Source-linked AI summary
Maximum likelihood estimation for social network dynamics
Tom A. B. Snijders, Johan Koskinen, Michael Schweinberger
TL;DR
The paper develops maximum-likelihood estimation for longitudinal network data modeled as discrete observations of a continuous-time Markov process. It uses data augmentation, MCMC, and stochastic approximation, and reports that ML appears more advantageous than MoM for small data sets, while emphasizing that the simulations are very limited.
Problem
Versatile statistical methods for realistic longitudinal network modeling are still developing, while existing actor-oriented models lack closed-form likelihood calculations.
Method
The paper approximates the ML estimator with an MCMC-based Markov Chain Stochastic Approximation algorithm using augmented data and simulated unobserved network changes.
Results
ML estimation shows strong advantages over MoM in root mean squared error and test power for small data sets, based on very limited simulations.
Takeaways & Limitations
Likelihood-based estimation provides a basis for likelihood-based model selection and extensions to more complicated network models.
Takeaways & Limitations
The reported simulation evidence is very limited, and further simulation studies are necessary.
Abstract
from arXiv · showhide
A model for network panel data is discussed, based on the assumption that the observed data are discrete observations of a continuous-time Markov process on the space of all directed graphs on a given node set, in which changes in tie variables are independent conditional on the current graph. The model for tie changes is parametric and designed for applications to social network analysis, where the network dynamics can be interpreted as being generated by choices made by the social actors represented by the nodes of the graph. An algorithm for calculating the Maximum Likelihood estimator is presented, based on data augmentation and stochastic approximation. An application to an evolving friendship network is given and a small simulation study is presented which suggests that for small data sets the Maximum Likelihood estimator is more efficient than the earlier proposed Method of Moments estimator.
1. Introduction.
The paper addresses the need for versatile statistical methods for longitudinal social-network data and presents maximum-likelihood estimation alongside existing stochastic-approximation methods.
- Longitudinal studies are important for understanding social networks, but versatile methods for realistic longitudinal modeling are still emerging.
- The model represents repeated network observations as discrete observations of a continuous-time stochastic process on directed graphs.
- Existing actor-oriented models are difficult to estimate in closed form but can be simulated, motivating a stochastic-approximation Method of Moments estimator.
- Maximum-likelihood estimation is expected to improve statistical efficiency and enables likelihood-based model selection.
- The article presents an MCMC algorithm for approximating the ML estimator and evaluates it with an empirical example and a small simulation study.
2. Model definition.
The model treats network evolution as an actor-oriented continuous-time Markov process in which social actors control their outgoing ties under one-change-at-a-time constraints.
- Networks are represented as n × n adjacency matrices observed at repeated panel-design times, with self-ties structurally excluded.
- The network develops as a continuous-time Markov process even though it is observed only at discrete time points.
- Actors have control, subject to constraints, over their outgoing ties and do not coordinate their changes.
- The model focuses on actor-oriented processes, while tie-oriented models can be treated similarly.
1. Opportunities for change
Change opportunities arrive actor by actor according to exponential waiting times, with actor-specific rates and covariate- or position-dependent specifications.
- Each actor receives stochastic opportunities to change one outgoing tie variable.
- The Markov assumption implies exponentially distributed waiting times between opportunities.
- Actor-specific rate functions determine how quickly opportunities arise when the current digraph is x.
- Conditional on an opportunity, the model specifies the probability that actor i receives it.
- Rate functions may be constant or depend on covariates and positional characteristics such as outdegree.
2. Options for change
When an actor can change a tie, the model defines permitted successor networks and assigns transition probabilities through an objective function representing relative attractiveness.
- A permitted set A_i(x0) restricts the networks actor i may reach from the current network x0.
- The usual permitted set includes the unchanged network and networks differing in one outgoing tie, allowing actors to retain a satisfactory state.
- Alternative specifications exclude structurally impossible ties or require a change whenever an opportunity occurs.
