STERGM approximation formulas

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

The current implementation is a separable dyadic-logistic temporal approximation. It is not a full MCMC-MLE implementation equivalent to R tergm or stergm.

Two-snapshot setup

Given two graph snapshots:

$$G_t$$

and:

$$G_{t+1}$$

each dyad has two observed states:

$$Y_{ij,t}$$

and:

$$Y_{ij,t+1}$$

Formation risk set

The formation model is fit only on dyads absent at time t:

$$Y_{ij,t}=0$$

The formation outcome is:

$$F_{ij,t+1}=Y_{ij,t+1}$$

The fitted dyadic-logistic approximation is:

$$Pr(F_{ij,t+1}=1\mid Y_{ij,t}=0)={\text{logit}}^{-1}\left(\sum _{k=1}^p{\beta }_k^F{x}_{ijk,t}^F\right)$$

Dissolution risk set

The dissolution model is fit only on dyads present at time t:

$$Y_{ij,t}=1$$

The dissolution outcome is:

$$D_{ij,t+1}=1-Y_{ij,t+1}$$

The fitted dyadic-logistic approximation is:

$$Pr(D_{ij,t+1}=1\mid Y_{ij,t}=1)={\text{logit}}^{-1}\left(\sum _{k=1}^p{\beta }_k^D{x}_{ijk,t}^D\right)$$

Next-period simulation

A simulated next-period graph starts from G_t.

For existing ties, dissolution is sampled from:

$$D_{ij,t+1}\sim \text{Bernoulli}(p_{ij}^D)$$

For absent ties, formation is sampled from:

$$F_{ij,t+1}\sim \text{Bernoulli}(p_{ij}^F)$$

where:

$$p_{ij}^D={\text{logit}}^{-1}\left(\sum _{k=1}^p{\beta }_k^D{x}_{ijk,t}^D\right)$$

and:

$$p_{ij}^F={\text{logit}}^{-1}\left(\sum _{k=1}^p{\beta }_k^F{x}_{ijk,t}^F\right)$$

Scope note

The current model captures the separable formation/dissolution framing, but it does not implement full temporal ERGM MCMC-MLE estimation or full equivalence with R tergm / stergm.