# Model Terms in iglm

## Overview

This vignette describes all model terms available in `iglm` (version 1.2.4) for specifying the sufficient statistics of joint network-attribute models. Terms are passed on the right-hand side of the `formula` argument in `iglm()` and govern how individual attributes and network connections jointly determine the log-linear probabilities of the model.

A model in `iglm` decomposes its sufficient statistics into two families:

- Unit-level terms \(g_i(x_i, y_i)\): depend only on unit \(i\)’s own attributes.
- Pair-level terms \(h_{i,j}(x, y, z)\): depend on the connection \(z_{i,j}\) and the attributes of units \(i\) and \(j\) as well as the wider network.

The total sufficient statistic of the model is then \[ S(x, y, z) = \sum_i g_i(x_i, y_i) + \sum_{i \ne j} h_{i,j}(x, y, z). \]

## Key Definitions

Before presenting the individual terms, we introduce the notation used throughout this vignette.

1. Connection Indicators. Let \(z_{i,j} \in \{0,1\}\) denote the binary connection from unit \(i\) to unit \(j\), and let \(c_{i,j}\) indicate whether units \(i\) and \(j\) share a local neighbourhood (i.e., \(\mathbf{N}_i \cap \mathbf{N}_j \neq \emptyset\)).

   - Overlapping connection \(u_{i,j} = c_{i,j}\, z_{i,j}\): a connection between units whose neighbourhoods overlap.
   - Non-overlapping connection \(k_{i,j} = (1 - c_{i,j})\, z_{i,j}\): a connection between units with disjoint neighbourhoods.
   - Mode-selected connection \(e_{i,j}^{(\mathtt{s})}\) for mode `s` \(\in \{\texttt{global},\, \texttt{local},\, \texttt{alocal}\}\): \[ e_{i,j}^{(\mathtt{s})} \;=\; \begin{cases} z_{i,j} & \text{if } \mathtt{s} = \texttt{global} \\ u_{i,j} & \text{if } \mathtt{s} = \texttt{local} \\ k_{i,j} & \text{if } \mathtt{s} = \texttt{alocal} \end{cases} \] > Note: For `gwesp`, `gwdsp`, `gwodegree`, `gwidegree`, `edges_x_match`, and `edges_y_match`, only `mode %in% c("global", "local")` is implemented.

2. Degree Statistics. For unit \(i \in \mathbf{P}\) and mode \(\mathtt{s} \in \{\texttt{global},\, \texttt{local}\}\):

   - Out-degree: \(\operatorname{deg}(i, \mathtt{s}) = \sum_{j \in \mathbf{P} \setminus \{i\}} e_{i,j}^{(\mathtt{s})}\), with shorthand \(\operatorname{deg}(i) = \operatorname{deg}(i, \mathtt{global})\).
   - In-degree: \(\operatorname{ideg}(i, \mathtt{s}) = \sum_{j \in \mathbf{P} \setminus \{i\}} e_{j,i}^{(\mathtt{s})}\), with shorthand \(\operatorname{ideg}(i) = \operatorname{ideg}(i, \mathtt{global})\).

3. Common Partners (CP). For a dyad \((i,j)\) and mode \(\mathtt{s}\):

   - OTP (Outgoing Two-Paths): \(\operatorname{CP}(i, j, \mathtt{s}, \texttt{OTP}) = \sum_{h \notin \{i,j\}} e_{i,h}^{(\mathtt{s})}\, e_{h,j}^{(\mathtt{s})}\)
   - ISP (Incoming Shared Partners): \(\operatorname{CP}(i, j, \mathtt{s}, \texttt{ISP}) = \sum_{h \notin \{i,j\}} e_{h,i}^{(\mathtt{s})}\, e_{h,j}^{(\mathtt{s})}\)
   - OSP (Outgoing Shared Partners): \(\operatorname{CP}(i, j, \mathtt{s}, \texttt{OSP}) = \sum_{h \notin \{i,j\}} e_{i,h}^{(\mathtt{s})}\, e_{j,h}^{(\mathtt{s})}\)
   - ITP (Incoming Two-Paths): \(\operatorname{CP}(i, j, \mathtt{s}, \texttt{ITP}) = \sum_{h \notin \{i,j\}} e_{h,i}^{(\mathtt{s})}\, e_{j,h}^{(\mathtt{s})}\)
   - Undirected (symmetric): \(\operatorname{CP}(i, j, \mathtt{s}) = \sum_{h \notin \{i,j\}} e_{i,h}^{(\mathtt{s})}\, e_{h,j}^{(\mathtt{s})}\)

