iglm-terms: Model Specification for iglm Terms

In iglm: Regression under Interference in Connected Populations

Model Specification for iglm Terms

Description

The help pages of iglm describe the model with details on model fitting and estimation. Generally, a model is specified via it's sufficient statistics, that can be further decomposed into two parts:

• \mathbf{g}_i(x_i^*,y_i^*) = \mathbf{g}_i(x_i,y_i)= (g_i(x_i,y_i)): A vector of unit-level functions (or "g-terms") that describe the relationship between an individual actor i's predictors (x_i) and their own response (y_i).

• \mathbf{h}_{i,j}(x_i^*,x_j^*, y_i^*, y_j^*, z) = \mathbf{h}_{i,j}(x,y,z)= (h_{i,j}(x,y,z)): A vector of pair-level functions (or "h-terms") that specify how the connections (z) and responses (y_i, y_j) of a pair of units \{i,j\} depend on each other and the wider network structure.

Each term defines a component for the model's features, which are a sum of unit-level components, \sum_i g_i(x_i,y_i), and/or pair-level components, \sum_{i \ne j} h_{i,j}(x,y,z). The implemented terms are grouped into three categories:

1. Attribute Terms: Depend only on individual attributes x_i or y_i.

2. Network Terms: Depend only on the connections z_{i,j}.

3. Joint Attribute/Network Terms: Depend on both individual attributes and connections.

degrees: Degrees: Specifies node-level fixed effects. Estimation requires an MM algorithm constraint.

edges(mode = "global"): Edges: Captures the baseline propensity of tie formation z_{i,j}, partitioned by structural boundary c_{i,j}.

• global: h_{i,j}(x,y,z) = z_{i,j}

• local: h_{i,j}(x,y,z) = c_{i,j} z_{i,j}

• alocal: h_{i,j}(x,y,z) = (1 - c_{i,j}) z_{i,j}

mutual(mode = "global"): Mutual Reciprocity: Evaluates reciprocal tie formation in directed networks.

• global: h_{i,j}(x,y,z) = z_{i,j} z_{j,i} (for i < j)

• local: h_{i,j}(x,y,z) = c_{i,j} z_{i,j} z_{j,i} (for i < j)

• alocal: h_{i,j}(x,y,z) = (1 - c_{i,j}) z_{i,j} z_{j,i} (for i < j)

cov_z(data, mode = "global"): Dyadic Covariate: Exogenous dyadic covariate w_{i,j} influence on edge formation.

• global: h_{i,j}(x,y,z) = w_{i,j} z_{i,j}

• local: h_{i,j}(x,y,z) = c_{i,j} w_{i,j} z_{i,j}

• alocal: h_{i,j}(x,y,z) = (1 - c_{i,j}) w_{i,j} z_{i,j}

cov_z_out(data, mode = "global"): Covariate Sender: Exogenous monadic covariate v_{i} influence on generating an outgoing tie.

• global: h_{i,j}(x,y,z) = v_i z_{i,j}

• local: h_{i,j}(x,y,z) = c_{i,j} v_i z_{i,j}

• alocal: h_{i,j}(x,y,z) = (1 - c_{i,j}) v_i z_{i,j}

cov_z_in(data, mode = "global"): Covariate Receiver: Exogenous monadic covariate v_{j} influence on receiving an incoming tie.

• global: h_{i,j}(x,y,z) = v_j z_{i,j}

• local: h_{i,j}(x,y,z) = c_{i,j} v_j z_{i,j}

• alocal: h_{i,j}(x,y,z) = (1 - c_{i,j}) v_j z_{i,j}

cov_x(data = v): Nodal Covariate (X): Effect of a unit-level exogenous covariate v_i on endogenous attribute x_i. g_i(x_i,y_i) = v_i x_i

cov_y(data = v): Nodal Covariate (Y): Effect of a unit-level exogenous covariate v_i on endogenous attribute y_i. g_i(x_i,y_i) = v_i y_i

attribute_xy(mode = "global"): Nodal Attribute Interaction (X-Y): Interaction of attributes x_i and y_i.

