ergm (version 3.9.4)

# ergm-terms: Terms used in Exponential Family Random Graph Models

## Description

The function ergm is used to fit exponential random graph models, in which the probability of a given network, $$y$$, on a set of nodes is $$h(y) \exp\{\eta(\theta) \cdot g(y)\}/c(\theta)$$, where $$h(y)$$ is the reference measure (for valued network models), $$g(y)$$ is a vector of network statistics for $$y$$, $$\eta(\theta)$$ is a natural parameter vector of the same length (with $$\eta(\theta)=\theta$$ for most terms), and $$c(\theta)$$ is the normalizing constant for the distribution.

The network statistics $$g(y)$$ are entered as terms in the function call to ergm.

This page describes the possible terms (and hence network statistics) included in ergm package. Other packages may add their own terms, and package ergm.userterms provides tools for implementing them.

The current recommendation for any package implementing additional terms is to create a help file with a name or alias ergm-terms, so that help("ergm-terms") will list ERGM terms available from all loaded packages.

## Specifying models

Terms to ergm are specified by a formula to represent the network and network statistics. This is done via a formula, that is, an R formula object, of the form y ~ <term 1> + <term 2> ..., where y is a network object or a matrix that can be coerced to a network object, and <term 1>, <term 2>, etc, are each terms chosen from the list given below. To create a network object in R, use the network function, then add nodal attributes to it using the %v% operator if necessary.

## Binary and valued ERGM terms

ergm functions such as ergm and simulate (for ERGMs) may operate in two modes: binary and weighted/valued, with the latter activated by passing a non-NULL value as the response argument, giving the edge attribute name to be modeled/simulated.

Binary ERGM statistics cannot be used in valued mode and vice versa. However, a substantial number of binary ERGM statistics --- particularly the ones with dyadic indepenence --- have simple generalizations to valued ERGMs, and have been adapted in ergm. They have the same form as their binary ERGM counterparts, with an additional argument: form, which, at this time, has two possible values: "sum" (the default) and "nonzero". The former creates a statistic of the form $$\sum_{i,j} x_{i,j} y_{i,j}$$, where $$y_{i,j}$$ is the value of dyad $$(i,j)$$ and $$x_{i,j}$$ is the term's covariate associated with it. The latter computes the binary version, with the edge considered to be present if its value is not 0.

Valued version of some binary ERGM terms have an argument threshold, which sets the value above which a dyad is conidered to have a tie. (Value less than or equal to threshold is considered a nontie.)

## Covariate transformations

Some terms taking nodal or dyadic covariates take optional transform and transformname arguments. transform should be a function with one argument, taking a data structure of the same mode as the covariate and returning a similarly structured data structure, transforming the covariate as needed.

For example, nodecov("a", transform=function(x) x^2) will add a nodal covariate having the square of the value of the nodal attribute "a".

transformname, if given, will be added to the term's name to help identify it.

## Nodal attribute levels

Terms taking a categorical nodal covariate also take levels argument. This can be used to control the set and the ordering of attribute levels.

## Terms to represent network statistics included in the <code><a rd-options="=ergm-package" href="/link/ergm?package=ergm&version=3.9.4&to=%3Dergm-package" data-mini-rdoc="=ergm-package::ergm">ergm</a></code> package

A cross-referenced html version of the term documentation is is available via vignette('ergm-term-crossRef') and terms can also be searched via search.ergmTerms.

