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Dynamic Latent Space Modeling of Inhomogeneous Poisson Network Processes with Applications to International Relations

Jie Jian, Jiguo Cao, Owen G. Ward

arXiv:2609.08813v2stat.APstat.MEstat.ML

TL;DR

The paper addresses how dyadic relationships evolve in continuous event streams, which static counts can obscure when similar totals conceal distinct histories. A continuous-time dynamic latent space model represents relational event streams with smooth latent trajectories and node-specific activity parameters. Applied to cooperative ICEWS events among 60 major economies over 1995–2022, the proposed method yields excellent predictive performance and latent trajectories of nations.

  • Problem

    The paper addresses how dyadic relationships evolve in continuous event streams, which static counts can obscure when similar totals conceal distinct histories.

    Source: §1 · pp. 1–2

  • Method

    A continuous-time dynamic latent space model represents relational event streams with smooth latent trajectories and node-specific activity parameters.

    Source: §1 · p. 3

  • Results

    Applied to cooperative ICEWS events among 60 major economies over 1995–2022, the proposed method yields excellent predictive performance and latent trajectories of nations.

    Source: §1 · p. 4

  • Takeaways & Limitations

    Separating node activity from spatial geometry prevents baseline interaction rates from masquerading as central positioning, so trajectories reflect shifts in dyadic relationships.

    Source: §7 · p. 25

    The paper identifies existing continuous latent-position formulations as limited by piecewise-linear paths and omission of node-specific baseline activity parameters.

    Source: §1 · p. 3

Abstract

from arXiv · show

We study continuous-time relational event data, where time-stamped dyadic interactions reflect both individual node propensities and evolving relational proximity. We propose a dynamic latent space model for inhomogeneous Poisson processes, where event intensities depend on node-specific activity parameters and time-varying latent distances modeled via flexible B-splines. We prove model identifiability by decoupling baseline activity from latent position, ensuring high interaction volumes do not warp the spatial map. For scalability, we develop a minibatch stochastic gradient algorithm with stable initialization and geometric anchoring, alongside an effective-degrees-of-freedom BIC for tuning model complexity. Simulations confirm accurate parameter recovery and out-of-sample prediction. Applied to cooperative diplomatic events among 60 major economies (1995--2022), the model uncovers shifting patterns of international cooperation and isolates mobile geopolitical actors from stationary institutional anchors.

1 Introduction

Relational events arrive as continuous streams, but static totals can hide dyadic histories with sharply different temporal patterns. The paper introduces a continuous-time latent-space Poisson model to track evolving proximity while separating it from node activity.

1 Introduction

  • Static totals can conceal contrasting dyadic histories, while snapshot frameworks lack an instantaneous event mechanism for fine-grained dynamics and smooth forecasting.Source: §1 · pp. 2–3, Figure 1 · p. 2
  • Existing continuous latent-position formulations use piecewise-linear trajectories, omit node-specific baseline activity, and face computational bottlenecks on large networks.Source: §1 · p. 3
  • The model represents dyadic interactions as independent inhomogeneous Poisson processes, with intensity determined by node activity and squared distance between smooth latent trajectories.Source: §1 · p. 3
  • Additive node parameters separate baseline activity from spatial distance, and the joint specification is fully identifiable without ad hoc constraints.Source: §1 · p. 3
  • Formal results identify continuous-time geometry and node-level activity, resolving rotational and translational invariance and ensuring unique activity estimates through strict concavity.Source: §1 · p. 4
  • Dyadic minibatch stochastic gradient descent supports large-scale inference, while a BIC uses generalized Gauss–Newton optimization and Hutchinson trace estimation of effective degrees of freedom.Source: §1 · p. 4
  • The ICEWS application to 60 major economies (1995–2022) yields excellent predictive performance and interpretable latent trajectories.Source: §1 · pp. 3–4

2 A dynamic latent space model for network event data

The section defines a continuous-time dyadic Poisson model whose event intensity combines node activity with time-varying latent distances. It represents trajectories with B-splines, regularizes their movement, and anchors the coordinate frame for estimation.

