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Fitting an ERGM and Viewing the Output with tabulergm

Source: vignettes/ergm-with-tabulergm.Rmd
ergm-with-tabulergm.Rmd

Overview

This vignette demonstrates how to:

  1. Fit an ERGM using the ergm package.
  2. Produce a publication-ready Markdown table with tabulergm_table(format = "markdown").
  3. Render the table inline in a Quarto or R Markdown document using results: asis.
  4. Replace the shipped term titles, descriptions, and citations for a single table.
  5. Interactively preview the table in the RStudio viewer (or a browser) with tabulergm_view().

Setup

Fitting the model

We use the Florentine marriage network, which ships with ergm, and fit a simple model with an edges term and a nodematch term for wealth quartile.

data(florentine)

model <- ergm(
  flomarriage ~ edges + nodematch("wealth"),
  control = control.ergm(seed = 42)
)
summary(model)
#> Call:
#> ergm(formula = flomarriage ~ edges + nodematch("wealth"), control = control.ergm(seed = 42))
#> 
#> Maximum Likelihood Results:
#> 
#>                  Estimate Std. Error MCMC % z value Pr(>|z|)    
#> edges             -1.5892     0.2454      0  -6.477   <1e-04 ***
#> nodematch.wealth     -Inf     0.0000      0    -Inf   <1e-04 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#>      Null Deviance: 166.4  on 120  degrees of freedom
#>  Residual Deviance: 107.4  on 118  degrees of freedom
#>  
#> AIC: 109.4  BIC: 112.2  (Smaller is better. MC Std. Err. = 0)
#> Warnings:
#> 
#>  * The following terms have infinite coefficient estimates due to an
#>    extreme sufficient statistic:
#> 
#>    nodematch.wealth

Creating a Markdown table

Calling tabulergm_table() with format = "markdown" returns a knitr_kable object. Adding the chunk option results: asis (or results = "asis" in R Markdown) causes knitr to emit the table verbatim, so the Markdown renderer (Quarto, Pandoc, GitHub, etc.) formats it properly.

Math notation in the math column is automatically wrapped in $...$ so that Pandoc can render it reliably in table cells across output formats, including Word. Network figures in the figure column are emitted with Markdown image syntax.

Quarto tip: use #| results: asis (or results = "asis" in R Markdown) on the chunk so that knitr emits the table verbatim instead of quoting it.

tabulergm_table(
  model,
  include_math        = TRUE,
  include_description = TRUE,
  format              = "markdown"
)
term figure estimate se pvalue description math
edges -1.59 0.25 <0.01 Counts the ties present in the network. Acts as the baseline density term, playing the role an intercept plays in a regression model. (holland1981) βˆ‘i<jyij\sum_{i<j} y_{ij}
nodematch -Inf 0.00 <0.01 Counts the ties joining nodes that share the same value of a categorical attribute, the standard measure of homophily. Setting diff = TRUE adds one statistic per attribute value (differential homophily). (wasserman1996; mcpherson2001) βˆ‘i<jyij𝟏(xi=xj)\sum_{i<j} y_{ij} \mathbf{1}(x_i = x_j)

Note: Orange nodes indicate nodes with a focal attribute.

holland1981holland1981doi:10.1080/01621459.1981.10477598
wasserman1996wasserman1996doi:10.1007/BF02294547
mcpherson2001mcpherson2001doi:10.1146/annurev.soc.27.1.415

Customizing titles, descriptions, and citations

Each term carries a short title and a plain-language description, taken from tabulergm’s term dictionary and falling back to the ergm term database for terms the dictionary does not cover. Add the title column with include_title = TRUE, and replace either field for a single table with the override.* arguments:

tabulergm_table(
  model,
  include_title       = TRUE,
  include_description = TRUE,
  override.title      = c(edges = "Density"),
  override.desc       = c(edges = "Baseline propensity to form ties."),
  format              = "markdown"
)
term title figure estimate se pvalue description
edges Density -1.59 0.25 <0.01 Baseline propensity to form ties. (holland1981)
nodematch Uniform homophily -Inf 0.00 <0.01 Counts the ties joining nodes that share the same value of a categorical attribute, the standard measure of homophily. Setting diff = TRUE adds one statistic per attribute value (differential homophily). (wasserman1996; mcpherson2001)