- The objective function f_i(β,x0,x) represents the relative attractiveness of moving from x0 to x, and transition probabilities are normalized over permitted alternatives.
- The notation p_ij(β,x0) identifies the probability that actor i changes the tie to j, while j = i denotes no change.
- The transition rule can be motivated by a random-utility model in which actors maximize the objective function plus a standard Gumbel disturbance.
- Allowing the objective function to depend on x0 represents path-dependence, including different effects for withdrawing and creating a tie.
- Objective functions can incorporate network structures such as reciprocated ties, transitive triplets, indirect ties, and persistent reciprocity, as well as covariates.
3. Intensity matrix; time-homogeneity
The model uses a continuous-time Markov process whose transition probabilities depend on the intensity matrix and observation-interval durations. Time heterogeneity can be represented through time-varying model components or period-specific rate multipliers.
- Time-homogeneity: The intensity matrix is time-homogeneous except for time dependence introduced through time-varying components in the model functions.
- Intensity matrix: The continuous-time process is represented by an intensity matrix Q, with transition probabilities over an interval obtained from e^(t_m−t_{m−1})Q.The matrix Q has elements determined by the model’s transition intensities.
- Time-homogeneity: Observation moments can mark time heterogeneity, including covariates whose values change between successive observation periods.
- Time-homogeneity: Period-specific multiplicative rate parameters can absorb changes in observation-interval durations, making the numerical time values less important.The parameter may differ freely between consecutive observation periods.
- Comparison with discrete-time models: Unlike discrete-time longitudinal ERG models, this continuous-time probability model is defined independently of the observational design and accommodates irregularly spaced intervals.Its dynamics are defined through conditional probabilities of individual tie changes.
- Comparison with discrete-time models: The actor-oriented process forms an incompletely observed exponential family and avoids the practical near-degeneracy problem affecting some ERG specifications.Both model classes require computationally intensive MCMC procedures, but the actor-oriented process can be simulated directly conditional on an initial network.
3. ML estimation.
Maximum likelihood estimation is made tractable by augmenting observed network panels with latent sample paths and applying MCMC-based stochastic approximation. The resulting procedure yields likelihood calculations or approximations conditional on the first observation and converges through tail averaging under stated conditions.
- ML estimation: The algorithm augments each interval between observations with simulated intermediate digraph sequences, then uses them in Robbins–Monro updates to solve the likelihood equation.Intermediate paths are simulated with Metropolis–Hastings within the MCMC procedure.
- Augmented data: Given the initial network and the ordered opportunity sequence, the augmented process completely determines the network trajectory within an observation interval.
- Augmented data: The augmentation records the number and identities of tie-change opportunities while integrating out their exact times.Retaining no-change opportunities simplifies likelihood computation.
- Augmented data: The likelihood approximations require large numbers of opportunities and sufficiently long observation intervals, together with bounded actor-level rate functions.
- Augmented data: The sample-path likelihood becomes directly expressible, exactly or approximately, after augmenting the observed endpoint networks.
- Missing data principle: The MCSA method replaces the difficult observed-data score with a computable augmented-data score generated from the conditional distribution of missing sample paths.The likelihood equation is solved through stochastic approximation under regularity conditions.
- Stochastic approximation: The stochastic-approximation update uses a diminishing gain sequence and a suitable matrix, with tail averaging converging to the likelihood-equation solution for a broad class of positive definite matrices.The stated gain sequence decreases asymptotically as aN ∼ N^-c for c < 1.
- Missing data principle: The missing-information formulation expresses observed Fisher information as complete-data information minus information attributable to the unobserved sample paths.This formulation supports calculation of standard errors.
4. Empirical example.
The empirical example analyzes friendship dynamics among 32 freshman students using actor-oriented models, estimating parameters separately with Method of Moments and Maximum Likelihood. Both methods produce the same substantive conclusions, while the single data example cannot establish their relative value.
- Data and model: The data comprise six friendship-network waves for 32 freshman students, with observation intervals increasing from three to six weeks.The relation studied is being friends or close friends.