%%%%%%%%%%%%%%%%%%%%%%555 % This section below is specially formatted so that it can be parsed to build additional % documentation about ergm terms. (parsing functions are in build_term_index.R) % The list of term tags must appear on the same line as the term name to be parsed correctly %%beginTerms%%
absdiff(attrname, pow=1) (binary) (dyad-independent) (frequently-used) (directed) (undirected) (quantitative nodal attribute), absdiff(attrname, pow=1, form ="sum") (valued) (dyad-independent) (directed) (undirected) (quantitative nodal attribute)

Absolute difference: The attrname argument is a character string giving the name of a quantitative attribute in the network's vertex attribute list. This term adds one network statistic to the model equaling the sum of abs(attrname[i]-attrname[j])^pow for all edges (i,j) in the network.

absdiffcat(attrname, base=NULL) (binary) (dyad-independent) (directed) (undirected) (categorical nodal attribute), absdiffcat(attrname, base=NULL, form="sum") (valued) (dyad-independent) (directed) (undirected) (categorical nodal attribute)

Categorical absolute difference: The attrname argument is a character string giving the name of a quantitative attribute in the network's vertex attribute list. This term adds one statistic for every possible nonzero distinct value of abs(attrname[i]-attrname[j]) in the network; the value of each such statistic is the number of edges in the network with the corresponding absolute difference. The optional base argument is a vector indicating which nonzero differences, in order from smallest to largest, should be omitted from the model (i.e., treated like the zero-difference category). The base argument, if used, should contain indices, not differences themselves. For instance, if the possible values of abs(attrname[i]-attrname[j]) are 0, 0.5, 3, 3.5, and 10, then to omit 0.5 and 10 one should set base=c(1, 4). Note that this term should generally be used only when the quantitative attribute has a limited number of possible values; an example is the "Grade" attribute of the faux.mesa.high or faux.magnolia.high datasets.

altkstar(lambda, fixed=FALSE) (binary) (undirected) (curved) (categorical nodal attribute)

Alternating k-star: This term adds one network statistic to the model equal to a weighted alternating sequence of k-star statistics with weight parameter lambda. This is the version given in Snijders et al. (2006). The gwdegree and altkstar produce mathematically equivalent models, as long as they are used together with the edges (or kstar(1)) term, yet the interpretation of the gwdegree parameters is slightly more straightforward than the interpretation of the altkstar parameters. For this reason, we recommend the use of the gwdegree instead of altkstar. See Section 3 and especially equation (13) of Hunter (2007) for details. The optional argument fixed indicates whether the decay parameter is fixed at the given value, or is to be fit as a curved exponential family model (see Hunter and Handcock, 2006). The default is FALSE, which means the scale parameter is not fixed and thus the model is a CEF model. This term can only be used with undirected networks.

asymmetric(attrname=NULL, diff=FALSE, keep=NULL) (binary) (directed) (dyad-independent) (triad-related)

Asymmetric dyads: This term adds one network statistic to the model equal to the number of pairs of actors for which exactly one of $$(i{\rightarrow}j)$$ or $$(j{\rightarrow}i)$$ exists. This term can only be used with directed networks. If the optional attrname argument is used, only asymmetric pairs that match on the named vertex attribute are counted. The optional modifiers diff and keep are used in the same way as for the nodematch term; refer to this term for details and an example.

atleast(threshold=0) (valued) (directed) (undirected) (dyad-independent)

Number of dyads with values greater than or equal to a threshold Adds one statistic equaling to the number of dyads whose values equal or exceed threshold.

atmost(threshold=0) (valued) (directed) (undirected) (dyad-independent)

Number of dyads with values less than or equal to a threshold Adds one statistic equaling to the number of dyads whose values equal or are exceeded by threshold.

b1concurrent(by=NULL, levels=NULL) (binary) (bipartite) (undirected) (categorical nodal attribute)

Concurrent node count for the first mode in a bipartite (aka two-mode) network: This term adds one network statistic to the model, equal to the number of nodes in the first mode of the network with degree 2 or higher. The first mode of a bipartite network object is sometimes known as the "actor" mode. The optional argument by is a character string giving the name of an attribute in the network's vertex attribute list; it functions just like the by argument of the b1degree term. Without the optional argument, this statistic is equivalent to b1mindegree(2). This term can only be used with undirected bipartite networks.

