Model Terms in iglm

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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.

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.
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 \(\operatorname{ideg}(i) = \operatorname{ideg}(i, \mathtt{global})\).
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): \(\operatorname{CP}(i, j, \mathtt{s}) = \sum_{h \notin \{i,j\}} e_{i,h}^{(\mathtt{s})}\, e_{h,j}^{(\mathtt{s})}\)
Geometrically-Weighted Weight. The decay function used by geometrically weighted statistics is: \[ w_k(\alpha) = \exp(\alpha)\Bigl[1 - \bigl(1 - \exp(-\alpha)\bigr)^k\Bigr]. \]
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.

These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.