4. Geometrically-Weighted Weight. The decay function used by geometrically weighted statistics is: \[ w_k(\alpha) = \exp(\alpha)\Bigl[1 - \bigl(1 - \exp(-\alpha)\bigr)^k\Bigr]. \]

5. Auxiliary Indicators.

   - Undirected indicator \(\mathbb{I}_U(\mathbf{z})\): takes value \(1\) if connections in \(\mathbf{z}\) are undirected, \(0\) otherwise.
   - Transitive connection indicator \(d_{i,j}(\mathbf{z}) = \mathbb{I}(\exists\, k \in \mathbf{N}_i \cap \mathbf{N}_j : z_{i,k} = z_{k,j} = 1)\): equals \(1\) when a locally transitive path from \(i\) to \(j\) exists.

## Category 1: Attribute Dependence Terms (\(g_i\) Terms)

These terms capture how individual predictors \(x_i\) (exogenous) and \(y_i\) (endogenous) relate to each other, without reference to the network.

### `attribute_x`

Description: Intercept for the endogenous \(x\)-attribute.

\[ g_i(x_i, y_i) = x_i \]

    formula <- object ~ attribute_x

### `attribute_y`

Description: Intercept for the endogenous \(y\)-attribute.

\[ g_i(x_i, y_i) = y_i \]

    formula <- object ~ attribute_y

### `cov_x(data = v)`

Description: Effect of a unit-level exogenous covariate \(v_i\) on attribute \(x_i\).

\[ g_i(x_i, y_i) = v_i\, x_i \]

    formula <- object ~ cov_x(data = v)

### `cov_y(data = v)`

Description: Effect of a unit-level exogenous covariate \(v_i\) on attribute \(y_i\).

\[ g_i(x_i, y_i) = v_i\, y_i \]

    formula <- object ~ cov_y(data = v)

### `attribute_xy(mode = "global" | "local" | "alocal")`

Description: Interaction between the two attributes \(x_i\) and \(y_i\), optionally mediated by the neighbourhood structure.

Mode | Formula
--- | ---
`global` | \(x_i\, y_i\)
`local` | \(x_i \sum_{j \in \mathbf{N}_i} y_j + y_i \sum_{j \in \mathbf{N}_i} x_j\)
`alocal` | \(x_i \sum_{j \notin \mathbf{N}_i} y_j + y_i \sum_{j \notin \mathbf{N}_i} x_j\)

    formula <- object ~ attribute_xy(mode = "local")

## Category 2: Network Dependence Terms (\(h_{i,j}\) Terms)

These terms capture how the network topology \(z\) drives edge formation. All are pair-level statistics.

### `degrees`

Description: Node-level degree fixed effects. One parameter per unit, capturing heterogeneity in activity not explained by other terms. Estimation relies on an MM algorithm constraint.

    formula <- object ~ degrees

### `edges(mode = "global" | "local" | "alocal")`

Description: Baseline propensity for a tie \(z_{i,j}\) to form; the network analogue of an intercept.

\[ h_{i,j}(x, y, z) = e_{i,j}^{(\mathtt{s})} \]

Suitable for both directed and undirected networks.

    formula <- object ~ edges(mode = "global")
    formula <- object ~ edges(mode = "local")
    formula <- object ~ edges(mode = "alocal")

### `mutual(mode = "global" | "local" | "alocal")`

Description: Reciprocity in directed networks. Counts pairs where \(i \to j\) and \(j \to i\) both exist (counted once per unordered pair, hence the factor \(1/2\)).

\[ h_{i,j}(x, y, z) = \frac{e_{i,j}^{(\mathtt{s})}\, e_{j,i}^{(\mathtt{s})}}{2} \]

Only valid for directed networks.

    formula <- object ~ mutual(mode = "global")

