Multivariate Peer Effect Modeling

• Multivariate peer effect models are frameworks that capture how multiple behaviors or covariates jointly influence social outcomes via latent class structures.
• The model extends traditional ERGMs using a finite mixture approach to incorporate unobserved heterogeneity in sender and receiver effects within networks.
• Empirical and simulation studies demonstrate that correctly modeling latent classes improves estimation accuracy and uncovers subgroup-specific peer mechanisms.

A multivariate peer effect model specifies how the behaviors, attributes, or outcomes of multiple agents in a social network jointly respond to the corresponding features of their peers, with “multivariate” denoting that either multiple covariates, multiple behaviors, or several latent subpopulations moderate the effect. Such models generalize classical peer influence specifications by allowing either (i) multiple types of peer influence in a single parametric framework, (ii) unobserved heterogeneity in peer effects, or (iii) the simultaneous modeling of several outcomes. Central challenges for multivariate peer effect modeling include identification in the presence of unobserved confounding, handling endogenous network (peer group) formation, estimating heterogeneous effect structures, and scaling computation to large or dense networks.

1. Model Structure: Sender/Receiver Finite Mixture ERGMs

The Sender/Receiver Finite Mixture ERGM (SRFM-ERGM) extends traditional exponential random graph models (ERGMs) to permit latent heterogeneity in how nodal covariates and network features affect tie formation. In the canonical ERGM, the probability of observing a network with adjacency matrix $A$ (given nodal covariates $X$) is

$$P(G = g \mid X, \theta) = \frac{\exp\left\langle \theta, s(A, X) \right\rangle}{\psi(\theta)}$$

where $s(A, X)$ is a vector of sufficient statistics and $\psi(\theta)$ is the intractable normalizing constant.

SRFM-ERGMs introduce a latent class structure indexed by variable $Z$. Each node belongs to one latent class $q \in \{1,\ldots,Q\}$, so that node-specific effects enter into the sufficient statistics conditional on $Z$: $$P(G = g \mid Z, \theta, X) = \frac{\exp\left(\sum_i \sum_{q=1}^{Q} z_{iq} \langle \theta_q, s_{i\cdot} \rangle \right)}{\psi(\theta, Z, X)}$$ Here, $\theta_q$ is the parameter vector for class $q$ and $s_{i\cdot}$ represents the statistics for node $i$, either for sending or receiving ties.

There are two principal variants:

• Sender Finite Mixture ERGM: latent class moderates the effect of covariates on the probability that a sender nominates friends.
• Receiver Finite Mixture ERGM: latent class moderates the “popularity” effect (being nominated).

Multivariate peer effects are incorporated by parameterizing $s_{i\cdot}$ to include multiple nodal covariates (e.g., substance use, antisocial behavior), network features (e.g., mutuality, GWESP), their interactions, and (optionally) to allow only a subset of effects to vary by class.

2. Heterogeneity, Latent Class Assignment, and Estimation

A central innovation is the ability to flexibly model unobserved heterogeneity in peer effects, distinguishing the SRFM-ERGM from conventional blockmodels or random effects ERGMs. Some parameters (e.g., the sender effect of alcohol use) can be allowed to vary by latent class, while others are constrained to be homogeneous. This specification permits the modeling of subgroup-specific peer selection or influence mechanisms without imposing a strong community structure that would require strong within-block connection density.

Estimation is computationally challenging because the likelihood contains both an intractable normalizing constant and latent class assignments for all nodes. The SRFM-ERGM adopts a data augmentation strategy using an expectation–maximization (EM) algorithm:

1. E-step: Update latent class probabilities for each node using a hard or soft assignment, based on the current parameter estimates. The expected assignment for node $i$ to class $q$ is:

$$E[z_{iq} \mid \theta, \pi, A, X] = \frac{\pi_q \exp\left(\sum_{j \neq i} [a_{ij} \langle \theta_q, s_{ij} \rangle - \log(1 + \exp(\langle \theta_q, s_{ij}\rangle)) ]\right)}{ \sum_{q'} \ldots }$$

1. M-step: Given (possibly hard) class assignments, update class-specific ERGM parameters by maximizing a conditional likelihood approximated by logistic regression (i.e., pseudolikelihood).
2. Final step: Once class assignments stabilize, full MLE estimation is attempted—if feasible—with latent classes treated as known.

This approach scales to moderate-size networks (tested up to $n=151$ in the adolescent network example) and accommodates both small true effects and more complex structural statistics.

