- As function of individual covariates
- e.g. Are girls more popular than boys?

- As function of network structures
- e.g. If Adam is friends with Bill, and Bill is friends with Carl, what can we say about the chances of Adam and Carl being friends?

Suppose we…

- Want to predict dichotomous ties
- e.g. two countries at war

- Predictors are actor attributes
- e.g. government types and difference in defense budgets

- What’s wrong with estimating a logistic regression from an
*n*x*k*matrix like this?

regime 1 | regime 2 | budget diff | war? |
---|---|---|---|

dem | dem | $5e7 | 0 |

dem | theo | $1e9 | 0 |

dem | dict | $2e9 | 1 |

- Even if one is not explicitly interested in how other ties affect the likelihood of a tie, network effects imply a correlation of residuals, so omitting them will produce biased estimates.
- e.g. If the U.S. and U.K. are both at war with Iraq, that fact must be considered for an unbiased estimate of the odds of the U.S. and U.K. being at war with each other.

Cranmer and Desmarais (2011) used ERGM to rebut conventional international relations wisdom on democracies going to war:

Simultaneously model effect of network structures, actor attributes, and relational attributes

Get estimates and uncertainty for each effect

Much like a logit: Change in the (log-odds) likelihood of a tie for a unit change in a predictor

Predictors are network-level statistics, but we think about their changes locally

(See Pierre-Andre’s talk for details)

- Define joint-likelihood of all ties
- Observed set is expectation

Choose set of (\(k\)) sufficient statistics (\(\Gamma\))

- Use MCMC to find parameter values (\(\theta\)) for statistics that maximize the likelihood of the set of observed ties (\(Y_m\))
- MCMC provides confidence intervals

Maximize \(\theta\):

\[ P(Y_m) = \dfrac{exp(-\sum_{j=1}^k\Gamma_{mj}\theta_j)}{\sum_{m=1}^Mexp(-\sum_{j=1}^k\Gamma_{mj}\theta_j)} \]

- From parameter estimates, can estimate the probability of any edge:

\[ P(Y_{ij} | Y_{-ij}, \theta) = logistic \sum_{h=1}^{k}\theta_{h}\delta_{h}^{ij}(Y) \]

That allows us to back out probabilities of interest, e.g.