### `cov_z(data = w, mode = "global" | "local" | "alocal")`

Description: Dyadic covariate — exogenous edge-level covariate \(w_{i,j}\) influences tie formation.

\[ h_{i,j}(x, y, z) = w_{i,j}\, e_{i,j}^{(\mathtt{s})} \]

Suitable for both directed and undirected networks.

    formula <- object ~ cov_z(data = W, mode = "global")

### `cov_z_out(data = v, mode = "global" | "local" | "alocal")`

Description: Sender covariate — exogenous nodal attribute \(v_i\) influences the propensity to send a tie.

\[ h_{i,j}(x, y, z) = v_i\, e_{i,j}^{(\mathtt{s})} \]

Only valid for directed networks.

    formula <- object ~ cov_z_out(data = v, mode = "global")

### `cov_z_in(data = v, mode = "global" | "local" | "alocal")`

Description: Receiver covariate — exogenous nodal attribute \(v_j\) influences the propensity to receive a tie.

\[ h_{i,j}(x, y, z) = v_j\, e_{i,j}^{(\mathtt{s})} \]

Only valid for directed networks.

    formula <- object ~ cov_z_in(data = v, mode = "global")

### `isolates`

Description: Captures the proportion of units with no connections at all (total degree zero).

\[ h_{i,j}(x, y, z) = \mathbb{I}\!\left(\sum_{j \in \mathbf{P} \setminus \{i\}} z_{i,j} + z_{j,i} = 0\right) \]

Suitable for both directed and undirected networks.

    formula <- object ~ isolates

### `nonisolates`

Description: Captures the proportion of units that have at least one connection.

\[ h_{i,j}(x, y, z) = \mathbb{I}\!\left(\sum_{j \in \mathbf{P} \setminus \{i\}} z_{i,j} + z_{j,i} \ne 0\right) \]

Suitable for both directed and undirected networks.

    formula <- object ~ nonisolates

### `gwdegree(mode = "global" | "local", decay = α)`

Description: Geometrically Weighted Degree — captures the overall degree distribution with exponential decay parameter \(\alpha\).

\[ h_{i,j}(x, y, z) = w_{\operatorname{deg}(i)}(\alpha) + w_{\operatorname{deg}(j)}(\alpha) \]

Suitable for both directed and undirected networks. Only `mode %in% c("global", "local")` is available.

    formula <- object ~ gwdegree(mode = "global", decay = 0.5)

### `gwodegree(mode = "global" | "local", decay = α)`

Description: Geometrically Weighted Out-Degree — captures the out-degree distribution in directed networks.

\[ h_{i,j}(x, y, z) = w_{\operatorname{deg}(i,\,\mathtt{s})}(\alpha) \]

Only valid for directed networks. Only `mode %in% c("global", "local")` is available.

    formula <- object ~ gwodegree(mode = "global", decay = 0.5)

### `gwidegree(mode = "global" | "local", decay = α)`

Description: Geometrically Weighted In-Degree — captures the in-degree distribution in directed networks.

\[ h_{i,j}(x, y, z) = w_{\operatorname{ideg}(i,\,\mathtt{s})}(\alpha) \]

Only valid for directed networks. Only `mode %in% c("global", "local")` is available.

    formula <- object ~ gwidegree(mode = "global", decay = 0.5)

### `transitive`

Description: Transitivity indicator — rewards edges that close a locally transitive triple.

\[ h_{i,j}(x, y, z) = d_{i,j}(\mathbf{z})\, z_{i,j} \]

Suitable for both directed and undirected networks.

    formula <- object ~ transitive

### `gwesp_symm(mode = "global" | "local", decay = α)`

Description: Geometrically Weighted Edgewise Shared Partners (undirected) — the classic GWESP statistic for undirected networks.

\[ h_{i,j}(x, y, z) = e_{i,j}^{(\mathtt{s})}\, w_{\operatorname{CP}(i,j,\mathtt{s})}(\alpha) \]

Suitable for undirected networks only.

    formula <- object ~ gwesp_symm(mode = "global", decay = 0.5)

### `gwesp(mode = "global" | "local", type = "OTP" | "ISP" | "OSP" | "ITP", decay = α)`

Description: Geometrically Weighted Edgewise Shared Partners (directed) — conditions shared partners on a specific path type.

