Hoff 2002 - Monks Florentine Families and Classroom Examples

Summary

Three standard network datasets: Sampson’s 18 monks (distance model in ${\mathbb{R}}^2$ has log-lik -66.02 with 34 parameters vs -82.12 for the Wang-Wong SBM with 37), Padgett-Ansell Florentine marriages (15 families after dropping an isolate), and Hansell’s classroom friendship (27 students; projection model with a same-sex covariate fits best, log-lik -224.58).

Overview

Each analysis fits ML estimates (MDS start) and then Bayesian MCMC; positions are plotted with marginal posterior clouds.

Main Content

4.1 Monk data (Sampson 1968)

18 monks, directed positive affect; groups: loyal opposition (monks 2-6), young Turks (8-14), outcasts (16-18), waverers (1, 7, 15). Number of nonvacuously transitive ordered triples is 49; in 500 random reallocations of ties (holding out-degrees fixed) the largest was 35, so the data are more transitive than chance. Model: $\text{logit}p(y_{ij}=1)=\alpha -|z_i-z_j|$, $z\in {\mathbb{R}}^2$; free parameters $18\times 2-3+1=34$ ($z$ fixed up to the 3 rigid-motion degrees of freedom, plus $\alpha$).

Model	Maximised log-likelihood	Parameters
Distance model (${\mathbb{R}}^3$)	-34.04	50
Distance model (${\mathbb{R}}^2$)	-66.02	34
Stochastic blockmodel	-82.12	37

Interpretation: the ${\mathbb{R}}^2$ latent-space model beats Wang-Wong’s SBM with fewer parameters, and does not presuppose groups; ${\mathbb{R}}^3$ gains 16 parameters for a large log-lik gain (-34.04). The paper’s argument that the fit cannot be improved by going to higher dimensions: the best-fitting symmetric model has $p_{ij}=1$ for mutual ties, $0$ for null dyads and $1/2$ for asymmetric dyads, so its log-likelihood is $-a\log 4$ with $a$ the number of asymmetric dyads; for the monks $a=26$, giving $-36.04$, which the distance model in ${\mathbb{R}}^3$ essentially achieves (in the paper’s wording, there exist positions $\hat{z}$ and $\hat{\alpha }$ with the limit equal to $-26\log 4$ as $c\to \infty$). Bayesian fit: diffuse $N(0,100^2)$ priors, $2.5\times 10^6$ scans (output every 2,000); the chain mixes reasonably fast in $z_i$ and distances but slowly in $\alpha$ (Fig. 2). Posterior clouds validate the loyal/Turks/outcast categorisation; monk 15 lies between groups.

4.2 Florentine families (Padgett & Ansell 1993)

16 families; undirected marriage ties; one family with no ties dropped (15 analysed; its ML distance would be infinite). In ${\mathbb{R}}^2$, ML positions can make the likelihood arbitrarily close to 1 as $\alpha \to \infty$ (overfitting: fitted tie probabilities all 0 or 1); a prior on $\alpha$ fixes this: exponential with mean 2, $N(0,100^2)$ on $Z$, $5\times 10^6$ scans (5,000 saved), mixes faster than the monk chain (Fig. 4). The Medici (family 9) have above-average distance to others; two equivalent $d_2$ representations (Fig. 3a-b) differ in the position of families 1, 9, 14, 10 (bimodal posterior); marginal clouds in Fig. 3c.

4.3 Classroom data (Hansell 1984)

13 boys, 14 girls; strong friendship ties; out-ties range 0-19 (mean 5.8, sd 4.7; sd of in-ties 3.2); 72% of ties are same-sex; transitive ordered triples 400 (max 347 in 500 reallocations). Projection models used because of variable send rates.

Model	Maximised log-likelihood	Parameters
Projection, with covariate (same sex)	-224.58	55
Stochastic blockmodel	-227.57	55
Projection, no covariate	-229.05	54

Interpretation: the sex covariate is nominally significant (LRT); the median angle between boys’ and girls’ latent vectors is 76 degrees without covariate (two roughly orthogonal groups), 60 degrees after adjusting for same-sex, and 13 degrees with a full model with four baseline rates. Naive logistic regression gives log-odds ratio 1.3 (se 0.2) for same-sex vs different-sex ties; the Bayesian posterior for $\beta$ ($5\times 10^6$ scans, saved every 1,000) has a wider interval (ties are not independent given latent positions), Fig. 5c.

Examples

Posterior predictive reciprocity check (Sec. 5)

Expected numbers of (reciprocal, nonreciprocal, null) dyads from the fitted conditional-independence model vs observed: monk ${\mathbb{R}}^2$ model (17.3, 21.4, 114.3) vs observed (15, 26, 112); classroom model (26.7, 103.6, 220.7) vs observed (24, 109, 218). A reasonable fit, though both datasets show slightly less reciprocity than the model predicts.

Workflow for new data (E21-style): draw $Z$ with PyMC, align with Procrustes (Hoff 2002 - Likelihood Estimation and Procrustes MCMC), then run network posterior predictive checks (degree, transitivity, shared partners, geodesics; see Network Outcome Models - Stochastic Block Models and Network Checks).

Connections

Social Network Models has a simpler network model in PyMC; Hoff 2002 - Discussion Extensions and Stochastic Blockmodel Contrast.

See Also

Hoff 2002 - Latent Space Models Overview