Note: Orange nodes indicate nodes with a focal attribute.

holland1981holland1981doi:10.1080/01621459.1981.10477598
wasserman1996wasserman1996doi:10.1007/BF02294547
mcpherson2001mcpherson2001doi:10.1146/annurev.soc.27.1.415

override.math, override.figure, and override.citation work the same way, and the single override argument sets several fields at once:

tabulergm_table(
  model,
  override = list(
    edges     = list(title = "Density", desc = "Baseline tie propensity."),
    nodematch = list(citation = "doi:10.1016/S0378-8733(01)00029-6")
  )
)

Override names are matched against the term name first and the coefficient name second, so an expanded coefficient such as nodematch.wealth.3 can be targeted on its own.

Terms with a citation show a (key) marker next to their description, and the matching [key] identifier line is appended below the table. Citations are stored as a DOI, arXiv id, PubMed id, or URL rather than a formatted reference, so readers can import them into their own bibliography software:

tabulergm_table(
  flomarriage ~ gwesp(0.5, fixed = TRUE) + gwdegree(0.5, fixed = TRUE),
  format = "markdown"
)
term figure math description
gwesp exp⁑(Ο„)βˆ‘i=1nβˆ’2[1βˆ’(1βˆ’exp(βˆ’Ο„))i]EPi(y)\exp{(\tau)} \sum_{i=1}^{n-2} \left[1 - \left(1 - \exp{(-\tau)}\right)^i\right] EP_i(y) Summarizes how many partners tied nodes share, weighting each additional shared partner geometrically less than the last. Provides a better-behaved measure of transitive closure than a raw triangle count; the decay parameter controls how fast the weights fall off. (snijders2006; hunter2007)
gwdegree exp⁑(Ο„)βˆ‘i=1nβˆ’1[1βˆ’(1βˆ’exp(βˆ’Ο„))i]Di(y)\exp{(\tau)} \sum_{i=1}^{n-1} \left[1 - \left(1 - \exp{(-\tau)}\right)^i\right] D_i(y) Summarizes the degree distribution with geometrically decreasing weights. Captures whether ties concentrate on a few high-degree nodes or spread evenly, and stabilizes models that would otherwise degenerate. (snijders2006; hunter2007)

snijders2006snijders2006doi:10.1111/j.1467-9531.2006.00176.x
hunter2007hunter2007doi:10.1016/j.socnet.2006.08.005

Reorganizing a table with styles

The default table uses one column per metadata component. For a compact dictionary-style layout, pipe a table through with_style_name_over_formula(). It places the curated title over the formula in a Name cell and puts the term drawing in Representation, while retaining model statistics and any columns explicitly requested from tabulergm_table().

tabulergm_table(
  flomarriage ~ edges + nodematch("wealth") + triangle,
  include_description = FALSE,
  format = "markdown"
) |>
  with_style_name_over_formula()
Name Representation
Number of edges (holland1981)
βˆ‘i<jyij\sum_{i\lt{}j} y_{ij} 		term figure
Uniform homophily (wasserman1996; mcpherson2001)
βˆ‘i<jyij𝟏(xi=xj)\sum_{i\lt{}j} y_{ij} \mathbf{1}(x_i = x_j) 		term figure
Triangles (frank1986)
βˆ‘i<j<kyijyjkyik\sum_{i\lt{}j\lt{}k} y_{ij} y_{jk} y_{ik} 		term figure

Note: Orange nodes indicate nodes with a focal attribute.

[holland1981] doi:10.1080/01621459.1981.10477598
[wasserman1996] doi:10.1007/BF02294547
[mcpherson2001] doi:10.1146/annurev.soc.27.1.415
[frank1986] doi:10.1080/01621459.1986.10478342

Styled Markdown is emitted as a raw HTML table because pipe tables cannot reliably hold the multi-line name/formula cells. This layout is therefore for HTML-capable Markdown targets; use the default plain style for portable pipe tables. with_style_plain() restores the original layout.