b1cov(attrname, transform, transformname) (binary) (undirected) (bipartite) (dyad-independent) (quantitative nodalattribute) (frequently-used), b1cov(attrname, transform, transformname, form="sum") (valued) (undirected) (bipartite) (dyad-independent) (quantitative nodal attribute) (frequently-used)

Main effect of a covariate for the first mode in a bipartite (aka two-mode) network: The attrname argument is a character string giving the name of a numeric (not categorical) attribute in the network's vertex attribute list. This term adds a single network statistic to the model equaling the total value of attrname(i) for all edges $$(i,j)$$ in the network. This term may only be used with bipartite networks. For categorical attributes, see b1factor.

b1degrange(from, to=+Inf, by=NULL, homophily=FALSE, levels=NULL) (binary) (bipartite) (undirected)

Degree range for the first mode in a bipartite (a.k.a. two-mode) network: The from and to arguments are vectors of distinct integers (or +Inf, for to (its default)). If one of the vectors has length 1, it is recycled to the length of the other. Otherwise, they must have the same length. This term adds one network statistic to the model for each element of from (or to); the $$i$$th such statistic equals the number of nodes of the first mode ("actors") in the network of degree greater than or equal to from[i] but strictly less than to[i], i.e. with edge count in semiopen interval [from,to). The optional argument by is a character string giving the name of an attribute in the network's vertex attribute list. If this is specified and homophily is TRUE, then degrees are calculated using the subnetwork consisting of only edges whose endpoints have the same value of the by attribute. If by is specified and homophily is FALSE (the default), then separate degree range statistics are calculated for nodes having each separate value of the attribute.

This term can only be used with bipartite networks; for directed networks see idegrange and odegrange. For undirected networks, see degrange, and see b2degrange for degrees of the second mode ("events").

b1degree(d, by=NULL, levels=NULL) (binary) (bipartite) (undirected) (categorical nodal attribute) (frequently-used)

Degree for the first mode in a bipartite (aka two-mode) network: The d argument is a vector of distinct integers. This term adds one network statistic to the model for each element in d; the $$i$$th such statistic equals the number of nodes of degree d[i] in the first mode of a bipartite network, i.e. with exactly d[i] edges. The first mode of a bipartite network object is sometimes known as the "actor" mode. The optional argument by is a character string giving the name of an attribute in the network's vertex attribute list. If this is specified then each node's degree is tabulated only with other nodes having the same value of the by attribute. This term can only be used with undirected bipartite networks.

b1factor(attrname, base=1, levels=NULL) (binary) (bipartite) (undirected) (dyad-independent) (frequently-used) (categorical nodal attribute), b1factor(attrname, base=1, levels=NULL, form="sum") (valued) (bipartite) (undirected) (dyad-independent) (frequently-used) (categorical nodal attribute)

Factor attribute effect for the first mode in a bipartite (aka two-mode) network: The attrname argument is a character string giving the name of a categorical attribute in the network's vertex attribute list. This term adds multiple network statistics to the model, one for each of (a subset of) the unique values of the attrname attribute. Each of these statistics gives the number of times a node with that attribute in the first mode of the network appears in an edge. The first mode of a bipartite network object is sometimes known as the "actor" mode. To include all attribute values is usually not a good idea, because the sum of all such statistics equals the number of edges and hence a linear dependency would arise in any model also including edges. Thus, the base argument tells which value(s) (numbered in order according to the sort function) should be omitted. The default value, base=1, means that the smallest (i.e., first in sorted order) attribute value is omitted. For example, if the “fruit” factor has levels “orange”, “apple”, “banana”, and “pear”, then to add just two terms, one for “apple” and one for “pear”, then set “banana” and “orange” to the base (remember to sort the values first) by using nodefactor("fruit", base=2:3). This term can only be used with undirected bipartite networks.

b1mindegree(d) (binary) (bipartite) (undirected)

Minimum degree for the first mode in a bipartite (aka two-mode) network: The d argument is a vector of distinct integers. This term adds one network statistic to the model for each element in d; the $$i$$th such statistic equals the number of nodes in the first mode of a bipartite network with at least degree d[i]. The first mode of a bipartite network object is sometimes known as the "actor" mode. This term can only be used with undirected bipartite networks.

