Exponential Random Graph Models (ERGMs). The probability of an edge is determined by the change in network statistics (e.g., triangles, $k$ -stars, shared partners) caused by adding that edge. This allows for complex structural dependencies like transitivity and homophily.

D. Theoretical Guarantees

Well-Posedness: The authors prove the existence, pathwise uniqueness, and non-explosion (stability) of the process.
Stability Conditions: They establish sufficient conditions for stability using a contraction argument on marked histories. Key requirements include:
- Decay: The influence of past events must decay over time.
- Saturation/Boundedness: The effect of network statistics on intensity must be bounded (e.g., using logistic links) to prevent supercritical cascades where the network explodes infinitely in finite time.

E. Inference and Simulation

Likelihood: They derive a tractable log-likelihood function. By using the decomposition $\lambda = \lambda_g q$ , the compensator term (integral over the mark space) simplifies to an integral over the ground rate only, avoiding the intractable sum over the massive dynamic mark space.
$\ell(\theta) = \sum \log q(m_i) + \sum \log \lambda_g(t_i) - \int \lambda_g(t) dt$
Simulation: They utilize Ogata's thinning algorithm adapted for marked processes with dynamic mark spaces to generate exact simulations.

3. Key Contributions

Unified Framework: Provides the first principled, likelihood-based framework for continuous-time network evolution that jointly models update timing and topology with non-separable dependence.
Mark Path Dependence: Formalizes and proves stability for Hawkes processes where the excitation kernel depends on the evolving graph state (not just the event time), capturing long-term structural dependencies.
Computational Tractability: Demonstrates that despite the complexity of the dynamic mark space, likelihood evaluation is feasible by separating the ground rate and mark probability, enabling Maximum Likelihood Estimation (MLE).
Software Implementation: Introduces the open-source R package hawkesNet to facilitate simulation and inference.

4. Results

Simulation Study:
- The authors simulated 100 realizations for both Preferential Attachment and Change Statistic models.
- Parameter Recovery: MLE successfully recovered true parameters for mark distributions (structural parameters) and ground rates (temporal parameters) in stable regimes.
- Consistency: As the observation window $T$ increased, the variance of estimators decreased, and RMSE decayed, confirming empirical consistency.
- Stability: The study highlighted the difference between stable (linear growth) and supercritical (exponential explosion) regimes, validating the theoretical stability conditions.
Real-World Application (ACM Hypertext 2009):
- Applied to a dynamic face-to-face contact network of conference attendees.
- Model Fit: The Change Statistic HawkesNet model (using GWESP and GWDEG terms) successfully captured the observed network structure, including degree distribution, edgewise shared partners (triadic closure), and geodesic distances.
- Interpretation: Results confirmed that social closure (triangles) and popularity (degree) drive edge formation, and that inactive nodes become less "attractive" over time (time decay).
- Goodness-of-Fit: Simulation-based diagnostics showed the model accurately reproduced complex structural features that simpler models (like standard ERGMs or linear Hawkes) failed to capture.

5. Significance

Statistical Rigor: Moves beyond descriptive network models to provide a generative, likelihood-based statistical inference tool for continuous-time networks.
Handling Complexity: Solves the "curse of dimensionality" in network modeling by avoiding the need to treat every potential edge as a separate dimension (as in multivariate Hawkes processes) while still capturing complex structural dependencies.
Realism: By allowing the network state to influence the rate of updates (not just the type), the model better reflects real-world phenomena like "bursty" activity and the feedback loops inherent in social and biological networks.
Future Directions: The framework opens the door to modeling edge dissolution (birth-death), weighted networks, and integrating exogenous covariates, providing a flexible foundation for future research in dynamic network analysis.