The same styled data frame can be saved or previewed through a pipe:

tabulergm_table(model, include_description = FALSE) |>
  with_style_name_over_formula() |>
  tabulergm_save("compact-ergm")

tabulergm_table(model, include_description = FALSE) |>
  with_style_name_over_formula() |>
  tabulergm_view()

The compact LaTeX snippet requires the array, booktabs, and graphicx packages for multiline cells and figures. Fitted-model estimates, standard errors, and p-values display two decimal places by default; use digits = NULL to retain full precision or set another number of decimal places.

Compact-style presentation settings are carried by the table object, so the same layout is used for previews and saved output. column_widths takes named fractions of the table width and figure_height is measured in inches:

tabulergm_table(model, include_description = FALSE) |>
  with_style_name_over_formula(
    column_widths = c(Name = .5, Representation = .2),
    figure_height = .8
  ) |>
  tabulergm_save("compact-ergm")

Inspecting a formula without a fitted model

You can also pass a bare formula to inspect term metadata before fitting:

tabulergm_table(
  flomarriage ~ edges + nodematch("wealth") + triangle,
  format = "markdown"
)
term figure math description
edges βˆ‘i<jyij\sum_{i<j} y_{ij} Counts the ties present in the network. Acts as the baseline density term, playing the role an intercept plays in a regression model. (holland1981)
nodematch βˆ‘i<jyij𝟏(xi=xj)\sum_{i<j} y_{ij} \mathbf{1}(x_i = x_j) Counts the ties joining nodes that share the same value of a categorical attribute, the standard measure of homophily. Setting diff = TRUE adds one statistic per attribute value (differential homophily). (wasserman1996; mcpherson2001)
triangle βˆ‘i<j<kyijyjkyik\sum_{i<j<k} y_{ij} y_{jk} y_{ik} Counts the sets of three mutually connected nodes, the basic measure of local clustering in an undirected network. (frank1986)

Note: Orange nodes indicate nodes with a focal attribute.

holland1981holland1981doi:10.1080/01621459.1981.10477598
wasserman1996wasserman1996doi:10.1007/BF02294547
mcpherson2001mcpherson2001doi:10.1146/annurev.soc.27.1.415
frank1986frank1986doi:10.1080/01621459.1986.10478342

The term dictionary

tabulergm ships math and network drawings for commonly used ERGM terms, including directed variants and mode-specific bipartite terms (b1* terms summarize the first mode, and b2* terms summarize the second mode). The table below covers every term currently included in the dictionary; terms with both directed and undirected definitions (edges, gwesp, gwdsp, isolates) display the undirected version:

dictionary_terms <- network ~
  edges + mutual + triangle +
  gwesp(0.5, fixed = TRUE) + gwdsp(0.5, fixed = TRUE) +
  gwdegree(0.5, fixed = TRUE) + altkstar(2, fixed = TRUE) +
  nodematch("attr") + nodefactor("attr") + nodemix("attr") +
  nodecov("attr") + absdiff("attr") + edgecov("cov") +
  transitiveties + cyclicalties +
  nodeicov("attr") + nodeocov("attr") +
  gwidegree(0.5, fixed = TRUE) + gwodegree(0.5, fixed = TRUE) +
  nodeifactor("attr") + nodeofactor("attr") +
  kstar(2) + istar(2) + ostar(2) +
  isolates + degree(1) + concurrent +
  dgwesp(0.5, fixed = TRUE) + dgwdsp(0.5, fixed = TRUE) +
  gwb1dsp(0.5, fixed = TRUE) + gwb2dsp(0.5, fixed = TRUE) +
  gwb1degree(0.5, fixed = TRUE) + gwb2degree(0.5, fixed = TRUE) +
  b1factor("type") + b2factor("group") +
  b1nodematch("type") + b2nodematch("group") +
  b1starmix(2, "type") + b2starmix(2, "group")

