Hoff 2002 - Latent Space Models Overview

Summary

Hoff, Raftery and Handcock (JASA 2002) model a network $Y$ by giving each actor a latent position $z_i$ in a low-dimensional Euclidean “social space” and letting tie probability depend on distance, conditionally independent given positions: $\text{logit}P(y_{ij}=1)=\alpha +{\beta }^{\top }{x}_{ij}-|z_i-z_j|$. The latent geometry automatically yields reciprocity and transitivity, supports MLE or MCMC with Procrustes post-processing, and quantifies uncertainty in positions. It fits three classic datasets better than the stochastic blockmodel.

Overview

Question. How can network data (an $n\times n$ sociomatrix) be modelled so that transitivity and clustering are explained by unobserved characteristics, avoiding the degeneracy and instability of exponential random graph ($p^{\ast }$) models and the rigid groups of stochastic blockmodels?

Contributions. (i) Distance and projection (directional, activity-weighted) models; (ii) estimation: logistic likelihood, MLE via MDS start, Bayesian MCMC with Procrustes alignment; (iii) analyses of 18 monks, 16 Florentine families, 27 classmates; (iv) comparison with the Wang-Wong stochastic blockmodel; (v) extensions (dyad dependence $\gamma$, covariates, MDS contrast).

Motivating PyMC examples

https://redam94.github.io/pymc-challenges/examples.html: E21 - Network data as the outcome: latent-space and stochastic block models (Lazega lawyers, 71 actors, 2,485 dyads; dyad logistic vs sociality random effects, Hoff latent space + Procrustes, mixed-membership SBM, network PPCs: degree, transitivity, shared partners, geodesics).

Source Map

Paper section	Covered in
Abstract, 1 Introduction (random graphs, $p_1$, $p^{\ast }$, SBM)	Hoff 2002 - Distance and Projection Models
2.1 Distance models (eq. 1-3, $d_k$ representability)	Hoff 2002 - Distance and Projection Models
2.2 Projection models	Hoff 2002 - Distance and Projection Models
3 Estimation (eq. 4, Procrustes MCMC)	Hoff 2002 - Likelihood Estimation and Procrustes MCMC
4.1 Monk data, Table 1, Figs. 1-2	Hoff 2002 - Monks Florentine Families and Classroom Examples
4.2 Florentine families, Figs. 3-4	Hoff 2002 - Monks Florentine Families and Classroom Examples
4.3 Classroom, Table 2, Fig. 5	Hoff 2002 - Monks Florentine Families and Classroom Examples
5 Discussion (extensions, MDS)	Hoff 2002 - Discussion Extensions and Stochastic Blockmodel Contrast
SBM background (Wang-Wong, Nowicki-Snijders)	Hoff 2002 - Discussion Extensions and Stochastic Blockmodel Contrast, Network Outcome Models - Stochastic Block Models and Network Checks

Examples

See Hoff 2002 - Likelihood Estimation and Procrustes MCMC (code) and Hoff 2002 - Monks Florentine Families and Classroom Examples.

Connections

• Relevance to marketing measurement: networks as outcomes appear in referral/co-purchase graphs, creator-brand affiliation, and customer similarity; a latent space gives interpretable “positions” and uncertainty for segmentation. The paper itself concerns social networks only.
• Existing vault: Social Network Models (basic network modelling in PyMC), Reinhart 2018 - Crime, Epidemic and Network Applications (events on networks).

See Also

Hoff 2002 - Latent Space Approaches to Social Network Analysis