Causes of Model Feedback

  • Changes to the node set
    • Demographic churn (birth, death, migration)
      • Deaths and out-migration result in inactive nodes, which also dissolve edges
      • Births and in-migration result in newly active nodes, open for new edges
    • Sometimes, entry and exit from the epidemic-relevant network means something other than birth and death
      • e.g., initiation and cessation of sexual activity
      • We use the terms arrival and departure accordingly
  • Changes to nodal attributes
    • Simulating from an ERGM involves evaluating the current nodal attributes referenced in the formula
      • e.g., preferential mixing on age and disease status with absdiff and nodematch terms
    • These attributes may change over time, in different ways
  • Broader temporal shifts in behavior or biology
    • Monotonic increases in sexual partnership rates
    • Social distancing!

Changing Network Size and Composition

As social networks change in size (say, a village of \(n = 5{,}000\) nodes grows to \(n = 10{,}000\)), which of the following do you think is generally preserved?

  • Number of edges?   \(e\)
  • Mean degree?   \(2e/n\)
  • Density?   \(e / \binom{n}{2}\)

Changing Network Size and Composition

  • Applying the coefficients as-is from a TERGM fit to a network of changing size preserves density across time.
  • For one-mode networks, preserving mean degree instead requires a simple transformation of the edges coefficient in the formation model:

\[\theta_{\text{new}} = \theta_{\text{old}} + \log(N_{\text{old}}) - \log(N_{\text{new}})\]

  • This is mathematically equivalent to partitioning the original edges term into an offset equal to \(\log(N)\) and a residual, then updating the offset as \(N\) changes.

Relational Dissolution through Death

  • We fit our dynamic network using static data, with a process for dissolving relationships governed by a coefficient derived from relational duration.
  • All of this was done in a context with no information about death, another process that terminates relationships.
  • If we simply layer death onto the model (even with the size correction from the previous slide), two measures drop below the values we want: relationship durations and the number of relationships.
  • Some of this might be desired: if we could interview deceased people, we might find their past relationships shorter than those of the same birth cohort still alive in our sample. But others are likely not.

Relational Dissolution through Death

An approximate correction:

  1. Calculate dissolution coefficients as before, ignoring death.
  2. Estimate formation coefficients conditional on those dissolution coefficients.
  3. Recalculate the dissolution coefficients as the log-odds of a relationship persisting conditional on both actors living:

\[\operatorname{logit}\left[\, 1 - \frac{P(E_t) - P(N_t)}{P(\neg N_t)} \,\right]\]

where \(P(E_t) = 1/D\) is the overall probability of a tie dissolving at time \(t\) from any cause, and \(P(N_t)\) is the probability of either incident node dying at time \(t\).

Review of Offsets and Corrections

When approximating the fit of a formation STERGM conditional on a dissolution STERGM… …subtract the dissolution coefficients from the corresponding formation ones (edapprox = TRUE)
When network size \(N\) changes and you want to preserve mean degree… …add the ln of the old \(N\) and subtract the ln of the new \(N\) on the edges coefficient in the formation model (or, equivalently, use an edges offset and update it with the ln of the new \(N\))
To adjust for node departures when simulating from a STERGM estimated from a cross-sectional network and durations \(\operatorname{logit}\left[\, 1 - \dfrac{P(E_t) - P(N_t)}{P(\neg N_t)} \,\right]\)