SD-Score: Dynamic ERGMs in Temporal Networks

• SD-Score is a modeling paradigm that applies score-driven updating mechanisms to allow ERGM parameters to evolve over time in temporal networks.
• It integrates static network statistics with time series methods by using the score of the log-likelihood for dynamic parameter adjustments.
• Empirical applications in interbank credit and U.S. Congress networks demonstrate improved link prediction and formal instability testing compared to static models.

The Static–Dynamic Score (SD-Score) framework refers to a modeling paradigm in which score-driven updating mechanisms are applied to temporal networks, allowing the parameters of Exponential Random Graph Models (ERGMs) to evolve over time. This approach merges time series methodologies with network analysis by utilizing the gradient of the log-likelihood—known as the score—to dynamically adjust latent states associated with network features, thereby bridging static network statistics and their temporal evolution.

1. Foundational Concepts and Motivation

SD-Score arises from the need to model networks whose structure and underlying generative processes are subject to temporal change. Traditional ERGMs describe the probability of observing a static network configuration given fixed parameters and sufficient statistics. However, the assumption of stationarity is inappropriate for many real-world systems, such as financial and political networks, which exhibit pronounced nonstationarity. SD-Score builds on Dynamic Conditional Score (DCS) or Generalized Autoregressive Score (GAS) models by allowing ERGM parameters to become latent dynamic processes updated recursively according to the score—i.e., the gradient of the model log-likelihood evaluated at each time point (Gangi et al., 2019).

2. Technical Mechanism and Updating Rule

At the heart of the SD-Score methodology is the score-driven recursion for parameter updates. Let $\theta$ denote the vector of network model parameters associated with statistics $h$, and let $Y$ denote the observed network at time $t$. The ERGM log-likelihood takes the form:

$$\ln P(Y | \theta) = \sum_{s} \theta_s h_s(Y) - \ln \mathcal{K}(\theta)$$

where $\mathcal{K}(\theta)$ is the normalizing constant.

The score component for parameter $\theta_s$ is:

$$\nabla_s(\theta) = h_s(Y) - \frac{\partial \ln \mathcal{K}(\theta)}{\partial \theta_s}$$

Parameter evolution is governed by:

$$f_t = w + \beta f_{t-1} + \alpha \cdot S \nabla_{t-1}$$

with:

• $w$: intercept vector,
• $\beta$: persistence matrix,
• $\alpha$: sensitivity matrix,
• $S$: scaling matrix (typically a function of the inverse Fisher information),
• $\nabla_{t-1}$: score from time $t-1$.

As a concrete example, in the SD-beta model for the interbank credit network, node-specific parameters evolve as:

$$\theta_s^{(\mathrm{in})}(t+1) = w_s^{(\mathrm{in})} + \beta_s^{(\mathrm{in})} \theta_s^{(\mathrm{in})}(t) + \alpha_s^{(\mathrm{in})} \left[ \sum_{i} \frac{Y_{is} - p_{is}}{\sqrt{ \sum_{i} p_{is}(1 - p_{is})}} \right]$$

where $p_{is}$ is the current model-implied probability of a link. The update reflects the degree to which newly observed links deviate from the model’s expectation.

3. Comparative Analysis: Static vs. Score-Driven Dynamic Models

Static ERGMs estimate parameters independently at each time point or pool them under stationarity assumptions. This approach fails to leverage temporal continuity and is unable to efficiently track gradual shifts or abrupt structural changes. In contrast, SD-Score drives parameter updates via direct feedback from new observations, adapting smoothly to both persistent trends and regime changes. Filtering experiments demonstrate reduced root mean square error for recovering dynamic parameter paths in simulation studies. Forecasting exercises—such as future link prediction—reveal that SD-ERGMs outperform static ERGM or AR(1) models due to their direct incorporation of “surprise” information via the score (Gangi et al., 2019).

4. Practical Implementations and Empirical Applications

Two empirical applications illustrate the effectiveness of SD-Score methodologies.

a. Interbank Credit Network (Italian e-MID Data):

• SD-beta model applied to weekly aggregated interbank loan networks.
• Rolling-window training (over 100 weeks) followed by forward link prediction.
• Performance evaluated with ROC and AUC metrics.
• SD-Score-driven filtering yields enhanced link prediction compared to naive and static AR(1) baselines.

b. U.S. Congress Co-Voting Network:

• Construction of temporal networks linking senators who vote together on >75% of bills per Congress.
• SD-ERGMs use edge count and GWESP (geometrically weighted edgewise shared partners) statistics to capture link density and transitivity.
• Filtering reveals time variation in network tendencies not detectable via static models.
• LM tests enable formal assessment of parameter instability.

Application	Network Statistic(s)	SD-ERGMs Use Case
Interbank Credit Network (e-MID)	Node-specific in-/out-degree	Link prediction, dynamic filtering
U.S. Congress Co-Voting Network	Edge count, GWESP (transitivity)	Temporal evolution assessment, LM testing

5. Integrative Framework and Statistical Implications

SD-Score generalizes the notion of observation-driven updating across the domain of temporal networks. By using the score to energize parameter evolution, the framework creates a direct statistical link between static network features (such as degree distributions or transitivity) and their latent dynamics. Filtered parameter paths allow for the computation of confidence bands and statistical testing of stationarity, supporting rigorous inquiry into whether network tendencies are time-varying or constant.

A plausible implication is that SD-Score offers a unified methodology bridging network analysis and time series econometrics, enabling informed model selection and change detection in systems characterized by both complex interactions and temporal dependency.

6. Implications for Applied Network Science

In applied settings, SD-Score enables practitioners to test for structural changes and accommodate nonstationarity in networks encountered in finance (e.g., shifts in lending topology) and politics (e.g., realignment of legislative coalitions). The framework facilitates both synthetic network sequence generation and signal extraction, improving the fidelity and interpretability of dynamic network models. Even in cases where stationarity is initially assumed, SD-Score provides tools to identify latent dynamics and supports robust adaptation to evolving systems.

7. Methodological Significance and Future Prospects

SD-Score-driven ERGMs represent a substantive advance in temporal network modeling by embedding observation-driven dynamics into the ERGM framework. The methodology enhances forecasting accuracy, enables formal instability testing, and supports comprehensive filtering of parameter paths over time. A plausible implication is that future research will extend score-driven principles to more complex network types, multivariate statistics, and hybrid models, further integrating dynamic score methodologies with modern network analysis.