2.1 Data structure and notation

  • Observed interactions are recorded as event times for unordered dyads over a shared observation window with a fixed node set.Treating dyads as unordered is a modeling choice, not an assumed property of the data.Source: §2.1 · pp. 4–5

2.2 Intensity model

  • The model combines node-specific activity parameters with squared latent distance to govern dyadic event intensity over time.Its two reduced specifications omit either the latent geometry or the activity parameters; later comparisons demonstrate that both are needed.Source: §2.2 · p. 5

2.3 Functional representation of the trajectories

  • B-spline basis functions turn the trajectories’ infinite-dimensional representation into a finite-dimensional estimation problem.The resulting squared latent distances are quadratic forms in coefficient differences.Source: §2.2 · p. 6, §2.3 · pp. 6–7

2.4 Roughness penalties and the penalized objective

  • Penalties on integrated squared velocity and curvature regulate trajectory travel and smoothness, limiting overfitting to localized event noise.The paper distinguishes path length from abruptness of movement as substantive features of trajectories.Source: §2.4 · p. 7
  • A weak ridge term makes the combined penalty proper, while the velocity and curvature penalties correspond to continuous-time analogues of first- and second-order random-walk priors.The model-selection criterion replaces raw parameter counts with an effective-dimension penalty.Source: §2.4 · pp. 7–8

2.5 Anchored parameterization

  • Anchoring two nodes resolves translation, rotation, and reflection ambiguity by fixing one at the origin and constraining the other to the x-axis.The sign of the second node’s x-coordinate remains free; the anchored parameterization also reduces the optimization to free spline coefficients.Source: §2.5 · p. 8

3 Identifiability

The paper establishes identifiability on the anchored coefficient space, separating activity parameters from latent geometry. Its results reduce trajectory ambiguity to a time-invariant finite symmetry and establish local non-degeneracy under stated conditions.

3 Identifiability

  • Under trajectory conditions A1–A3, restricting coefficients to the anchored set C0 yields identifiability of (c,β).A1 requires the two anchors not to collide, A2 prevents residual ambiguity from varying over time, and A3 excludes a specified collinearity pattern.Source: §3 · p. 9

3.1 Identifiability of the trajectories

  • Under A1 and A2, equal dynamic distance matrices recover trajectories up to one time-invariant element of V4, a group of coordinate sign flips.The anchoring constraints reduce the usual time-varying isometric ambiguity to this discrete symmetry; reported distances and intensities are invariant under V4.Source: §3 · p. 9, §3.1 · pp. 9–10

3.2 Identifiability of the activity parameters

  • For fixed trajectories, the activity vector β is identified without explicit normalization, and its intensity map is injective.The conditional activity maximizer exists and is unique when every node participates in an event without monopolizing the process.Source: §3.2 · p. 10

3.3 Joint identifiability

  • Joint identifiability decouples baseline activity from spatial geometry: under A1–A3, equivalent configurations coincide up to V4.The proof uses the rank bound that at most four activity parameters can differ, then rules out nonzero shifts using A3.Source: §3.3 · p. 11
  • For n ≥7 under A1–A3, the joint log-intensity map has an injective differential and the expected Fisher information is strictly positive definite.This establishes local non-degeneracy on the free coordinates and supports the well-posedness of the BIC approximation.Source: §3.3 · pp. 11–12

4 Estimation

The estimation procedure separates trajectory and activity updates, evaluates cumulative intensities numerically, and uses an aligned warm start. It selects smoothing parameters with a curvature-adjusted BIC and randomized trace estimation.

4 Estimation

  • The procedure alternates stochastic trajectory updates with a profile update for activity parameters, using quadrature and a curvature-adjusted BIC for tuning.A piecewise-linear initializer is aligned across time before projection onto the B-spline basis.Source: §4 · p. 12

4.1 Stochastic gradient ascent for the trajectories

  • Minibatching reduces trajectory-gradient cost from O(n^2QK) to O(BQK) by sampling dyads uniformly without replacement.The estimator uses the sampled dyads’ likelihood contributions while evaluating the penalty gradient in full.Source: §4.1 · pp. 12–13

4.2 Numerical evaluation of the cumulative intensity

  • Gauss–Legendre quadrature evaluates cumulative dyadic intensities because the exponential of the piecewise-polynomial integrand has no closed-form integral.The quadrature grid and basis matrix are fixed across iterations.Source: §4.2 · p. 13

4.3 Profile update for the activity parameters

  • Given trajectories, activity parameters are updated by a parallel fixed-point iteration that converges to the unique maximizer when every node has participated in an event.The score equations have no closed-form solution, and the update uses log-sum-exp to avoid overflow.Source: §4.3 · p. 13

4.4 Initialization

  • The initializer fits separate static models on two time blocks, aligns their orientations to minimize displacement, then projects the joined paths onto the B-spline basis.This warm start captures broad movement trends while preventing erratic early path oscillations.Source: §4.4 · p. 14

4.5 Selection of tuning parameters

  • Smoothing parameters are selected by minimizing BIC with effective degrees of freedom estimated using a positive-semidefinite Gauss–Newton information matrix.A Hutchinson randomized trace estimator, accelerated by preconditioned conjugate gradients, scales the calculation.Source: §4.5 · pp. 14–15

5 Simulation studies

The simulations test recovery and prediction in sparse fluid-drift and dense bipolar-consolidation settings. DLS-PP achieves good predictive performance across most metrics in both settings, while the tuning and trajectory results illustrate the role of smoothing.