\[ h_{i,j}(x, y, z) = e_{i,j}^{(\mathtt{s})}\, w_{\operatorname{CP}(i,j,\mathtt{s},\mathtt{type})}(\alpha) \]

Only valid for directed networks. Only `mode %in% c("global", "local")` is available.

    formula <- object ~ gwesp(mode = "global", type = "OTP", decay = 0.5)

### `gwdsp_symm(mode = "local", decay = α)`

Description: Geometrically Weighted Dyadwise Shared Partners (undirected) — models triadic potential irrespective of the closing edge.

\[ h_{i,j}(x, y, z) = w_{\operatorname{CP}(i,j,\mathtt{local})}(\alpha) \]

Suitable for undirected networks only.

    formula <- object ~ gwdsp_symm(mode = "local", decay = 0.5)

### `gwdsp(mode = "global" | "local", type = "OTP" | "ISP" | "OSP" | "ITP", decay = α)`

Description: Geometrically Weighted Dyadwise Shared Partners (directed) — models directed triadic potential irrespective of the closing edge.

\[ h_{i,j}(x, y, z) = w_{\operatorname{CP}(i,j,\mathtt{s},\mathtt{type})}(\alpha) \]

Only valid for directed networks. Only `mode %in% c("global", "local")` is available.

    formula <- object ~ gwdsp(mode = "global", type = "OTP", decay = 0.5)

## Category 3: Joint Attribute/Network Dependence Terms (\(h_{i,j}\) Terms)

These terms capture the interplay between nodal attributes and network position. They are the key building blocks for studying spillover effects.

### `attribute_xz(mode = "local")`

Description: Additive effect of \(x_i\) and \(x_j\) on local edge formation.

\[ h_{i,j}(x, y, z) = (x_i + x_j)\, u_{i,j} \]

Suitable for both directed and undirected networks.

    formula <- object ~ attribute_xz(mode = "local")

### `attribute_yz(mode = "local")`

Description: Additive effect of \(y_i\) and \(y_j\) on local edge formation.

\[ h_{i,j}(x, y, z) = (y_i + y_j)\, u_{i,j} \]

Suitable for both directed and undirected networks.

    formula <- object ~ attribute_yz(mode = "local")

### `edges_x_match(mode = "global" | "local")`

Description: Homophily on \(x\) — rewards edges between units with equal \(x\)-values.

\[ h_{i,j}(x, y, z) = \mathbb{I}(x_i = x_j)\, e_{i,j}^{(\mathtt{s})} \]

Suitable for both directed and undirected networks.

    formula <- object ~ edges_x_match(mode = "global")

### `edges_y_match(mode = "global" | "local")`

Description: Homophily on \(y\) — rewards edges between units with equal \(y\)-values.

\[ h_{i,j}(x, y, z) = \mathbb{I}(y_i = y_j)\, e_{i,j}^{(\mathtt{s})} \]

Suitable for both directed and undirected networks.

    formula <- object ~ edges_y_match(mode = "global")

### `outedges_x(mode = "global" | "local" | "alocal")`

Description: Effect of sender attribute \(x_i\) on out-degree formation.

\[ h_{i,j}(x, y, z) = x_i\, e_{i,j}^{(\mathtt{s})} \]

Only valid for directed networks.

    formula <- object ~ outedges_x(mode = "global")

### `inedges_x(mode = "global" | "local" | "alocal")`

Description: Effect of receiver attribute \(x_j\) on in-degree reception.

\[ h_{i,j}(x, y, z) = x_j\, e_{i,j}^{(\mathtt{s})} \]

Only valid for directed networks.

    formula <- object ~ inedges_x(mode = "global")

### `outedges_y(mode = "global" | "local" | "alocal")`

Description: Effect of sender attribute \(y_i\) on out-degree formation.

\[ h_{i,j}(x, y, z) = y_i\, e_{i,j}^{(\mathtt{s})} \]

Only valid for directed networks.

    formula <- object ~ outedges_y(mode = "global")

### `inedges_y(mode = "global" | "local" | "alocal")`

Description: Effect of receiver attribute \(y_j\) on in-degree reception.

