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A Separable Model for Dynamic Networks
Pavel N. Krivitsky, Mark S. Handcock
TL;DR
Dynamic-network modeling needs to distinguish prevalence from the incidence and duration processes that generate it. The paper introduces STERGM, a discrete-time ERGM-based model with separate formation and dissolution components, develops likelihood-based inference and computational estimation, and demonstrates interpretable application to school friendship ties. Its separable parameterization improves attribution of incidence and duration while retaining ERGM flexibility, though within-step independence may be inappropriate for some network settings.
Problem
Dynamic-network models need realistic temporal structure, while conventional parameters can confound tie incidence with duration and limit attribution of prevalence to these processes.
Method
STERGM extends discrete-time ERGM-based network evolution with separate formation and dissolution components, supported by likelihood-based inference and computational algorithms for estimation.
Results
STERGM provides more interpretable attribution of incidence and duration without sacrificing the ability to incorporate effects of specific features of past networks.
Takeaways & Limitations
The framework supports realistic dynamic-network simulation and is applicable to directed and undirected networks with relatively light computational burden.
Takeaways & Limitations
STERGM assumes within-step independence of formation and dissolution, and some settings do not allow a separable formulation.
Abstract
from arXiv · showhide
Models of dynamic networks --- networks that evolve over time --- have manifold applications. We develop a discrete-time generative model for social network evolution that inherits the richness and flexibility of the class of exponential-family random graph models. The model --- a Separable Temporal ERGM (STERGM) --- facilitates separable modeling of the tie duration distributions and the structural dynamics of tie formation. We develop likelihood-based inference for the model, and provide computational algorithms for maximum likelihood estimation. We illustrate the interpretability of the model in analyzing a longitudinal network of friendship ties within a school.
1 Introduction
Dynamic networks require realistic, tractable models that capture both cross-sectional structure and temporal evolution. Existing approaches model network series or event histories, while this paper targets clearer attribution of prevalence to tie incidence and duration.
- Dynamic networks arise across fields, creating demand for realistic and tractable models of networks that evolve over time.
- Epidemiological network models must represent both cross-sectional structure and temporal structure because partnership timing affects disease spread.
- TERGMs extend ERGMs by modeling the transition probability from a network at time t to a network at time t + 1.
- Existing dynamic-network approaches commonly fit repeated network observations or use precisely timed relational events.
- A single network snapshot identifies prevalence, whereas incidence and duration describe the dynamic process producing that prevalence.
- The paper extends ERGM-based models, develops conditional maximum likelihood estimation, and applies the methodology to longitudinal school friendship data.
2 Discrete-Time ERGM-Based Models for Network Evolution
The paper formulates discrete-time network evolution through ERGM-based transitions and examines how statistics affect formation, dissolution, and interpretation. It motivates separability by showing that conventional terms can confound incidence with duration and produce unrealistic churning.
- The discrete-time model draws the network at time t from an ERGM conditional on one or more previous networks.
- Transition models use sufficient statistics evaluated on current and prior networks, allowing model terms to depend on network history.
- TERGMs are stepwise ERGMs whose transition probabilities require a normalizing constant.
- Model interpretation: A coefficient on an edge-count statistic simultaneously increases the incidence of new ties and the duration of extant ties.
- Model interpretation: Stability statistics likewise reduce new-tie formation while reducing dissolution, coupling lower incidence with longer duration.
- Tie dynamics: For an extant tie, the model separately considers the probability that the tie will be removed, yielding a geometric duration distribution.
- Tie dynamics: A higher mixing coefficient simultaneously increases tie incidence between two groups and the duration of those ties.
- Model interpretation: Selective mixing terms can create churning models in which ties between groups toggle more frequently than realistic processes would imply.
3 Separable Parametrisation
STERGM separates tie formation from dissolution within each discrete time step, yielding independently interpretable incidence and duration processes while retaining ERGM-based flexibility. The model also supports likelihood-based transition probabilities and adapts familiar network statistics, with trade-offs for effects requiring prior-network dependence.
- Generative mechanism: The model constructs formation and dissolution networks from the previous network, then applies both to produce the next network.Observed endpoint networks identify the intermediate states as y+ = yt−1 ∪yt and y− = yt−1 ∩yt.
