Hoff 2002 - Distance and Projection Models

Summary

Given latent positions $z_i\in {\mathbb{R}}^k$, ties are conditionally independent with log-odds $\alpha +{\beta }^{\top }{x}_{ij}-|z_i-z_j|$ (distance model). Reciprocity and transitivity follow from the triangle inequality. A projection model uses unit vectors $v_i$ and activity $a_i$ to allow asymmetry: log-odds $\alpha +{\beta }^{\top }{x}_{ij}+z_i^{\top }{z}_j/|z_j|$.

Overview

Background: $p_1$ (Holland & Leinhardt 1981) assumes dyad independence; $p^{\ast }$/ERGMs (Frank & Strauss 1986) suffer degeneracy (Handcock 2000, Snijders 2002); the stochastic blockmodel (Wang & Wong 1987) needs known groups; Nowicki & Snijders (2001) use latent classes. The latent-space view says ties become likelier when actors’ unobserved characteristics are close.

Main Content

Conditional independence latent position model

$Y$ is the $n\times n$ sociomatrix, $x_{ij}$ pair covariates, $\theta$ parameters, $Z=(z_1,\dots ,z_n)$ latent positions:

$$P(Y\mid Z,X,\theta )=\prod _{i\ne j}P(y_{ij}\mid z_i,z_j,x_{ij},\theta ).$$

Distance model (eq. 1)

$${\eta }_{ij}=\text{log odds}(y_{ij}=1\mid z_i,z_j,x_{ij},\alpha ,\beta )=\alpha +{\beta }^{\top }{x}_{ij}-|z_i-z_j|.$$

Interpretation: for $j$ and $k$ equidistant from $i$ the odds ratio of $i\to j$ vs $i\to k$ is $\exp \{{\beta }^{\top }(x_{ij}-x_{ik})\}$. Distances could be any set satisfying the triangle inequality; Euclidean in low dimension is chosen for parsimony and interpretability. Reciprocity and transitivity: if $i\to j$ and $j\to k$ then $d_{ij},d_{jk}$ are probably small, so $d_{ji}$ (reciprocity) and $d_{ik}$ (transitivity) are small.

Uncovariate parameterisation and $d_k$-representability (eq. 2-3)

For undirected ties with no covariates, $\text{logodds}(y_{ij}=1\mid d_{ij},\alpha )=\alpha (1-d_{ij})$. A set of distances represents $Y$ if $d_{ij}>1$ whenever $y_{ij}=0$ and $d_{ij}<1$ whenever $y_{ij}=1$; as $\alpha \to \infty$ the data probability tends to 1. A network is $d_k$ representable if there are $z_i\in {\mathbb{R}}^k$ with these distance properties, i.e. $i\sim j$ iff $i,j$ lie within $k$-dimensional unit balls centred at each other.

Examples of low-dimensional representability: an $n$-star is trivially $d_2$ representable (periphery on two opposite sides of the hub); an $n$-chain $1\sim 2\sim \cdots \sim n$ is $d_2$ representable for every $n$ (equidistant positions on a spiral/line segment). Model complexity: $[nk]$ position parameters versus $n(n-1)$ data.

Projection model (Sec. 2.2)

To allow $p(i\to j)\ne p(j\to i)$, give each actor an unit-length $k$-vector $v_i$ and activity $a_i>0$, $z_i=a_i{v}_i$ ($v_i^{\top }{v}_j>0$, $=0$, $<0$ means the characteristics point in the same, orthogonal, opposite direction). Then

$$\text{logodds}(y_{ij}=1\mid z_i,z_j,x_{ij},\alpha ,\beta )=\alpha +{\beta }^{\top }{x}_{ij}+\frac{z_i^{\top }{z}_j}{|z_j|}.$$

The term is the signed magnitude of the projection of $z_i$ on the direction of $z_j$. Variant: acceptance rates via $z_i^{\top }{z}_j/|z_i|$.

Examples

Simulate from a latent distance model and check transitivity

Setup: 30 actors, $z_i\sim N(0,I_2)$, $\alpha =2$; compare transitive triad count to an Erdos-Renyi graph with the same density.

import numpy as np
rng = np.random.default_rng(1); n = 30; Z = rng.normal(size=(n, 2)); alpha = 2.0
D = np.linalg.norm(Z[:, None] - Z[None], axis=2); p = 1/(1 + np.exp(-(alpha - D)))
A = (rng.uniform(size=(n, n)) < np.triu(p, 1)); A = (A | A.T).astype(int)
tri = np.trace(np.linalg.matrix_power(A, 3))/6
dens = A.sum()/(n*(n-1)); ER = (rng.uniform(size=(n, n)) < dens); ER = np.triu(ER, 1); ER = (ER | ER.T).astype(int)
print("triangles latent:", tri, " ER:", np.trace(np.linalg.matrix_power(ER, 3))/6)

PyMC sketch of the distance model (untested):

import pymc as pm, pytensor.tensor as pt
with pm.Model():
    alpha = pm.Normal("alpha", 0, 3); Z = pm.Normal("Z", 0, 1, shape=(n, 2))
    D = pt.sqrt(((Z[:, None, :] - Z[None, :, :])**2).sum(-1) + 1e-9)
    eta = alpha - D
    pm.Bernoulli("y", logit_p=eta[iu], observed=y[iu])    # iu: i<j indices for undirected

Remember positions are only identified up to rotation/reflection/translation (see estimation note).

Connections

Hoff 2002 - Likelihood Estimation and Procrustes MCMC handles non-identifiability. For ICAR/graph smoothing the contrast is Riebler 2016 - Scaling and the BYM2 Construction; basic network GLMs: Social Network Models.

See Also

Hoff 2002 - Latent Space Models Overview