\[ h_{i,j}(x, y, z) = y_j\, e_{i,j}^{(\mathtt{s})} \]

Only valid for directed networks.

    formula <- object ~ inedges_y(mode = "global")

### `spillover_xx(mode = "local")`

Description: Symmetric \(x\)-to-\(x\) spillover — the product \(x_i x_j\) along local connections, capturing peer effects in the \(x\) attribute.

\[ h_{i,j}(x, y, z) = x_i\, x_j\, u_{i,j} \]

Suitable for both directed and undirected networks.

    formula <- object ~ spillover_xx(mode = "local")

### `spillover_xx_scaled(mode = "global" | "local")`

Description: Degree-normalised \(x\)-to-\(x\) spillover, accounting for the number of neighbours.

\[ h_{i,j}(x, y, z) = \left(\frac{x_i\, x_j}{\operatorname{deg}(i,\mathtt{s})} + \frac{x_j\, x_i}{\operatorname{deg}(j,\mathtt{s})}\,\mathbb{I}_U(\mathbf{z})\right) e_{i,j}^{(\mathtt{s})} \]

Suitable for both directed and undirected networks.

    formula <- object ~ spillover_xx_scaled(mode = "global")

### `spillover_yy(mode = "local")`

Description: Symmetric \(y\)-to-\(y\) spillover — the product \(y_i y_j\) along local connections.

\[ h_{i,j}(x, y, z) = y_i\, y_j\, u_{i,j} \]

Suitable for both directed and undirected networks.

    formula <- object ~ spillover_yy(mode = "local")

### `spillover_yy_scaled(mode = "global" | "local")`

Description: Degree-normalised \(y\)-to-\(y\) spillover.

\[ h_{i,j}(x, y, z) = \left(\frac{y_i\, y_j}{\operatorname{deg}(i,\mathtt{s})} + \frac{y_j\, y_i}{\operatorname{deg}(j,\mathtt{s})}\,\mathbb{I}_U(\mathbf{z})\right) e_{i,j}^{(\mathtt{s})} \]

Suitable for both directed and undirected networks.

    formula <- object ~ spillover_yy_scaled(mode = "global")

### `spillover_xy(mode = "local")`

Description: Symmetric cross-attribute spillover — \(x_i \to y_j\) and \(x_j \to y_i\) along local connections. For undirected networks both directions are summed.

\[ h_{i,j}(x, y, z) = x_i\, y_j\, u_{i,j} + x_j\, y_i\, u_{i,j}\, \mathbb{I}_U(\mathbf{z}) \]

Suitable for both directed and undirected networks.

    formula <- object ~ spillover_xy(mode = "local")

### `spillover_xy_scaled(mode = "global" | "local")`

Description: Degree-normalised symmetric cross-attribute spillover (\(x \to y\)).

\[ h_{i,j}(x, y, z) = \left(\frac{x_i\, y_j}{\operatorname{deg}(i,\mathtt{s})} + \frac{x_j\, y_i}{\operatorname{deg}(j,\mathtt{s})}\,\mathbb{I}_U(\mathbf{z})\right) e_{i,j}^{(\mathtt{s})} \]

Suitable for both directed and undirected networks.

    formula <- object ~ spillover_xy_scaled(mode = "global")

### `spillover_yx(mode = "local")`

Description: Directed cross-attribute spillover — \(y_i \to x_j\) only (no symmetrisation). Only for directed networks.

\[ h_{i,j}(x, y, z) = y_i\, x_j\, u_{i,j} \]

Only valid for directed networks.

    formula <- object ~ spillover_yx(mode = "local")

### `spillover_yx_scaled(mode = "global" | "local")`

Description: Degree-normalised cross-attribute spillover (\(y \to x\)), with symmetrisation for undirected networks.

\[ h_{i,j}(x, y, z) = \left(\frac{y_i\, x_j}{\operatorname{deg}(i,\mathtt{s})} + \frac{y_j\, x_i}{\operatorname{deg}(j,\mathtt{s})}\,\mathbb{I}_U(\mathbf{z})\right) e_{i,j}^{(\mathtt{s})} \]

Suitable for both directed and undirected networks.

    formula <- object ~ spillover_yx_scaled(mode = "global")