tabulergm_table(dictionary_terms, format = "markdown")
term figure math description
edges βˆ‘i<jyij\sum_{i<j} y_{ij} Counts the ties present in the network. Acts as the baseline density term, playing the role an intercept plays in a regression model. (holland1981)
mutual βˆ‘i<jyijyji\sum_{i<j} y_{ij} y_{ji} Counts the dyads in which both directed ties are present, capturing the tendency for ties to be returned. (holland1981)
triangle βˆ‘i<j<kyijyjkyik\sum_{i<j<k} y_{ij} y_{jk} y_{ik} Counts the sets of three mutually connected nodes, the basic measure of local clustering in an undirected network. (frank1986)
gwesp exp⁑(Ο„)βˆ‘i=1nβˆ’2[1βˆ’(1βˆ’exp(βˆ’Ο„))i]EPi(y)\exp{(\tau)} \sum_{i=1}^{n-2} \left[1 - \left(1 - \exp{(-\tau)}\right)^i\right] EP_i(y) Summarizes how many partners tied nodes share, weighting each additional shared partner geometrically less than the last. Provides a better-behaved measure of transitive closure than a raw triangle count; the decay parameter controls how fast the weights fall off. (snijders2006; hunter2007)
gwdsp exp⁑(Ο„)βˆ‘i=1nβˆ’2[1βˆ’(1βˆ’exp(βˆ’Ο„))i]DPi(y)\exp{(\tau)} \sum_{i=1}^{n-2} \left[1 - \left(1 - \exp{(-\tau)}\right)^i\right] DP_i(y) Summarizes shared partners over every dyad, tied or not, with geometrically decreasing weights. Commonly paired with gwesp to separate shared partnership from closure itself. (snijders2006; hunter2007)
gwdegree exp⁑(Ο„)βˆ‘i=1nβˆ’1[1βˆ’(1βˆ’exp(βˆ’Ο„))i]Di(y)\exp{(\tau)} \sum_{i=1}^{n-1} \left[1 - \left(1 - \exp{(-\tau)}\right)^i\right] D_i(y) Summarizes the degree distribution with geometrically decreasing weights. Captures whether ties concentrate on a few high-degree nodes or spread evenly, and stabilizes models that would otherwise degenerate. (snijders2006; hunter2007)
altkstar βˆ‘k=2nβˆ’1(βˆ’1)kSk(y)Ξ»kβˆ’2\sum_{k=2}^{n-1} (-1)^k \frac{S_k(y)}{\lambda^{k-2}} Alternating sum of the k-star counts, an equivalent parameterization of the geometrically weighted degree distribution used to model degree heterogeneity. (snijders2006; hunter2007)
nodematch βˆ‘i<jyij𝟏(xi=xj)\sum_{i<j} y_{ij} \mathbf{1}(x_i = x_j) Counts the ties joining nodes that share the same value of a categorical attribute, the standard measure of homophily. Setting diff = TRUE adds one statistic per attribute value (differential homophily). (wasserman1996; mcpherson2001)
nodefactor βˆ‘i<jyij[𝟏(xi=k)+𝟏(xj=k)]\sum_{i<j} y_{ij} \left[\mathbf{1}(x_i = k) + \mathbf{1}(x_j = k)\right] Counts the tie endpoints belonging to each level of a categorical attribute, measuring how active nodes with that value are in forming ties.
nodemix βˆ‘i<jyij𝟏({xi,xj}={k,l})\sum_{i<j} y_{ij} \mathbf{1}(\{x_i, x_j\} = \{k, l\}) Counts the ties for every pairing of attribute values, reproducing the full mixing matrix of a categorical attribute. (wasserman1996; morris1991)
nodecov βˆ‘i<jyij(xi+xj)\sum_{i<j} y_{ij} (x_i + x_j) Sums a quantitative attribute over both ends of each tie, measuring how strongly that attribute drives tie formation.
