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

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

Abstract

We propose a novel modeling framework for time-evolving networks allowing for long-term dependence in network features that update in continuous time. Dynamic network growth is functionally parameterized via the conditional intensity of a marked point process. This characterization enables flexible, joint modeling of both update timing and the network updates themselves, dependent on the entire left-continuous sample path. We propose a path dependent nonlinear marked Hawkes process as an expressive platform for modeling such data; its dynamic mark space embeds the time-evolving network. We prove well-posedness and establish sufficient stability conditions, demonstrate simulation and subsequent feasible likelihood-based inference through numerical study, and illustrate the methodology with an application to conference attendee social network data. The proposed formulation provides a flexible and principled foundation for statistical inference on complex network evolution in continuous time.

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

Abstract

We propose a novel modeling framework for time-evolving networks allowing for long-term dependence in network features that update in continuous time. Dynamic network growth is functionally parameterized via the conditional intensity of a marked point process. This characterization enables flexible, joint modeling of both update timing and the network updates themselves, dependent on the entire left-continuous sample path. We propose a path dependent nonlinear marked Hawkes process as an expressive platform for modeling such data; its dynamic mark space embeds the time-evolving network. We prove well-posedness and establish sufficient stability conditions, demonstrate simulation and subsequent feasible likelihood-based inference through numerical study, and illustrate the methodology with an application to conference attendee social network data. The proposed formulation provides a flexible and principled foundation for statistical inference on complex network evolution in continuous time.

Paper Structure

This paper contains 46 sections, 9 theorems, 120 equations, 13 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Dynamic Network Point Process Setup
  3. Marked point process notation
  4. Network evolution as a marked point process
  5. Network probability mass function induced models
  6. Mark path dependent Hawkes processes
  7. Poisson embedding and simulation via thinning
  8. Simulation via thinning
  9. Design principles for stability in induced HawkesNet models
  10. One-event influence and a sharp contraction envelope
  11. Likelihood and Parameter Estimation
  12. Simulation Study
  13. Application to Dynamic Human Contact Patterns
  14. Model specification
  15. Results
  16. ...and 31 more sections

Key Result

Theorem 4.5

Assume ass:H0--ass:HL. Then there exists a pathwise unique simple marked point process $N$ on $[0,\infty)\times\mathcal{M}$ satisfying the fixed-point thinning equation eq:thinning. Moreover, for every $T>0$, $\mathbb E[N([0,T]\times\mathcal{M})]<\infty$, hence $N([0,T]\times\mathcal{M})<\infty$ alm

Figures (13)

  • Figure 1: A mark path dependent (nonlinear) Hawkes representation of network-update data. Red denotes the current update mark and black the accumulated graph. Because the embedding of an old update can change over time (e.g. through triangle participation), its contribution to future intensity need not be fixed at birth.
  • Figure 2: Linear versus mark path dependent Hawkes structure. Left: linear Hawkes admits a cluster genealogy. Right: each event may depend on the full current history, so the arrows indicate history dependence rather than branching ancestry.
  • Figure 3: GOF diagnostics comparing observed (orange points) vs. simulated (blue boxplots): (a) degree distribution and (b) edgewise shared partner distribution.
  • Figure 4: BA HawkesNet: Distribution of parameter estimates by time window $T$. Red dashed lines indicate the true parameter values. The tightening of the boxplots around the truth demonstrates the decreasing variance of the MLE as data increases.
  • Figure 5: BA HawkesNet: RMSE versus time window $T$ by parameter. The monotonic decay in RMSE for all parameters confirms the consistency of the estimation procedure.
  • ...and 8 more figures

Theorems & Definitions (28)

  • Definition 2.1: Marked Hawkes process
  • Remark 2.2: Topology--timing feedback vs. the ground--mark decomposition
  • Definition 2.3: Admissible update marks
  • Example 1: Preferential Attachment HawkesNet
  • Example 2: Change Statistic HawkesNet
  • Definition 4.1: Mark path dependent Hawkes process
  • Remark 4.4: Evaluating $\lambda$ on counterfactual histories
  • Theorem 4.5: Existence, pathwise uniqueness, non-explosion
  • Corollary 4.7: Bounded mean activity and at-most-linear expected network growth
  • Lemma 4.8: Route to Assumption \ref{['ass:HL']} under $\lambda=\lambda_g q$
  • ...and 18 more