### `spillover_yc(mode = "local", data = v)`

Description: Interaction of endogenous attribute \(y\) with exogenous covariate \(v\) along overlapping connections, with symmetrisation for undirected networks.

\[ h_{i,j}(x, y, z) = c_{i,j}\bigl(v_j\, y_i + \mathbb{I}_U(\mathbf{z})\, v_i\, y_j\bigr)\, z_{i,j} \]

Suitable for both directed and undirected networks.

    formula <- object ~ spillover_yc(data = v, mode = "local")

## Quick-Reference Table

The table below summarises all terms, their mathematical definitions, and whether they support undirected networks.

Term | Definition | Undirected
--- | --- | ---
`attribute_x` | \(x_i\) | ✓
`attribute_y` | \(y_i\) | ✓
`cov_x` | \(v_i\, x_i\) | ✓
`cov_y` | \(v_i\, y_i\) | ✓
`attribute_xy(mode = "global")` | \(x_i\, y_i\) | ✓
`attribute_xy(mode = "local")` | \(x_i \sum_{j \in \mathbf{N}_i} y_j + y_i \sum_{j \in \mathbf{N}_i} x_j\) | ✓
`attribute_xy(mode = "alocal")` | \(x_i \sum_{j \notin \mathbf{N}_i} y_j + y_i \sum_{j \notin \mathbf{N}_i} x_j\) | ✓
`degrees` | Degree fixed effects | ✓
`edges(mode = "s")` | \(e_{i,j}^{(\mathtt{s})}\) | ✓
`mutual(mode = "s")` | \(e_{i,j}^{(\mathtt{s})}\,e_{j,i}^{(\mathtt{s})}/2\) | ✗
`cov_z(mode = "s")` | \(w_{i,j}\, e_{i,j}^{(\mathtt{s})}\) | ✓
`cov_z_out(mode = "s")` | \(v_i\, e_{i,j}^{(\mathtt{s})}\) | ✗
`cov_z_in(mode = "s")` | \(v_j\, e_{i,j}^{(\mathtt{s})}\) | ✗
`isolates` | \(\mathbb{I}(\sum_j z_{i,j}+z_{j,i}=0)\) | ✓
`nonisolates` | \(\mathbb{I}(\sum_j z_{i,j}+z_{j,i}\ne 0)\) | ✓
`gwdegree(mode = "global")` | \(w_{\deg(i)}(\alpha)+w_{\deg(j)}(\alpha)\) | ✓
`gwodegree(mode = "s")` | \(w_{\deg(i,\mathtt{s})}(\alpha)\) | ✗
`gwidegree(mode = "s")` | \(w_{\operatorname{ideg}(i,\mathtt{s})}(\alpha)\) | ✗
`transitive` | \(d_{i,j}(\mathbf{z})\,z_{i,j}\) | ✓
`gwesp_symm(mode = "s")` | \(e_{i,j}^{(\mathtt{s})}\,w_{\operatorname{CP}(i,j,\mathtt{s})}(\alpha)\) | ✓
`gwesp(mode = "s", type = "…")` | \(e_{i,j}^{(\mathtt{s})}\,w_{\operatorname{CP}(i,j,\mathtt{s},\mathtt{type})}(\alpha)\) | ✗
`gwdsp_symm(mode = "local")` | \(w_{\operatorname{CP}(i,j,\mathtt{local})}(\alpha)\) | ✓
`gwdsp(mode = "s", type = "…")` | \(w_{\operatorname{CP}(i,j,\mathtt{s},\mathtt{type})}(\alpha)\) | ✗
`attribute_xz(mode = "local")` | \((x_i+x_j)\,u_{i,j}\) | ✓
`attribute_yz(mode = "local")` | \((y_i+y_j)\,u_{i,j}\) | ✓
`edges_x_match(mode = "s")` | \(\mathbb{I}(x_i=x_j)\,e_{i,j}^{(\mathtt{s})}\) | ✓
`edges_y_match(mode = "s")` | \(\mathbb{I}(y_i=y_j)\,e_{i,j}^{(\mathtt{s})}\) | ✓
`outedges_x(mode = "s")` | \(x_i\,e_{i,j}^{(\mathtt{s})}\) | ✗
`inedges_x(mode = "s")` | \(x_j\,e_{i,j}^{(\mathtt{s})}\) | ✗
`outedges_y(mode = "s")` | \(y_i\,e_{i,j}^{(\mathtt{s})}\) | ✗
`inedges_y(mode = "s")` | \(y_j\,e_{i,j}^{(\mathtt{s})}\) | ✗
`spillover_xx(mode = "local")` | \(x_i\,x_j\,u_{i,j}\) | ✓
`spillover_xx_scaled(mode = "s")` | \(\left(\frac{x_i x_j}{\deg(i,\mathtt{s})}+\frac{x_j x_i}{\deg(j,\mathtt{s})}\mathbb{I}_U\right)e_{i,j}^{(\mathtt{s})}\) | ✓
`spillover_yy(mode = "local")` | \(y_i\,y_j\,u_{i,j}\) | ✓
`spillover_yy_scaled(mode = "s")` | \(\left(\frac{y_i y_j}{\deg(i,\mathtt{s})}+\frac{y_j y_i}{\deg(j,\mathtt{s})}\mathbb{I}_U\right)e_{i,j}^{(\mathtt{s})}\) | ✓
`spillover_xy(mode = "local")` | \(x_i\,y_j\,u_{i,j}+x_j\,y_i\,u_{i,j}\,\mathbb{I}_U\) | ✓
`spillover_xy_scaled(mode = "s")` | \(\left(\frac{x_i y_j}{\deg(i,\mathtt{s})}+\frac{x_j y_i}{\deg(j,\mathtt{s})}\mathbb{I}_U\right)e_{i,j}^{(\mathtt{s})}\) | ✓
`spillover_yx(mode = "local")` | \(y_i\,x_j\,u_{i,j}\) | ✗
`spillover_yx_scaled(mode = "s")` | \(\left(\frac{y_i x_j}{\deg(i,\mathtt{s})}+\frac{y_j x_i}{\deg(j,\mathtt{s})}\mathbb{I}_U\right)e_{i,j}^{(\mathtt{s})}\) | ✓
`spillover_yc(mode = "local")` | \(c_{i,j}(v_j\,y_i+\mathbb{I}_U\,v_i\,y_j)\,z_{i,j}\) | ✓

