Personalized Graph-Based Dynamical Models

• Personalized graph-based dynamical models are frameworks that combine evolving graph structures with individual latent parameters to capture subject-specific dynamics.
• They utilize advanced techniques like controlled differential equations, probabilistic graphical models, and network PDEs to model complex temporal changes.
• Robust training algorithms and empirical benchmarks demonstrate their effectiveness in applications such as social network analysis, neurodegeneration forecasting, and personality inference.

Personalized graph-based dynamical models generalize classical graph representations by coupling individual-specific latent parameters and evolving relational structure with continuous, stochastic, or discrete-time dynamics. These models rigorously encode subject-specific patterns, enabling high-fidelity tracking of personalized state trajectories in complex domains such as social networks, neurodegeneration, and personality inference.

1. Formalism and Key Mathematical Structures

At the core, personalized graph-based dynamical models specify both the graph structure and the nodal properties as functions of time and individual-specific latent variables. Given a set of nodes $V$ and edges $E$, the structure $A$ (adjacency matrix) and node states $Z$ or $X$ evolve jointly:

• Temporal graph observation: $\{(t_k, G_{t_k})\}_{k=0}^{N}$, with $G_{t_k} = (V, E_{t_k})$.
• Node-level dynamics: For node $n$ at time $t$, states may include embedding vectors $Z_t^n \in \mathbb{R}^d$ or interpretable quantities such as opinion $x_n^{<t>}\in\mathbb{R}^p$, neurodegeneration $N_v^{(i)}(t)$, or linguistic profile $h_u$.

Dynamics are typically modeled via either:

• Controlled differential equations on graphs (GN-CDE (Qin et al., 2023)):

$$Z_t = Z_{t_0} + \int_{t_0}^{t} f_\theta(Z_s, A_s, \alpha) d\hat{A}_s$$

where $\alpha$ encodes per-user latent features; $f_\theta$ is a neural (GNN) vector field.

• Probabilistic graphical generative models (PENO (Chen et al., 2018)):

$$p_{ij}^{<t>} = \exp\left(-\frac{||x_i^{<t>} - x_j^{<t>}||_2^2}{\xi^2 b_i b_j}\right)$$

$$\theta_i^{<t+1>} \sim \mathcal{N}\left(\theta_i^{<t>} + r_i \cdot \text{imp}_i^{<t>}, \sigma_i^2 \right)$$

where personality traits $(l_n, r_n, \sigma_n, b_n)$ parameterize evolution at the individual level.

• Network-discretized PDE systems (cortical atrophy model (Li et al., 12 Nov 2025)):

$$\frac{dx}{dt} + D_x L^{(i)} x^{(i)} = \text{reaction/coupling terms}$$

$L^{(i)}$ is the graph Laplacian for personalized network $A^{(i)}$.

Significantly, these frameworks enable incorporating:

• Per-individual parameters ($\alpha$, personality vectors, embedding profiles)
• Dynamic evolution of both graph topology and node states conditioned on personalized latent features.

2. Model Architectures and Parameterization

Graph Neural Controlled Differential Equations (GN-CDE)

GN-CDE (Qin et al., 2023) parameterizes the vector field $f_\theta$ via multi-layer GCNs conditioned on instantaneous adjacency $A_s$ and node embeddings $Z_s$, optionally extended to per-user personalization:

• $f_\theta(Z_s, A_s, \alpha)$ where $\alpha$ modulates the vector field with user history or profile.
• Initialization $Z_{t_0} = \zeta_\theta(t_0, A_{t_0}, \alpha)$ removes invariance to translation and allows personalized starting points.

Continuous-time integration is performed via standard ODE solvers (e.g., Dormand–Prince), with adjoint backpropagation optimizing all parameters end-to-end without temporal segmentation.

Personalized Evolving Network and Opinion (PENO)

PENO (Chen et al., 2018) integrates four interpretable personality dimensions at the node level:

• Leadership $l_n$ (strength of social influence)
• Agreeableness $r_n$ (propensity to shift opinion toward neighbors)
• Neuroticism $\sigma_n$ (volatility in opinion updates)
• Openness $b_n$ (affinity for tie formation)

Edges are sampled via Gaussian kernels of opinion distance scaled by openness. Opinion trends propagate through Gaussian transitions shifted by impacts from neighbors, weighted by leadership and agreeableness. Node positions evolve at a fixed speed along the current trend's direction.

Digital Twin Modeling of Alzheimer's Trajectories

Individualized, low-rank perturbations to the population-level functional connectivity matrix generate subject-specific Laplacians $L^{(i)}$ (Li et al., 12 Nov 2025). Dynamical systems at each node track regional biomarker levels governed by diffusion, growth, and coupling rates, with personalization in 14 distinct parameters per individual.