3. Empirical and Simulation Evidence

Empirical illustration is provided via a high school friendship network, with students nominating friends and self-reporting on several behavioral covariates. The primary modeling example focuses on sender effects of alcohol use: the sender effect and its associated interaction ("absolute difference") are allowed to vary by latent class, while other covariate effects (for tobacco, marijuana, antisocial behavior) and structural effects (reciprocity, GWESP) are held homogeneous.

Results reveal two distinct classes:

• Class 1: Strong negative sender effect for alcohol use (higher use predicts fewer outgoing ties).
• Class 2: No significant sender effect for alcohol.

Edges parameters and certain covariate effects differ significantly between classes, highlighting the presence of meaningful unobserved heterogeneity.

Simulation studies (nine conditions manipulating class-specific edge and covariate effects) further demonstrate:

• Homogeneous ERGMs fit to heterogeneous data yield misfit—biased estimates, often of unpredictable magnitude/direction, in both covariate and structural parameters.
• Correctly specified SRFM-ERGMs yield near-unbiased estimates (with only minor residual bias for small effects or for complex statistics such as GWESP).
• Latent class label recovery (measured by Adjusted Rand Index) improves with class separation and network size.

4. Implications for Peer Effect Inference and Network Analysis

The SRFM-ERGM framework clarifies that fitting a single-model peer effect specification to a heterogeneous population can mask genuine differences or induce bias. Explicit modeling of latent class heterogeneity in nodal covariate effects can reveal subpopulations responding differently to the same peer structures, which is particularly relevant in domains such as adolescent substance use, where social behaviors are likely to function differently across subtypes.

Most previous models emphasizing network heterogeneity focus on blockmodels or latent space models, which encode dense community structure. In contrast, SRFM-ERGM latent classes are not required to form dense blocks, but rather allow for differences in how nodal covariates or network features affect tie formation regardless of local network density.

The approach is operationally tractable due to:

• Hard classification EM algorithm for separating latent classes,
• Integration with logistic regression pseudolikelihood for efficient parameter updates.

5. Comparison with Alternative Approaches

The SRFM-ERGM situates itself relative to several prior methods:

• Stochastic Blockmodels / Latent Space Models: Primarily model community structure, not parameter heterogeneity.
• Mixture ERGMs (Koskinen 2009): Use Bayesian estimation with MCMC, but suffer from scalability limitations in larger networks.
• Random Effects ERGMs: Allow for sender or receiver intercept heterogeneity, but do not admit covariate effect heterogeneity by class.

SRFM-ERGM was found in simulation to outperform homogeneous ERGMs when true heterogeneity exists and does not introduce excess bias when the model is redundant (true homogeneity).

A current limitation is that latent classes are specified on only one side (sender or receiver, not both), in order to keep the combinatorial complexity tractable. This restriction is protective against the “combinatorial explosion” that joint latent class assignment would incur.

6. Practical Implementation, Limitations, and Extensions

SRFM-ERGM is implemented using a maximum pseudolikelihood approach in conjunction with an EM updating scheme. Practical challenges include choice of number of latent classes $Q$, identifiability issues for weak heterogeneity, and computational bottlenecks for very large $n$ or dense statistics (e.g., GWESP). Data augmentation and approximation methods mitigate some computational intractability associated with the normalizing constant in ERGMs.

The method is suitable for questions where unobserved subpopulations may respond differently to peer exposures, covariates, or endogenous network processes, and where simple block or random effect models are insufficient to explain observed tie formation or selection.

Future directions include:

• Further development for the joint sender/receiver finite mixture model.
• Model selection criteria and diagnostics for choosing heterogeneous vs. homogeneous specifications.
• Extension to dynamic or longitudinal network data and integration with high-dimensional covariate adjustment (Eckles et al., 2017).

7. Significance and Context in Broader Multivariate Peer Effect Literature

SRFM-ERGMs represent a substantial step forward in modeling peer effect heterogeneity. By allowing latent classes to modulate the effects of covariates and network features on relationship formation or peer nomination, the framework provides fine-grained insight into the mechanisms of assortative mixing, peer selection, and influence dynamics in complex social systems.

This modeling approach stands in contrast to aggregate homogeneous effect estimation or to methods that reduce peer behavior to average or sum-based statistics. SRFM-ERGM is positioned as complementary to nonparametric identification approaches, high-dimensional observational adjustment, and newer machine learning augmented peer effect models, offering a flexible, interpretable, and empirically tractable addition to the multivariate peer effect modeling toolkit.