- Separable model: STERGM assumes formation and dissolution are conditionally independent given the network at the beginning of a time step.Its parameters are partitioned into separate formation and dissolution spaces.
- STERGM: Formation and dissolution models use ERGM components, allowing many cross-sectional network statistics to be adapted with parameters interpreted as affecting incidence or duration.The paper specifically discusses edge counts, selective mixing, and degree-distribution terms.
- STERGM: STERGM gains ease of specification, tractability, and interpretability by separating incidence from duration, while losing within-step interaction between formation and dissolution.It is a subclass of first-order Markov TERGMs, although the broader TERGM class permits processes that STERGM does not retain.
- Formation: Statistics evaluated on y+ = yt∪yt−1 can be affected by dissolution, potentially creating interference with the ultimately observed network yt.This interference is less likely when the network changes little during each time step.
- Explicitly dynamic terms: Explicitly dynamic terms are needed when formation or dissolution effects depend on specific features of the prior network yt−1.For example, a punishment for actors with multiple partners cannot be represented by a generic isolate-count dissolution statistic.
4 Likelihood-Based Inference for TERGMs
The paper develops conditional maximum-likelihood inference for dynamic network models observed as regularly spaced network series. It extends simulation-based likelihood methods to curved exponential-family transition models and uses simulated expectations to evaluate normalizing-constant ratios.
- Inference setup: Inference is based on observing a series of T + 1 networks and estimating transition-model parameters by conditional maximum likelihood.The discussion focuses on first-order Markov dependence for simplicity.
- Estimator: The method extends prior work to curved exponential-family transition models, where the natural parameters satisfy η(θ) ≠ θ.The extension builds on Hunter and Handcock (2006) and Geyer and Thompson (1992).
- Computation: The principal computational difficulty is evaluating ratios of conditional normalizing constants that depend on networks at earlier times.The paper expresses these ratios as products of model-based expectations.
- Computation: Each expectation in the likelihood-ratio expression can be estimated by simulation, allowing a bridge-sampling algorithm to fit TERGMs to network series data.The approach extends the bridge sampler of Hunter and Handcock (2006) to this setting.
5 Application to the Dynamics of Friendship
The school friendship application uses a separable formation–dissolution model to explain structural change over time, with estimates interpreted through formation and persistence effects. The analysis finds strong sex homophily, reciprocity, transitive closure, and primary-school effects in formation, while dissolution is more strongly associated with reciprocal and time-varying hierarchical structure.
- Data and model: The application analyzes directed friendship nominations among 26 Dutch secondary-school students observed at four three-month intervals.The data include 17 girls and 9 boys, student covariates, sex, and shared-primary-school indicators; some observations were missing because of student absence.
- Data and model: The model includes sex homophily, reciprocity, primary-school co-attendance, popularity, transitive closure, and cyclical ties in the formation component.These terms represent exogenous and endogenous structural effects used to explain observed network change over the school year.
- Estimation: Conditional maximum likelihood estimation is implemented with a variant of the MCMC approach of Hunter and Handcock.The analysis uses the conditional MLE procedure developed for regularly spaced network series data and monitors the statistical properties of the MCMC algorithm.
- Formation results: Formation shows strong sex homophily, reciprocity, transitive closure, and greater tie formation among students who attended the same primary school.The positive transitive and negative cyclical effects indicate a strong hierarchical tendency in tie formation.
- Dissolution results: Dissolution is strongly retarded by reciprocal ties, while structural terms have less influence and boy-to-boy ties show modestly greater persistence.Dissolution parameters measure persistence, so negative parameters correspond to shorter durations.
- Temporal specification: The time-homogeneous model improves formation fit over null and Erdős–Rényi models, whereas dissolution shows some evidence favoring time-heterogeneous specifications driven mainly by increasing hierarchical tendency.Time-varying overall formation rates and fully time-heterogeneous formation parameters do not significantly improve fit; over time, transitive closure increasingly retards dissolution.
6 Discussion
The STERGM separates formation and dissolution parameters, improving incidence–duration interpretability while retaining ERGM flexibility and computational advantages. Its usefulness extends to varied network types and partially observed longitudinal data, but separability depends on the setting and time-step granularity.