5 Simulation studies

  • S1 evaluates model performance in unconstrained nonlinear fluid-multipolar interactions with 50 nodes in R2 over t in [0,1], averaging 4.8 interactions per dyad versus 27.1 in S2.S1 trajectories are smooth and continuous, generated via a Fourier basis expansion; S2 partitions nodes into two dynamic clusters.Source: §5 · p. 15, §5.1 · p. 18

5.1 S1: Fluid multipolar drift

  • In S1, BIC has an interior minimum, and the tuning surfaces vary more along ρ2 than ρ1 over the tested ranges.Both log-likelihood and effective degrees of freedom decrease as either penalty increases.Source: §5.1 · p. 18, Figure 5 · p. 18

5 Simulation studies

  • Across S1 and S2, DLS-PP achieves good predictive performance across most metrics.The studies compare in-sample estimation and held-out prediction against four baselines.Source: §5 · p. 16, §5.1 · p. 18, §5.2 · p. 20

5.1 S1: Fluid multipolar drift

  • In sparse S1, β and latent distances are recovered well, although a few β_i estimates are outliers.The reported β estimates span 30 replications; the distance illustration uses one trial.Source: §5.1 · p. 18, Figure 6 · p. 19

5.2 S2: Bipolar alliance consolidation

  • In S2, both components are recovered accurately with tight uncertainty across nodes; larger ρ2 produces straighter, smoother trajectories while preserving their true shapes.The trajectory comparison varies ρ2 with ρ1 fixed at 0.1; the ground-truth trajectories are straight lines.Source: Figure 7 · p. 19, §5.2 · p. 20, Figure 8 · p. 21

6 Application

The application analyzes cooperative diplomatic events among 60 major economies and compares DLS-PP with four competing methods. DLS-PP performs best across the reported metrics, while estimated activity levels and trajectories distinguish countries’ baseline interaction propensities from changes in their relational positions.

6.1 Data

  • Uneven news coverage and underreporting mean event volume reflects media attention as well as political activity, so estimates describe the reported interaction network.Events in less-covered regions are underreported.Source: §6.1 · p. 21
  • The analysis uses ICEWS records for cooperative CAMEO categories 01–08, pooling event dates by unordered country pair for 60 major economies.Countries are selected by median GDP-PPP rank over 1995–2022.Source: §6.1 · pp. 21–22

6.2 Baseline comparison

  • DLS-PP achieves the best performance across all reported in-sample and out-of-sample metrics.The out-of-sample evaluation holds out the final month across country pairs.Source: Table 3 · p. 23, §6.2 · p. 23

6.3 Estimation in DLS-PP

  • Estimated activity parameters generally decline with GDP rank, while Canada, the Netherlands, and Sweden have lower activity estimates than their economic rank suggests.The parameters measure each country’s baseline propensity for diplomatic interaction.Source: §6.3 · p. 23, Figure 9 · p. 24
  • China, Japan, and South Korea cluster and migrate in parallel toward the core, while Russia and Germany trace distinct paths in the estimated latent space.Figure 10 shows trajectories at four dates with the United States fixed at the origin.Source: §6.3 · pp. 23–24, Figure 10 · p. 24
  • Trajectory length measures movement from 1995 to 2022, while arc measures directional change; Ukraine, Iraq, and Poland rank high on both, whereas France barely moves.Bangladesh moves a comparable distance more smoothly; Canada, the United Kingdom, and the Netherlands also remain stable.Source: §6.3 · p. 24, Table 4 · p. 25

7 Conclusion

The conclusion presents DLS-PP as a continuous-time event-network model that separates node activity from dynamic spatial geometry. This separation distinguishes shifts in dyadic alignment from baseline interaction rates, while the computational architecture is described as adaptable to other network settings.

7 Conclusion

  • The model combines continuous-time latent trajectories with a scalable estimation pipeline and an effective-degrees-of-freedom selection criterion.The authors describe it as a framework for continuous-time event networks.Source: §7 · p. 25
  • Separating node-specific activity from dynamic spatial geometry prevents baseline interaction rates from masquerading as spatial centrality.The authors state that this eliminates degree-driven core-periphery artifacts and helps trajectories reflect shifts in dyadic alignment.Source: §7 · p. 25
  • The computational architecture is described as extending to discrete snapshots, weighted edge counts, and self-exciting Hawkes processes.The authors attribute this modularity to spline trajectories, dyadic minibatching, warm starts, and geometric penalties interacting through the log-intensity.Source: §7 · p. 25
  • Future extensions include time-varying activity parameters, gravity covariates in the intensity, and Laplace-based trajectory uncertainty quantification.Source: §7 · p. 25

A Gauss–Legendre quadrature

  • The cumulative intensities use a 40-point Gauss–Legendre rule, while Gram matrices use a 200-point grid computed before optimization.The quadrature rule is mapped to the observation window.Source: A Gauss–Legendre quadrature · p. 29

B Proofs for Section 3

The proof shows that equal latent distances constrain configurations to orthogonal transformations, then uses the model’s assumptions to restrict the remaining ambiguity to fixed sign choices. Those sign changes leave the distances and objective invariant.

