ERGM approximation formulas

This page documents the formulas used by the initial relationalstats.ergm implementation.

The current implementation is a dyadic-logistic approximation. It is not a full MCMC-MLE ERGM engine and should not be described as equivalent to R ergm.

Dyadic logistic approximation

For a candidate dyad (i, j), the model uses:

$$Pr(Y_{ij}=1)={\mathrm{logit}}^{-1}\left(\sum _{k=1}^p{\beta }_k{x}_{ijk}\right)$$

Equivalently:

$$\log \left(\frac{Pr(Y_{ij}=1)}{1-Pr(Y_{ij}=1)}\right)=\sum _{k=1}^p{\beta }_k{x}_{ijk}$$

The implementation uses add_intercept=False internally because the edges term acts as the baseline edge-propensity term.

Edges term

$$x_{ij,\mathrm{edges}}=1$$

Common-neighbor term

$$x_{ij,\mathrm{common}\mathrm{\_}\mathrm{neighbors}}=|N(i)\cap N(j)|$$

Transitivity proxy

$$x_{ij,\mathrm{transitiveties}}=|N(i)\cap N(j)|$$

This is a dyad-level approximation, not a full ERGM change statistic.

Degree-one term

$$x_{ij,\mathrm{degree}1}=\{\begin{array}{ll}1,& \text{if }{d}_i=1\text{ or }{d}_j=1\\ 0,& \text{otherwise}\end{array}$$

GWESP approximation

$$x_{ij,\mathrm{gwesp}}=1-(1-e^{-\alpha }{)}^{s_{ij}}$$

where:

$$s_{ij}=|N(i)\cap N(j)|$$

and the default decay parameter is:

$$\alpha =0.5$$

This approximates the intuition of geometrically weighted shared partners, but it is not the full curved ERGM GWESP term used by R ergm.

Nodematch term

A nodematch term has the form:

nodematch:<attribute>

The dyadic feature is:

$$x_{ij,\mathrm{nodematch}:\mathrm{attr}}=\{\begin{array}{ll}1,& \text{if }{a}_i=a_j\text{ and both values are non-missing}\\ 0,& \text{otherwise}\end{array}$$

GOF-style statistics

The approximate GOF workflow compares observed and simulated networks using:

• edges;
• density;
• average degree;
• triangle count;
• degree distribution;
• edge-wise shared partners;
• geodesic distance distribution with NR for unreachable pairs.