b1nodematch(attrname, diff=FALSE, keep=NULL, by=NULL, alpha=1, beta=1, byb2attr=NULL) (binary) (bipartite) (undirected) (dyad-independent) (categorical nodal attribute) (frequently-used)

Nodal attribute-based homophily effect for the first mode in a bipartite (aka two-mode) network: This term is introduced in Bomiriya et al (2014). The attrname argument is a character string giving the name of a categorical attribute in the network's vertex attribute list. Out of the two arguments (discount parameters) alpha and beta, both which takes values from [0,1], only one should be set at a time. If none is set to a value other than 1, this term will simply be a homophily based two-star statistic. This term adds one statistic to the model unless diff is set to TRUE, in which case the term adds multiple network statistics to the model, one for each of (a subset of) the unique values of the attrname attribute. To include only the attribute values you wish, use the keep argument. If an alpha discount parameter is used, each of these statistics gives the sum of the number of common second-mode nodes raised to the power alpha for each pair of first-mode nodes with that attribute. If a beta discount parameter is used, each of these statistics gives half the sum of the number of two-paths with two first-mode nodes with that attribute as the two ends of the two path raised to the power beta for each edge in the network. The byb2attr argument is a character string giving the name of a second mode categorical attribute in the network's attribute list. Setting this argument will separate the orginal statistics based on the values of the set second mode attribute--- i.e. for example, if diff is FALSE, then the sum of all the statistics for each level of this second-mode attribute will be equal to the original b1nodematch statistic where byb2attr set to NULL. This term can only be used with undirected bipartite networks.

b1star(k, attrname=NULL, levels=NULL) (binary) (bipartite) (undirected) (categorical nodal attribute)

k-Stars for the first mode in a bipartite (aka two-mode) network: The k argument is a vector of distinct integers. This term adds one network statistic to the model for each element in k. The $$i$$th such statistic counts the number of distinct k[i]-stars whose center node is in the first mode of the network. The first mode of a bipartite network object is sometimes known as the "actor" mode. A $$k$$-star is defined to be a center node $$N$$ and a set of $$k$$ different nodes $$\{O_1, \dots, O_k\}$$ such that the ties $$\{N, O_i\}$$ exist for $$i=1, \dots, k$$. The optional argument attrname is a character string giving the name of an attribute in the network's vertex attribute list. If this is specified then the count is over the number of $$k$$-stars (with center node in the first mode) where all nodes have the same value of the attribute. This term can only be used for undirected bipartite networks. Note that b1star(1) is equal to b2star(1) and to edges.

b1starmix(k, attrname, base=NULL, diff=TRUE, levels=NULL) (binary) (bipartite) (undirected) (categorical nodal attribute)

Mixing matrix for k-stars centered on the first mode of a bipartite network: Only a single value of $$k$$ is allowed. This term counts all k-stars in which the b2 nodes (called events in some contexts) are homophilous in the sense that they all share the same value of attrname. However, the b1 node (in some contexts, the actor) at the center of the k-star does NOT have to have the same value as the b2 nodes; indeed, the values taken by the b1 nodes may be completely distinct from those of the b2 nodes, which allows for the use of this term in cases where there are two separate nodal attributes, one for the b1 nodes and another for the b2 nodes (in this case, however, these two attributes should be combined to form a single nodal attribute called attrname. A different statistic is created for each value of attrname seen in a b1 node, even if no k-stars are observed with this value. Whether a different statistic is created for each value seen in a b2 node depends on the value of the diff argument: When diff=TRUE, the default, a different statistic is created for each value and thus the behavior of this term is reminiscent of the nodemix term, from which it takes its name; when diff=FALSE, all homophilous k-stars are counted together, though these k-stars are still categorized according to the value of the central b1 node. The base term may be used to control which of the possible terms are left out of the model: By default, all terms are included, but if base is set to a vector of indices then the corresponding terms (in the order they would be created when base=NULL) are left out.