absdiff βˆ‘i<jyij|xiβˆ’xj|\sum_{i<j} y_{ij} \left\lvert{}x_i - x_j\right\rvert{} Sums the absolute difference in a quantitative attribute across tied nodes. Negative estimates indicate homophily, since similar nodes contribute less.
edgecov βˆ‘i<jyijxij\sum_{i<j} y_{ij} x_{ij} Sums a fixed dyad-level covariate over the observed ties, letting an external matrix such as distance or a previously observed network predict tie formation. (wasserman1996)
transitiveties βˆ‘iβ‰ jyij𝟏(βˆƒk:yikykj=1)\sum_{i \neq j} y_{ij} \mathbf{1}\left(\exists k : y_{ik} y_{kj} = 1\right) Counts the ties closed by at least one two-path. Unlike a triple count, a tie contributes once no matter how many shared partners it has.
cyclicalties βˆ‘iβ‰ jyij𝟏(βˆƒk:yjkyki=1)\sum_{i \neq j} y_{ij} \mathbf{1}\left(\exists k : y_{jk} y_{ki} = 1\right) Counts the ties that take part in at least one cycle, capturing generalized exchange rather than hierarchy.
nodeicov βˆ‘iβ‰ jyijxj\sum_{i \neq j} y_{ij} x_j Sums the receiving node’s attribute value over all ties, measuring how a quantitative attribute drives incoming ties (popularity).
nodeocov βˆ‘iβ‰ jyijxi\sum_{i \neq j} y_{ij} x_i Sums the sending node’s attribute value over all ties, measuring how a quantitative attribute drives outgoing ties (activity).
gwidegree exp⁑(Ο„)βˆ‘i=1nβˆ’1[1βˆ’(1βˆ’exp(βˆ’Ο„))i]Diin(y)\exp{(\tau)} \sum_{i=1}^{n-1} \left[1 - \left(1 - \exp{(-\tau)}\right)^i\right] D^{\mathrm{in}}_i(y) Summarizes the in-degree distribution with geometrically decreasing weights. Captures whether incoming ties concentrate on a few popular nodes or spread evenly across receivers. (hunter2007; robins2009)
gwodegree exp⁑(Ο„)βˆ‘i=1nβˆ’1[1βˆ’(1βˆ’exp(βˆ’Ο„))i]Diout(y)\exp{(\tau)} \sum_{i=1}^{n-1} \left[1 - \left(1 - \exp{(-\tau)}\right)^i\right] D^{\mathrm{out}}_i(y) Summarizes the out-degree distribution with geometrically decreasing weights. Captures whether outgoing ties concentrate on a few highly active nodes or spread evenly across senders. (hunter2007; robins2009)
nodeifactor βˆ‘iβ‰ jyij𝟏(xj=k)\sum_{i \neq j} y_{ij} \mathbf{1}(x_j = k) Counts the incoming ties received by nodes at each level of a categorical attribute, measuring how popular nodes with that value are as receivers.
nodeofactor βˆ‘iβ‰ jyij𝟏(xi=k)\sum_{i \neq j} y_{ij} \mathbf{1}(x_i = k) Counts the outgoing ties sent by nodes at each level of a categorical attribute, measuring how active nodes with that value are as senders.
kstar βˆ‘i(βˆ‘jβ‰ iyijk)\sum_{i} \binom{\sum_{j \neq i} y_{ij}}{k} Counts the sets of k ties that share a common node, a Markov dependence measure of degree heterogeneity. Pass several values of k to include one statistic per star size. (frank1986)
istar βˆ‘j(βˆ‘iβ‰ jyijk)\sum_{j} \binom{\sum_{i \neq j} y_{ij}}{k} Counts the sets of k incoming ties that share a common receiver, capturing the spread of in-degrees (popularity). Pass several values of k to include one statistic per star size. (wasserman1996)