Dynamic Graph Transformers and Temporal Encoding

Dynamic Graph Transformer with Correlated Spatial-Temporal Positional Encoding (CorDGT) (Wang et al., 24 Jul 2024) introduces a parameter-free personalized interaction intensity computed using Poisson MLE and recentness, mapped through sinusoidal encodings in a multi-head Transformer framework. The model efficiently retains high-order proximity using correlated spatial-temporal positional encodings, enabling scalable and effective node representation learning in large-scale dynamic graphs.

Self-Supervised Graph Optimization for Personality Detection

LL4G (Shen et al., 2 Apr 2025) leverages LLM-derived semantic vectors to construct adaptive graphs reflecting both explicit and implicit textual relationships. A per-user "user node" aggregates post-level embeddings via attention, enabling personalized graph optimization and dynamic adaptation as new user data arrives.

3. Personalization Mechanisms and Latent Variables

Injection of personalization fundamentally distinguishes these models from classical homogeneous graph dynamics.

• Latent parameter integration: In GN-CDE, personalization is achieved by extending $f_\theta$ to accept latent parameters $\alpha$, which are either directly learnable per individual or derived from an encoder on user history (hypernetwork style).
• Explicit personality traits: PENO defines leadership, agreeableness, neuroticism, openness as global node-level constants, dynamically affecting both edge probabilities and opinion propagation.
• Patient-specific parameter inference: In digital twin frameworks for disease progression (Li et al., 12 Nov 2025), subject-specific parameters $(u^{(i)}, v^{(i)}, \Theta^{(i)})$ are optimized against longitudinal multimodal data, affording individual trajectory prediction and stratification.

A plausible implication is that such explicit personalization allows for model adaptation to individual variability, supporting finer-grained prediction and targeted interventions.

4. Training Algorithms and Optimization Strategies

Optimization routines vary by model formulation:

• Adjoint method for CDEs: GN-CDE uses the continuous adjoint for backpropagation, only requiring storage of terminal states and conferring scalability to long time horizons.
• Block-coordinate ascent/EM: PENO utilizes partial EM and gradient ascent across blocks (personality, opinions, trends) on observed temporal graph data.
• Multistage homotopy optimization: The digital twin model (Li et al., 12 Nov 2025) pursues a multi-phase training regime (homogenized, nonhomogenized, initial condition, network perturbation) to ensure stable and regularized parameter inference.
• Adam-driven joint optimization: LL4G self-supervised objectives jointly optimize GNN, attention, and linking components, demonstrating empirical improvements on Macro-F1 benchmarks.

5. Model Robustness, Calibration, and Scalability

Robustness to missing data and continuous integration are prominent features:

• GN-CDE achieves missing observation robustness by treating each adjacency channel as an independent coordinate and relying on continuous interpolation (e.g., spline imputation).
• Digital twin models impose constraints on adjacency matrix entries, ensuring stability in disease trajectory predictions.
• Self-supervised dynamic graph frameworks (LL4G, CorDGT) support fast online updating without full retraining, scaling gracefully to millions of nodes/edges.

6. Empirical Performance and Application Domains

Empirical evaluations highlight the competitive advantage and versatility of personalized graph-based dynamical models:

• Congressional voting prediction: PENO achieves AUC 0.93, outperforming neural/co-evolving baselines on edge prediction (Chen et al., 2018).
• Friendship prediction (Digg): PENO delivers AUC 0.89, superior to latent feature and phase-transition benchmarks.
• Personality detection: LL4G demonstrates improvements of 8.47 percentage points (Macro-F1) on Kaggle, and 4.8 pp on Pandora over published baselines (Shen et al., 2 Apr 2025).
• Neurodegeneration forecasting: Digital twin modeling yields >90% accuracy on 5-10 year forecasting of amyloid, tau, atrophy, and cognition, clustering patient-specific subtypes and identifying vulnerable regions (Li et al., 12 Nov 2025).
• Dynamic graph representation learning: CorDGT surpasses eight prior CTDG methods by up to 10 AP/AUC points and maintains high accuracy under large-scale inductive settings (Wang et al., 24 Jul 2024).

These results establish quantitative evidence for the efficacy of personalized graph-based dynamical models across a diverse range of complex systems.

7. Extensions and Future Directions

Advances are anticipated in:

• Multi-task and transfer learning: Reusing personalized latent variables (PENO) across relational tasks, with fine-tuning for new domains (Chen et al., 2018).
• Self-supervised and unsupervised optimization: Integrating increasingly powerful LLMs and Transformer architectures for expanding interpretability and adaptivity (Shen et al., 2 Apr 2025, Wang et al., 24 Jul 2024).
• Mechanistic disease modeling and digital twins: Enabling in silico intervention and stratification in precision medicine via network-PDE frameworks (Li et al., 12 Nov 2025).

It is plausible that further developments will center on deeper integration of domain-specific knowledge, richer personalization frameworks, and scalable optimization algorithms, reinforcing the foundational role of personalized graph-based dynamical models in computational science.