B.1 Proof of Proposition 1

  • Equal pairwise distances imply that the two latent configurations differ by an orthogonal map.Source: §B.1 · p. 29
  • Under the full-column-rank condition, the orthogonal map varies continuously over time but lies in a four-element set, so it must remain constant.Source: §B.1 · p. 29
  • The remaining sign changes generate distinct trajectories under the stated assumptions while preserving all pairwise distances and the objective.Source: §B.1 · p. 29

B.2 Proof of Proposition 2

The proof establishes strict concavity of the profile log-likelihood in the activity parameters and characterizes when its maximizer is finite. The finite-maximizer condition is expressed as a strict bound on each event count relative to the total.

B.2 Proof of Proposition 2

  • For n ≥ 3, the activity-parameter quadratic form is strictly concave because its only zero direction is the zero vector.Source: §B.2 · p. 30
  • Equal intensities imply equal activity parameters, and strict concavity makes a finite maximizer unique whenever the profile log-likelihood is coercive.Source: §B.2 · p. 30
  • A unique finite maximizer exists exactly when every m_i satisfies 0 < m_i < m.The proof establishes coercivity under this condition.Source: §B.2 · p. 30

B.3 Proof of Proposition 3

The proof bounds the rank of the centered distance representation and shows how eliminating node-activity offsets supports identification of latent positions.

B.3 Proof of Proposition 3

  • Multiplying the matrix equation by Q eliminates the activity vector a and yields Equation (19) of Proposition 3(i).Source: §B.3 · p. 31
  • rank G(t) is at most 2, as follows from its representation through the distance matrix.The proof also bounds rank(G*(t) − G(t)) by 4 and |S| by 4.Source: §B.3 · p. 31
  • The proof establishes a = 0 and, under the anchoring system, identifies the positions up to V4.It uses the non-collinearity of at least three unshifted nodes to show the remaining activity offsets vanish.Source: §B.3 · pp. 31–32

B.4 Proof of Proposition 4

The appendix establishes local identifiability and describes computational choices for node-activity updates, curvature, effective degrees of freedom, and tuning.

B.4 Proof of Proposition 4

  • Under the stated conditions, a zero directional derivative forces the tangent vector to vanish, establishing an injective differential and strictly positive definite expected Fisher information.Source: §B.4 · pp. 32–33

C Node activity updates in reduced models

D.1 Gauss–Newton curvature substitution

  • Replacing the potentially indefinite likelihood Hessian with positive-semidefinite Gauss–Newton curvature removes the negative-curvature distance terms and keeps the penalized operator invertible.The resulting effective dimension is defined in (0, dfree].Source: §D.1 · pp. 33–34

D.2 Matrix-free Hutchinson trace estimation

  • The matrix-free trace estimator avoids constructing dense curvature matrices, using Rademacher probes and conjugate-gradient solutions to linear systems.Fixed probe seeds and warm starts across tuning-grid cells remove Monte Carlo noise and produce a continuous surface.Source: §D.2 · p. 34

D.3 Optimization surface characteristics

  • The criterion surface is more sensitive to ρ2 than ρ1, motivating a broad two-parameter grid followed by a one-dimensional ρ2 search at fixed moderate ρ1.Source: §D.3 · p. 34

E Details of simulations and application

The supplementary details specify evaluation and extrapolation procedures and explain how the 60-country application panel was selected.

E Details of simulations and application

  • The two out-of-sample rules are persistence and constant velocity, with velocity defined using each model’s fitted trajectory or intensity behavior.For the trajectory-free Poisson β-model the rules coincide, so only one held-out block is reported.Source: E Details of simulations and application · p. 35
  • For constant-velocity extrapolation, one cell is reported as unstable and its distance IMSE is replaced by conditional-intensity IMSE.The endpoint derivative is large and becomes exponentiated in that fit.Source: E Details of simulations and application · p. 36

F 60 selected countries in application

  • The 60-country panel includes eligible sovereign economies ranked by annual PPP GDP, selected by the lowest median annual rank over 1995–2022.Eligibility required at least 23 years of non-missing GDP and population data; regional aggregates were excluded.Source: F 60 selected countries in application · p. 36
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