ostar βˆ‘i(βˆ‘jβ‰ iyijk)\sum_{i} \binom{\sum_{j \neq i} y_{ij}}{k} Counts the sets of k outgoing ties that share a common sender, capturing the spread of out-degrees (activity). Pass several values of k to include one statistic per star size. (wasserman1996)
isolates βˆ‘i𝟏(βˆ‘jβ‰ iyij=0)\sum_{i} \mathbf{1}\left(\sum_{j \neq i} y_{ij} = 0\right) Counts the nodes with no ties, capturing an excess (or shortage) of isolated nodes relative to the rest of the degree distribution.
degree βˆ‘i𝟏(βˆ‘jβ‰ iyij=d)\sum_{i} \mathbf{1}\left(\sum_{j \neq i} y_{ij} = d\right) Counts the nodes with exactly d ties. Pass several values of d to include one statistic per degree, e.g.Β to model low-degree nodes explicitly.
concurrent βˆ‘i𝟏(βˆ‘jβ‰ iyijβ‰₯2)\sum_{i} \mathbf{1}\left(\sum_{j \neq i} y_{ij} \geq 2\right) Counts the nodes with two or more ties, the number of actors holding concurrent partnerships. Common in models of sexual networks and disease transmission.
dgwesp exp⁑(Ο„)βˆ‘i=1nβˆ’2[1βˆ’(1βˆ’exp(βˆ’Ο„))i]EPiOTP(y)\exp{(\tau)} \sum_{i=1}^{n-2} \left[1 - \left(1 - \exp{(-\tau)}\right)^i\right] EP^{\mathrm{OTP}}_i(y) Directed counterpart of the edgewise shared partner statistic, measuring transitive closure with geometrically decreasing weights. Outgoing two-paths are counted by default; the term’s type argument selects a different two-path orientation. (hunter2007; robins2009)
dgwdsp exp⁑(Ο„)βˆ‘i=1nβˆ’2[1βˆ’(1βˆ’exp(βˆ’Ο„))i]DPiOTP(y)\exp{(\tau)} \sum_{i=1}^{n-2} \left[1 - \left(1 - \exp{(-\tau)}\right)^i\right] DP^{\mathrm{OTP}}_i(y) Directed counterpart of the dyadwise shared partner statistic, computed over every ordered dyad. Outgoing two-paths are counted by default; the term’s type argument selects a different two-path orientation. (hunter2007; robins2009)
gwb1dsp exp⁑(Ο„)βˆ‘i=1nB2[1βˆ’(1βˆ’exp(βˆ’Ο„))i]DPi(y)\exp{(\tau)} \sum_{i=1}^{n_{B_2}} \left[1 - \left(1 - \exp{(-\tau)}\right)^i\right] DP_i(y) Summarizes how many second-mode nodes each pair of first-mode nodes has in common, weighting additional shared partners geometrically less. (wang2009; hunter2007)
gwb2dsp exp⁑(Ο„)βˆ‘i=1nB1[1βˆ’(1βˆ’exp(βˆ’Ο„))i]DPi(y)\exp{(\tau)} \sum_{i=1}^{n_{B_1}} \left[1 - \left(1 - \exp{(-\tau)}\right)^i\right] DP_i(y) Summarizes how many first-mode nodes each pair of second-mode nodes has in common, weighting additional shared partners geometrically less. (wang2009; hunter2007)
gwb1degree exp⁑(Ο„)βˆ‘i=1nB2[1βˆ’(1βˆ’exp(βˆ’Ο„))i]DiB1(y)\exp{(\tau)} \sum_{i=1}^{n_{B_2}} \left[1 - \left(1 - \exp{(-\tau)}\right)^i\right] D^{B_1}_i(y) Summarizes the degree distribution of first-mode nodes with geometrically decreasing weights. Captures whether ties to the second mode concentrate on a few highly active first-mode nodes or spread evenly across them. (wang2009; hunter2007)
gwb2degree exp⁑(Ο„)βˆ‘i=1nB1[1βˆ’(1βˆ’exp(βˆ’Ο„))i]DiB2(y)\exp{(\tau)} \sum_{i=1}^{n_{B_1}} \left[1 - \left(1 - \exp{(-\tau)}\right)^i\right] D^{B_2}_i(y) Summarizes the degree distribution of second-mode nodes with geometrically decreasing weights. Captures whether ties from the first mode concentrate on a few popular second-mode nodes or spread evenly across them. (wang2009; hunter2007)