• global: g_i(x_i,y_i) = x_i y_i

• local: g_i(x_i,y_i) = x_i \sum_{j \in \mathcal{N}_i} y_j + y_i \sum_{j \in \mathcal{N}_i} x_j

• alocal: g_i(x_i,y_i) = x_i \sum_{j \notin \mathcal{N}_i} y_j + y_i \sum_{j \notin \mathcal{N}_i} x_j

attribute_yz(mode = "local"): Attribute Sum (Y-Z): Models the additive effect of y_i and y_j on edge formation within local neighborhoods.

attribute_xz(mode = "local"): Attribute Sum (X-Z): Models the additive effect of x_i and x_j on edge formation within local neighborhoods.

inedges_y(mode = "global"): Attribute In-Degree (Y-Z): Influence of endogenous y_j on in-degree reception.

• global: h_{i,j}(x,y,z) = y_j z_{i,j}

• local: h_{i,j}(x,y,z) = c_{i,j} y_j z_{i,j}

• alocal: h_{i,j}(x,y,z) = (1 - c_{i,j}) y_j z_{i,j}

outedges_y(mode = "global"): Attribute Out-Degree (Y-Z): Influence of endogenous y_i on out-degree formation.

• global: h_{i,j}(x,y,z) = y_i z_{i,j}

• local: h_{i,j}(x,y,z) = c_{i,j} y_i z_{i,j}

• alocal: h_{i,j}(x,y,z) = (1 - c_{i,j}) y_i z_{i,j}

inedges_x(mode = "global"): Attribute In-Degree (X-Z): Influence of endogenous x_j on in-degree reception.

• global: h_{i,j}(x,y,z) = x_j z_{i,j}

• local: h_{i,j}(x,y,z) = c_{i,j} x_j z_{i,j}

• alocal: h_{i,j}(x,y,z) = (1 - c_{i,j}) x_j z_{i,j}

outedges_x(mode = "global"): Attribute Out-Degree (X-Z): Influence of endogenous x_i on out-degree formation.

• global: h_{i,j}(x,y,z) = x_i z_{i,j}

• local: h_{i,j}(x,y,z) = c_{i,j} x_i z_{i,j}

• alocal: h_{i,j}(x,y,z) = (1 - c_{i,j}) x_i z_{i,j}

attribute_x: Attribute (X): Intercept for attribute x. g_i(x_i,y_i) = x_i

attribute_y: Attribute (Y): Intercept for attribute y. g_i(x_i,y_i) = y_i

edges_x_match(mode = "global"): Attribute Match (X-Z): Models homophily/matching on the binary attribute x.

• global: h_{i,j}(x,y,z) = \mathbb{I}(x_i = x_j) z_{i,j}

• local: h_{i,j}(x,y,z) = c_{i,j} \mathbb{I}(x_i = x_j) z_{i,j}

edges_y_match(mode = "global"): Attribute Match (Y-Z): Models homophily/matching on the binary attribute y.

• global: h_{i,j}(x,y,z) = \mathbb{I}(y_i = y_j) z_{i,j}

• local: h_{i,j}(x,y,z) = c_{i,j} \mathbb{I}(y_i = y_j) z_{i,j}

spillover_yy_scaled(mode = "global"): Scaled Y-Y-Z Outcome Spillover: Normalizes the y-outcome spillover influence by the relevant out-degree topology.

• global: h_{i,j}(x,y,z) = y_i y_j z_{i,j} / \text{deg}(i)

• local: h_{i,j}(x,y,z) = c_{i,j} y_i y_j z_{i,j} / \text{deg}(i, \text{local})

spillover_xx_scaled(mode = "global"): Scaled X-X-Z Outcome Spillover: Normalizes the x-outcome spillover influence by the relevant out-degree topology.

• global: h_{i,j}(x,y,z) = x_i x_j z_{i,j} / \text{deg}(i)

• local: h_{i,j}(x,y,z) = c_{i,j} x_i x_j z_{i,j} / \text{deg}(i, \text{local})

spillover_yx_scaled(mode = "global"): Scaled Y-X-Z Treatment Spillover: Normalizes cross-attribute y_i \to x_j spillover influence.

