Social Network Models

Summary

Social network analysis in the Statistical Rethinking framework models dyadic relationships between individuals as latent generalized social states. The model captures within-dyad reciprocity and between-dyad variation, enabling inference about sharing behaviour and social structure.

The Problem of Dyadic Data

In social networks, observations come in dyads (pairs of individuals). Key structure:

• $y_{AB}$: what A does to/for B (e.g., food shared by A with B)
• $y_{BA}$: what B does to/for A
• These are not independent: reciprocity means $y_{AB}$ and $y_{BA}$ are correlated

Standard regression ignores this dyadic dependence. Social network models treat it explicitly.

Generalized Giving Model (Statistical Rethinking, Lecture 15)

Based on McElreath’s Koster & Leckie-style household food sharing model:

$$y_{AB}\sim \text{Poisson}({\lambda }_{AB})$$ $$\log {\lambda }_{AB}=\alpha +\underset{\text{giving: A→B}}{\underbrace{g_{AB}}}+\underset{\text{A’s general giving}}{\underbrace{r_A}}+\underset{\text{B’s general receiving}}{\underbrace{r_B}}$$

Where the key latent structure is the dyadic giving term $g_{AB}$:

$$\left(\begin{array}{c}{g}_{AB}\\ g_{BA}\end{array}\right)\sim \text{MVN}\left(0,\left(\begin{array}{cc}{\sigma }^2& \rho {\sigma }^2\\ \rho {\sigma }^2& {\sigma }^2\end{array}\right)\right)$$

• $\rho$: reciprocity — how strongly giving in one direction predicts giving in return
• ${\sigma }^2$: variance of dyadic effects (heterogeneity in relationship strength)
• $r_A$, $r_B$: generalised giving/receiving rates for each individual (not dyad-specific)

Key insight

By estimating $\rho$, we can quantify whether relationships are reciprocal (positive $\rho$) or one-directional (zero/negative $\rho$). This cannot be done with standard regression.

Network as a Latent Variable

The dyadic effects $g_{AB}$ can be visualised as a weighted network — edges between each pair of individuals with weights proportional to the posterior mean of $g_{AB}$. This gives:

• Posterior uncertainty over network structure
• Natural interpretation: the network is a latent variable inferred from observed behaviour, not directly measured

PyMC Sketch

with pm.Model() as network_model:
    # Reciprocity correlation matrix for each dyad
    rho = pm.Beta("rho", 2, 2)
    sigma_g = pm.Exponential("sigma_g", 1)
    cov_dyad = sigma_g**2 * pm.math.stack([[1, rho], [rho, 1]])
 
    # Generalised giving/receiving per individual
    r = pm.Normal("r", 0, 1, shape=(N_individuals, 2))  # [giving, receiving]
 
    # Dyadic effects for each (i, j) pair
    G = pm.MvNormal("G", mu=0, cov=cov_dyad, shape=(N_dyads, 2))
 
    # Likelihood
    log_lambda = alpha + G[dyad_idx, direction] + r[sender_idx, 0] + r[receiver_idx, 1]
    pm.Poisson("y", mu=pm.math.exp(log_lambda), observed=y)

Connections

• Dyadic correlation: related to Copula Estimation (modelling joint distributions with correlation structure)
• Hierarchical Linear Models — individuals as random effects; dyads as nested structure
• Spatial Models - BYM — also uses graph structure to model dependence between units
• Statistical Rethinking lectures: Lecture 18 (Missing Data) is from the same series

Source

• Social Networks — PyMC port of Statistical Rethinking 2023, Lecture 15 (McElreath)
• Video: Lecture 15 — Social Networks