b1factor βˆ‘i∈B1βˆ‘j∈B2yij𝟏(xi=k)\sum_{i \in B_1} \sum_{j \in B_2} y_{ij} \mathbf{1}(x_i = k) Counts the ties incident on first-mode nodes at each level of a categorical attribute, measuring how active those nodes are.
b2factor βˆ‘i∈B1βˆ‘j∈B2yij𝟏(xj=k)\sum_{i \in B_1} \sum_{j \in B_2} y_{ij} \mathbf{1}(x_j = k) Counts the ties incident on second-mode nodes at each level of a categorical attribute, measuring how active those nodes are.
b1nodematch βˆ‘k∈B2βˆ‘i<j∈B1𝟏(xi=xj)yikyjk\sum_{k\in B_2} \sum_{i<j \in B_1} \mathbf{1}(x_i = x_j) y_{ik} y_{jk} Counts the pairs of first-mode nodes that share an attribute value and are both tied to the same second-mode node. The alpha and beta discount parameters temper the count when nodes share many partners. (bomiriya2023)
b2nodematch βˆ‘k∈B1βˆ‘i<j∈B2𝟏(xi=xj)yikyjk\sum_{k\in B_1} \sum_{i<j \in B_2} \mathbf{1}(x_i = x_j) y_{ik} y_{jk} Counts the pairs of second-mode nodes that share an attribute value and are both tied to the same first-mode node. The alpha and beta discount parameters temper the count when nodes share many partners. (bomiriya2023)
b1starmix βˆ‘i∈B1𝟏(xi=p)βˆ‘j1<β‹―<jk∈B2∏l=1kyijl𝟏(xjl=q)\sum_{i \in B_1} \mathbf{1}(x_i = p) \sum_{j_1 < \cdots < j_k \in B_2} \prod_{l=1}^{k} y_{i j_l} \mathbf{1}(x_{j_l} = q) Counts the k-stars centered on a first-mode node with one attribute value whose second-mode neighbors all share another, capturing mixing and degree together.
b2starmix βˆ‘j∈B2𝟏(xj=p)βˆ‘i1<β‹―<ik∈B1∏l=1kyilj𝟏(xil=q)\sum_{j \in B_2} \mathbf{1}(x_j = p) \sum_{i_1 < \cdots < i_k \in B_1} \prod_{l=1}^{k} y_{i_l j} \mathbf{1}(x_{i_l} = q) Counts the k-stars centered on a second-mode node with one attribute value whose first-mode neighbors all share another, capturing mixing and degree together.

Note: Orange nodes indicate nodes with a focal attribute. Orange and teal nodes represent nodes with different values of the focal attribute. Square nodes represent nodes in the first mode and circle nodes in the second mode.

holland1981holland1981doi:10.1080/01621459.1981.10477598
frank1986frank1986doi:10.1080/01621459.1986.10478342
snijders2006snijders2006doi:10.1111/j.1467-9531.2006.00176.x
hunter2007hunter2007doi:10.1016/j.socnet.2006.08.005
wasserman1996wasserman1996doi:10.1007/BF02294547
mcpherson2001mcpherson2001doi:10.1146/annurev.soc.27.1.415
morris1991morris1991doi:10.1016/0025-5564(91)90014-A
robins2009robins2009doi:10.1016/j.socnet.2008.10.006
wang2009wang2009doi:10.1016/j.socnet.2008.08.002
bomiriya2023bomiriya2023arXiv:2312.05673

Interactive preview with tabulergm_view()

During an interactive session you can call tabulergm_view() to open the table in the RStudio viewer pane or the system browser:

tabulergm_view(model, include_math = TRUE, include_description = TRUE)

tabulergm_view() builds a self-contained HTML page that loads MathJax from a CDN, so LaTeX math and embedded network figures render immediately without any additional setup.