• global: h_{i,j}(x,y,z) = y_i x_j z_{i,j} / \text{deg}(i)

• local: h_{i,j}(x,y,z) = c_{i,j} y_i x_j z_{i,j} / \text{deg}(i, \text{local})

spillover_xy_scaled(mode = "global"): Scaled X-Y-Z Treatment Spillover: Normalizes cross-attribute x_i \to y_j spillover influence.

• global: h_{i,j}(x,y,z) = x_i y_j z_{i,j} / \text{deg}(i)

• local: h_{i,j}(x,y,z) = c_{i,j} x_i y_j z_{i,j} / \text{deg}(i, \text{local})

gwesp(data, mode = "global", variant = "OSP", decay = 0): Geometrically Weighted Edgewise-Shared Partners: Models triadic closure propensity conditioning on existing edges. Types dictate path constraint: OTP, ITP, OSP, ISP for directed; symm for undirected.

gwdsp(data, mode = "global", variant = "OSP", decay = 0): Geometrically Weighted Dyadwise-Shared Partners: Models triadic potential irrespective of the closing edge. Types dictate path constraint: OTP, ITP, OSP, ISP for directed; symm for undirected.

gwdegree(mode = "global", decay = 0): Geometrically Weighted Degree: Captures the degree distribution utilizing an exponential decay parameter.

gwidegree(mode = "global", decay = 0): Geometrically Weighted In-Degree: Captures the in-degree distribution utilizing an exponential decay parameter.

gwodegree(mode = "global", decay = 0): Geometrically Weighted Out-Degree: Captures the out-degree distribution utilizing an exponential decay parameter.

spillover_yc_symm(data = v, mode = "local"): Symmetric Y-C-Z Treatment Spillover: Bidirectional mapping of exogenous covariate v and endogenous trait y interaction.

spillover_xy(mode = "local"): Directed X-Y-Z Treatment Spillover: Maps cross-attribute x_i \to y_j treatment assignment.

spillover_yc(mode = "local"): Directed Y-C-Z Treatment Spillover: Exogenous covariate v interacting with endogenous trait y.

spillover_yx(mode = "local"): Directed Y-X-Z Treatment Spillover: Maps cross-attribute y_i \to x_j treatment assignment.

spillover_yy(mode = "local"): Symmetric Y-Y-Z Outcome Spillover: Propagates y-outcome spillover effects.

spillover_xx(mode = "local"): Symmetric X-X-Z Outcome Spillover: Propagates x-outcome spillover effects.

transitive: Transitivity (Local): Indicator evaluating the presence of a local transitive triad configuration.

nonisolates: Non-Isolates: Captures frequency of nodes with degree strictly greater than zero.

isolates: Isolates: Captures frequency of nodes with degree zero.

Category 1

Attribute Terms:

Below is a detailed description of terms that depend only on nodal attributes:

• attribute_x-term, attribute_y-term

• cov_x-term, cov_y-term

• attribute_xy-term

Category 2

Network Terms:

Below is a detailed description of terms that depend only on the network structure:

• edges-term, mutual-term

• cov_z-term, cov_z_in-term, cov_z_out-term

• degrees-term

• gwdegree-term, gwidegree-term, gwodegree-term

• gwesp-term, gwdsp-term

• transitive-term, nonisolates-term, isolates-term

Category 3

Joint Attribute/Network Terms:

Below is a detailed description of terms that depend on both attributes and the network:

• attribute_xz-term, attribute_yz-term

• inedges_x-term, inedges_y-term, outedges_x-term, outedges_y-term

• edges_x_match-term, edges_y_match-term

• spillover_xx-term, spillover_yy-term

• spillover_yx-term, spillover_xy-term, spillover_yc-term, spillover_yc_symm-term

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.

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 (Statistical Methodology), 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.

Stewart, J.R. and M. Schweinberger (2025). Pseudo-Likelihood-Based M-Estimation of Random Graphs with Dependent Edges and Parameter Vectors of Increasing Dimension. The Annals of Statistics, to appear.

iglm documentation built on April 23, 2026, 5:07 p.m.