## Example: Combining Multiple Terms

The example below illustrates how several terms from all three categories can be combined in a single formula for a directed network:

    n_actor <- 100

    # Simulate attributes and neighbourhood
    set.seed(42)
    attribute_info <- rnorm(n_actor)
    block          <- matrix(1, nrow = 10, ncol = 10)
    neighborhood   <- as.matrix(Matrix::bdiag(replicate(10, block, simplify = FALSE)))

    object <- iglm.data(
      neighborhood = neighborhood,
      directed     = TRUE,
      type_x       = "binomial",
      type_y       = "binomial",
      n_actor      = n_actor
    )

    # Formula combining attribute, network, and spillover terms
    formula <- object ~
      # Category 1: attribute dependence
      attribute_x + attribute_y +
      # Category 2: network dependence
      edges(mode = "local") + mutual(mode = "local") +
      gwodegree(mode = "global", decay = 0.5) +
      gwesp(mode = "global", type = "OTP", decay = 0.5) +
      # Category 3: joint attribute/network dependence
      edges_x_match(mode = "local") +
      outedges_y(mode = "local") +
      spillover_yy_scaled(mode = "local")

For further details on model fitting, simulation, and assessment see `vignette("iglm")` and `?iglm-terms`.

## References

Fritz, C., Schweinberger, M., Bhadra, S., and D.R. Hunter (2025). A Regression Framework for Studying Relationships among Attributes under Network Interference. Journal of the American Statistical Association, to appear. doi:10.1080/01621459.2025.2565851

Schweinberger, M. and M.S. Handcock (2015). Local Dependence in Random Graph Models: Characterization, Properties, and Statistical Inference. Journal of the Royal Statistical Society, Series B, 7, 647–676.

Schweinberger, M. and J.R. Stewart (2020). Concentration and Consistency Results for Canonical and Curved Exponential-Family Models of Random Graphs. The Annals of Statistics, 48, 374–396.