- STERGMs use separate parameter sets for tie formation and dissolution, improving attribution of incidence and duration without excluding effects of past networks.
- Formation and dissolution are conditionally independent within a time step but dependent over time, allowing incidence structure to be identified alongside duration structure.
- The likelihood can be decomposed into components that are relatively easy to compute, providing computational advantages for inference.
- The model applies to directed and undirected networks, supports realistic dynamic-network simulation, and has tractable parameters with relatively light computational burden.
- Likelihood-based inference can accommodate partially observed relational information when the sampling design or missingness process is amenable to the model.
- The within-step independence assumption may be inappropriate for constrained processes, while separability is more plausible when discrete time steps represent relatively small changes.
- Formal tests of whether separability is appropriate are proposed as future work rather than developed in this paper.
- Potential extensions include changing network size, evolving actor attributes, and fitting egocentrically sampled data when tie-duration information is available.
A Separable TERGM Terms
The appendix introduces the fundamental model terms used in a STERGM and derives and discusses their roles.
- The appendix derives fundamental model terms for a STERGM.
- The appendix discusses how these terms can be used within the STERGM framework.
- The section provides groundwork for interpreting formation and dissolution terms in the model.
A.1.1 Formation
The formation statistic counts newly formed ties, and its parameter has a direct log-odds interpretation for gaining ties.
- The formation statistic g+ counts ties in the formation network, equivalently representing the union of networks across adjacent time points.
- θ+ represents the log-odds of a previously absent tie gaining a tie when no other formation terms are present.
- logit-1(θ+) gives the expected fraction of previously empty tie variables that gain a tie, while other terms yield conditional log-odds-ratios.
A.1.2 Dissolution
The dissolution statistic counts surviving ties, with its parameter describing tie survival or dissolution hazard and implying geometric durations under dyadic independence.
- The dissolution statistic g− counts ties surviving from the previous network, equivalently representing the intersection of adjacent networks.
- θ− represents the log-odds of an existing tie surviving to the next time point when no other dissolution terms are present.
- logit-1(θ−) is the expected fraction of extant ties surviving, whereas logit-1(−θ−) is the expected dissolution hazard.
- With dyadic independence in dissolution statistics, tie durations follow a geometric distribution with support N.
A.2 Selective Mixing
Selective mixing is represented by vector statistics for tie formation, with group-pair effects given a direction in the STERGM context.
- Selective mixing in the formation model is represented by a vector of statistics based on group memberships of dyad endpoints.
- In a STERGM, the selective-mixing statistics have a direction associated with formation.
A.2.1 Formation
The section explains how STERGM statistics affect formation and dissolution probabilities, including selective mixing, degree distributions, and standard ERGM terms. Degree-distribution terms introduce dyadic dependence, so some quantities lack closed forms and require heuristic interpretation.
- A.2.1 Formation: Adding a tie between groups k1 and k2 changes the selective-mixing formation statistic when the dyad was previously untied.
- A.2.1 Formation: θ+ k1,k2 is the conditional log-odds-ratio for a non-tied dyad gaining a tie based on its endpoint group memberships.
- A.2.2 Dissolution: Selective mixing in dissolution analogously represents the group memberships of extant ties being preserved to the next time step.
- A.3 Degree Distribution: Degree-distribution statistics count actors at particular degrees or degree ranges and introduce dyadic dependence, limiting closed-form quantities and the directiveness of conditional log-odds.
- A.3 Degree Distribution: These degree effects are interpreted conditionally on other terms, representing their incremental influence while other coefficients remain fixed.
- A.3.1 Formation: Formation degree-count terms track actors’ neighbor counts and distinguish transitions involving isolates, degree 1, and higher degrees.
- A.3.1 Formation: A negative d = 0 formation coefficient increases first-tie acquisition, whereas a positive coefficient reduces it; d = 1 has opposing effects on first and second ties.
- A.3.2 Dissolution: Dissolution degree terms affect loss of last or additional ties, while standard ERGM statistics can be used as implicitly dynamic formation or dissolution terms.