Source-linked AI summary
Modeling homophily and stochastic equivalence in symmetric relational data
Peter D. Hoff
TL;DR
The paper addresses how to model symmetric relational data exhibiting homophily, stochastic equivalence, or both. It develops an eigenmodel based on weighted inner products of node-specific latent factors, showing that it generalizes latent class and distance models. Across three datasets, the eigenmodel performs well for all three, while the other models perform well mainly on datasets matching their respective structures.
Problem
Many real networks combine homophily and stochastic equivalence, whereas latent class or distance models may represent only part of that structure.
Method
The eigenmodel represents pairwise relational variation using a weighted inner product of node-specific latent factors derived from an eigenvalue-decomposition perspective.
Results
The eigenmodel performs well across three datasets, while latent distance and class models perform well for the first and second datasets respectively.
Takeaways & Limitations
The eigenmodel provides a more general representation of symmetric relational data than latent class and distance models.
Takeaways & Limitations
The latent distance model assumes that strongly related nodes are also similar in how they relate to other nodes.
Abstract
from arXiv · showhide
This article discusses a latent variable model for inference and prediction of symmetric relational data. The model, based on the idea of the eigenvalue decomposition, represents the relationship between two nodes as the weighted inner-product of node-specific vectors of latent characteristics. This ``eigenmodel'' generalizes other popular latent variable models, such as latent class and distance models: It is shown mathematically that any latent class or distance model has a representation as an eigenmodel, but not vice-versa. The practical implications of this are examined in the context of three real datasets, for which the eigenmodel has as good or better out-of-sample predictive performance than the other two models.
1 Introduction
Relational data often combine homophily and stochastic equivalence, while latent distance and class models typically represent only one pattern. The paper introduces an eigenmodel that can represent both patterns and performs well across three datasets.
- Relational data are represented as symmetric sociomatrices whose entries describe pairwise measurements among nodes.
- Stochastic equivalence groups nodes with similar relationship patterns, using latent classes and class-level relationship probabilities.
- Homophily models stronger relationships between nodes with similar latent characteristics and accounts for transitivity, balance, and cohesive subgroups.
- Latent distance and class models can respectively capture homophily and stochastic equivalence, but either model may fail to represent the other pattern compactly.
- Many real networks exhibit both structural equivalence and homophily, so using either conventional model alone represents only part of their structure.
- The eigenmodel represents relational variation through a weighted inner product of node-specific latent factors and generalizes latent class and distance models.
- Mathematically, the eigenmodel can compactly represent the same network features as latent class and distance models, but those models cannot represent every eigenmodel.
- Across a social network, word-association data, and protein interactions, the eigenmodel performs well for all three datasets.
2 Latent variable modeling of relational data
The paper frames symmetric relational data with exchangeable latent-variable models, then uses a weighted latent-vector representation to encompass latent class and distance models while allowing broader dependence patterns.
- 2.1 Justification of latent variable modeling: Exchangeability requires relational models to remain unchanged under any permutation of node labels.This condition leads to latent-variable representations with i.i.d. node effects, i.i.d. pair-specific effects, and a function symmetric in its node arguments.
- 2.1 Justification of latent variable modeling: The model represents binary ties through a probit probability combining an intercept, observed covariates, and a symmetric latent interaction function.The resulting probability is Φ[µ + βT x_i,j + α(u_i,u_j)].
- 2.1 Justification of latent variable modeling: Ordered probit models retain full information from ordinal relational measurements instead of thresholding them into binary data.Threshold parameters are estimated for all values except the lowest value in the sample space.
- 2.2 Effects of nodal variation: The eigenmodel permits positive or negative homophily because signed eigenvalues determine whether similar latent coordinates contribute positively or negatively.Nodes with the same or similar latent vectors may therefore have either strong or weak relationships.
- 2.2 Effects of nodal variation: The eigenmodel contains every K-class latent-class matrix, with strict containment when K<n, and can weakly generalize distance models through rank and ordering.Distance matrices may require dimension K+1, while sufficiently large eigenmodel rank preserves their entry ordering under monotone transformations.
- 2.2 Effects of nodal variation: Distance models do not weakly generalize eigenmodels: as n grows, some eigenmodel entry orderings cannot be matched by any fixed-dimensional distance model.The supplied examples give n=3 for K=1 and n≤6 for K=2 before higher dimensions become necessary.
3 Model comparison on three different datasets
The authors compare three latent-variable models using five-fold cross-validation across three datasets and dimensions K ∈ {3, 5, 10}. Dataset structure determines which models perform best, while the eigenmodel performs competitively across settings.
- Cross validation: Five-fold cross-validation evaluates all three models across three datasets and dimensions K ∈ {3, 5, 10}.Each fold estimates parameters without one held-out subset and uses posterior predictive means for out-of-sample evaluation.
- Adolescent Health social network: Distance models perform best on the 247-student friendship network, with eigenmodels close behind and latent class models performing poorly.The friendship data exhibit substantial transitivity, which favors distance-based representations.
- Word neighbors in Genesis: Latent class models outperform distance models on Genesis word-neighbor data, while the eigenmodel performs equally well.These data are better characterized by classes of words connected across classes than by spatial proximity.
- Protein-protein interaction data: The protein-interaction network contains both hub-like stochastic equivalence and homophily, features that are difficult to capture jointly with the simpler models.The eigenmodel captures both patterns and is reported to outperform the other two models out of sample.
- Protein-protein interaction data: The K = 3 eigenmodel performs better than the other two models for every K considered in the protein-interaction analysis.This comparison concerns out-of-sample predictive performance.
4 Discussion
The discussion presents the eigenmodel as a flexible low-rank representation for symmetric relational data exhibiting homophily, stochastic equivalence, or both. It mathematically generalizes latent class and distance models while retaining broad representational capacity.
- Discussion: Latent class and distance models do not provide a complete picture of relational data combining homophily and stochastic equivalence.The eigenmodel is introduced to represent datasets exhibiting either or both patterns.
- Discussion: Any latent class or distance model can be represented by an eigenmodel, but the eigenmodel can represent additional network features.Its unrestricted low-rank approximation to the sociomatrix supports a wider array of patterns.
- Discussion: The eigenmodel is based on the eigenvalue decomposition of a symmetric matrix.For directed or rectangular relational data, the analogous construction would use a singular value decomposition.
- Discussion: The latent-vector formulation represents node-specific effects through vectors whose interactions can be inferred using model-based procedures for non-Gaussian relational data.The paper discusses ordered probit modeling and Bayesian inference for these latent quantities.