Hoff 2002 - Discussion Extensions and Stochastic Blockmodel Contrast

Summary

The authors list four advantages (visual representation of relationships, uncertainty quantification, flexibility via GLMs and time-varying positions, easy handling of missing ties), propose dyad-dependence extensions, suggest clustered priors on $Z$, and contrast the approach with MDS and blockmodels.

Overview

Compared with blockmodels the latent space avoids hard group membership and models transitivity directly; compared with MDS it models the observed ties (response) and gives uncertainty.

Main Content

• Advantages (Sec. 5). (1) model-based visual representation; (2) uncertainty in the social space is quantifiable and displayable; (3) generalises to multiple relations, tie strengths (GLMs), and time-varying positions as stochastic processes; (4) deals easily with missing data if ties are missing at random (likelihood uses only observed ties); (5) inherently transitive, so it should fit better than non-transitive models such as blockmodels.
• Dyad dependence. Extend to treat dyads $(y_{ij},y_{ji})$ as independent given $Z$, $\alpha$ and a parameter $\gamma$ for within-dyad dependence (reciprocity).
• Model checking. Compare expected reciprocal/nonreciprocal/null dyad counts with observed ones (see example note).
• Optimisation. Non-concavity in $Z$: look for multiple local maxima; MCMC helped (Florentine chain identified and mixed two separated modes). The log-likelihood is concave in $D$ if the distance matrix is not restricted to be Euclidean; one could fit in high dimension and then inspect subnetworks in lower dimensions.
• Priors. Diffuse normal priors may not reflect networks’ tendency to cluster; better: clustered point processes or mixtures of normals with unknown number of components, adding hierarchy for group tendencies without specifying membership.
• MDS contrast. Breiger, Boorman & Arabie (1975), Faust & Romney (1985): MDS uses ad hoc dissimilarities, does not model the stochastic response, large parameter counts, unknown asymptotics. Related Bayesian MDS: DeSarbo, Kim & Fong (1999), Oh & Raftery (2001); latent class MDS (Lazarsfeld & Henry 1968).
• Software. R code from the first author’s website (as stated in the paper).

Stochastic blockmodel contrast (Sec. 1)

Wang & Wong (1987): given known groups, tie rates depend on sender group, receiver group and actor effects; Nowicki & Snijders (2001): groups latent, dyads conditionally independent given class. These handle clusters of stochastically equivalent actors but fit poorly when many actors lie between clusters or when relations are transitive without clear clustering.

Examples

Missing ties: for a dyad with unobserved $y_{ij}$, drop its term from eq. 4; in a PyMC model y observed only at the index set of seen dyads (mask), the latent positions still fill in predictions for the missing ones.

Connections

Network Outcome Models - Stochastic Block Models and Network Checks explains SBM variants (mixed membership, degree-corrected) used in E21; link to marketing: missing-at-random assumption matters for sparsely observed referral/co-engagement graphs.

See Also

Hoff 2002 - Latent Space Models Overview | Social Network Models