- Data and model: The model includes network-structure effects for outdegree, reciprocity, transitive triplets, and 3-cycles, plus gender and gender similarity covariates.It illustrates both triadic dependence and covariate effects.
- Estimation: Parameters for the two transitions are estimated separately using both Method of Moments and Maximum Likelihood.The Method of Moments estimator equates observed and expected statistics and is obtained by stochastic approximation.
- Estimation: Simulation checks found every convergence ratio below 0.1, indicating adequate convergence for the fitted estimators.The checks used 2000 simulated runs for each component of the estimating equation.
- Scope: The single empirical example cannot support conclusions about the relative value of Method of Moments versus Maximum Likelihood.The authors reserve that comparison for simulation evidence.
5. Simulation examples.
The simulation studies compare Method of Moments and Maximum Likelihood under friendship-network designs with different information levels. They find similar performance with 32 actors, but clearer ML advantages with 20 actors, while emphasizing that the study is limited and not generalizable.
- Scope: The authors describe the simulation as a very limited exploration whose design is intended to resemble applications involving small friendship groups.Its limited nature does not allow generalization.
- 32 actors: The first study uses 1000 generated data sets modeled on 32 actors, three waves, and seven objective-function parameters.Five data sets were excluded because one or both estimators failed to converge adequately.
- 32 actors: For 32 actors, Method of Moments and Maximum Likelihood produce very similar results, with type-I error rates generally near nominal values.The main exception is inflated ML type-I error for the two rate parameters, associated with skewed rate-estimator distributions.
- 20 actors: The second study reduces the network to 20 actors and adds a reciprocity-by-gender-similarity interaction to create a more difficult estimation setting.The added effect increases parameter-estimator correlation, while the smaller network reduces information.
- 20 actors: Except for β2, tests based on ML have higher estimated power than MoM-based tests because of smaller mean squared errors and a less conservative test.The MoM test has especially low power for the gender-ego effect because β6 often has high standard errors.
6. Likelihood ratio tests.
The paper develops a likelihood-ratio procedure based on path sampling to support model comparison. Applied to the friendship data, it rejects equality of the network parameters across the two periods.
- Purpose: Likelihood-based estimation enables model-selection procedures, and the paper specifically explains a likelihood-ratio test.This is presented as an advantage over existing estimation methods.
- Method: The likelihood ratio is estimated by path sampling along a linear parameter path between two parameter values.The method approximates the relevant integral using simulations from conditional distributions along that path.
- Method: The MCMC initialization for each successive path point can use the preceding point's final state to reduce burn-in time.This is an implementation improvement for generating conditional draws.
- Application: 18.7, the estimated likelihood ratio for the friendship data, gives p < 0.01 under a χ2_7 reference distribution.The test compares equal versus period-specific network parameters while allowing rate parameters to differ under both hypotheses.
- Application: The test rejects the null hypothesis that the seven network parameters are equal across the two observation periods.The rejection occurs at conventional significance levels.
7. Discussion.
The article presents a maximum-likelihood algorithm for longitudinal network models and reports advantages over method-of-moments estimation, while identifying theoretical and computational limitations.
- The model represents triadic and other complex network dependencies, and the ML algorithm uses stochastic approximation with Monte Carlo simulations of unobserved changes.The simulation design integrates out waiting times between unobserved changes.
- No proof is available for consistency and asymptotic normality under the model’s nonstandard dependence assumptions.Limited simulations support the expectation of asymptotic normality, but a proof may be complicated.
- The algorithm generally converges well but is time-consuming, taking about 35 minutes per ML estimation versus about 2 minutes for MoM on a regular personal computer.Convergence can be problematic when data sets are small relative to the number of estimated parameters.
- The study reports strong ML advantages over MoM in root mean squared error and test power for small data sets.These conclusions are based on very limited simulations.
- The limited simulations suggest ML’s efficiency advantage is smaller for medium to large data sets, requiring further simulation studies.Likelihood-based estimation may nevertheless support extensions to more complicated models and model-selection procedures.