# Extensions to the Basic ERGM

Now that we have a general understanding with ERGMs and some familiarity with using them in R, we turn to a conceptual discussion of extensions to ERGMs. The purpose of this discussion is to provide you with a preview of what the basic ERGM can do with two tweaks.

# Table of contents

## Adding Constraints

Constraints are researcher-specified dyads where observed outcomes are not modelled as outcomes.

- These dyads are held at their observed values, tie or no tie (i.e. constant or treated as exogeneous)
- This removes the tie from being part of the outcome data (i.e. their rows are removed in the data)
- They still matter to local configuration counts/calculation of change statistics

### They can be used in a number of ways to account for the data generating process

- In temporally-aggregated network, sometimes actors never existed at the same time
- For example, some states from the post-Cold War network never existed at the same time in the global system
- Constraints can also be used to get rid of “outliers” in the data
- For example, in a scientific lab collaboration network, the lab responsible for producing a widely-used reagent might have their ties fixed at the observed state

### More important for the current discussion, constraints can be used systematically to produce specific network types

### Bipartite Networks

Bipartite networks have two different types of nodes; ties can form across types but not within types.

- People attending meetings (e.g. policy actors and shared policy forums)
- Legislators signing bills

#### Constrained to no ties in the main diagonal blocks.

Bipartite networks are very common, and the `ergm`

package has a lot of support for fitting these networks

### (Kind of) Directed Acyclic Graphs

Directed acyclic graphs are directed graphs/networks that have no directed cycles

- In the social sciences, academic citation networks and judicial court citation networks are (kind of) directed acyclic graphs
- In academic publishing, working papers/preprints can lead to reciprocol citations
- In the US Supreme Court, cases decided within the same term can cite each other

#### Constrained to no ties going back in order of publication unless they are within the same temporal period.

#### Resources:

- the
`cERGM`

package

#### Reference:

Schmid, CS et al. 2021. “Generative dynamics of supreme court citations: Analysis with a new statistical network model.” *Political Analysis*.

## Specifying Relational Contexts

Relational contexts are conditions that systematically govern the probability of edge formation in a complex network. In the current discussion, they are researcher-specified.

- Extension of “conditional network effects” discussed in the introduction
- In different relational contexts, network effects are allowed to have different coefficients
- They can be understood as interaction terms
- They can be combined with constraints

### Multilayer Networks

Networks that contain different types of relational contexts systematically organized into layers

- Flexible generalized framework that can account for many different types of network structures
- Network nodes are organized into
*layers*, which represent different kinds of ties - Temporal networks can be understood as multilayer networks where layers are temporal contexts

#### Three relational contexts with no constraints.

- By allowing for different kinds of ties to exist in a network, we can model different things with what are otherwise the same network configurations
- For example, the pile-on effect in conflicts are not likely to be the same across different combinations of actors

#### Resources:

#### Reference:

Chen, THY. 2021. “Statistical inference for multilayer networks in political science”. *Political Science Research and Methods*.

Wang, P et al. 2016 “Social selection models for multilevel networks.” *Social Networks*.

### Temporal Networks

Temporal networks are networks where ties are time-period specific

- They are intuitive to understand as it, but in the context of network analysis, temporal networks can be understood as a special case of the multilayer networks
- The time periods of the ties are the relational contexts

#### As many relational contexts as time periods, constrained to have no ties across periods.

- Only ties within each temporal layer (i.e. period) can exist while the others cells are held to no ties
- This formulation helps with more advanced ERGMs, but for temporal networks where each time period is relatively simple, existing formulations and packages perform very well already

#### Resources:

- the
`xergm`

package (specifically`btergm`

)

#### Reference:

Leifeld, et al. 2018. “Temporal exponential random graph models with btergm: Estimation and bootstrap confidence intervals”. *Journal of Statistical Software*.