b1twostar(b1attrname, b2attrname, base=NULL, b1levels=NULL, b2levels=NULL) (binary) (bipartite) (undirected) (categorical nodal attribute)

Two-star census for central nodes centered on the first mode of a bipartite network: This term takes two nodal attribute names, one for b1 nodes (actors in some contexts) and one for b2 nodes (events in some contexts). Only b1attrname is required; if b2attrname is not passed, it is assumed to be the same as b1attrname. Assuming that there are $$n_1$$ values of b1attrname among the b1 nodes and $$n_2$$ values of b2attrname among the b2 nodes, then the total number of distinct categories of two stars according to these two attributes is $$n_1(n_2)(n_2+1)/2$$. This model term creates a distinct statistic counting each of these categories. The base term may be used to leave some of these categories out; when passed as a vector of integer indices (in the order the statistics would be created when base=NULL), the corresponding terms will be left out.

b2concurrent(by=NULL) (binary) (bipartite) (undirected) (frequently-used)

Concurrent node count for the second mode in a bipartite (aka two-mode) network: This term adds one network statistic to the model, equal to the number of nodes in the second mode of the network with degree 2 or higher. The second mode of a bipartite network object is sometimes known as the "event" mode. The optional argument by is a character string giving the name of an attribute in the network's vertex attribute list; it functions just like the by argument of the b2degree term. Without the optional argument, this statistic is equivalent to b2mindegree(2). This term can only be used with undirected bipartite networks.

b2cov(attrname, transform, transformname) (binary) (undirected) (bipartite) (dyad-independent) (quantitative nodal attribute) (frequently-used), b2cov(attrname, transform, transformname, form="sum") (valued) (undirected) (bipartite) (dyad-independent) (quantitative nodal attribute) (frequently-used)

Main effect of a covariate for the second mode in a bipartite (aka two-mode) network: The attrname argument is a character string giving the name of a numeric (not categorical) attribute in the network's vertex attribute list. This term adds a single network statistic to the model equaling the total value of attrname(j) for all edges $$(i,j)$$ in the network. This term may only be used with bipartite networks. For categorical attributes, see b2factor.

b2degrange(from, to=+Inf, by=NULL, homophily=FALSE, levels=NULL) (binary) (bipartite) (undirected)

Degree range for the second mode in a bipartite (a.k.a. two-mode) network: The from and to arguments are vectors of distinct integers (or +Inf, for to (its default)). If one of the vectors has length 1, it is recycled to the length of the other. Otherwise, they must have the same length. This term adds one network statistic to the model for each element of from (or to); the $$i$$th such statistic equals the number of nodes of the second mode ("events") in the network of degree greater than or equal to from[i] but strictly less than to[i], i.e. with edge count in semiopen interval [from,to). The optional argument by is a character string giving the name of an attribute in the network's vertex attribute list. If this is specified and homophily is TRUE, then degrees are calculated using the subnetwork consisting of only edges whose endpoints have the same value of the by attribute. If by is specified and homophily is FALSE (the default), then separate degree range statistics are calculated for nodes having each separate value of the attribute.

This term can only be used with bipartite networks; for directed networks see idegrange and odegrange. For undirected networks, see degrange, and see b1degrange for degrees of the first mode ("actors").

b2degree(d, by=NULL) (binary) (bipartite) (undirected) (categorical nodal attribute) (frequently-used)

Degree for the second mode in a bipartite (aka two-mode) network: The d argument is a vector of distinct integers. This term adds one network statistic to the model for each element in d; the $$i$$th such statistic equals the number of nodes of degree d[i] in the second mode of a bipartite network, i.e. with exactly d[i] edges. The second mode of a bipartite network object is sometimes known as the "event" mode. The optional term by is a character string giving the name of an attribute in the network's vertex attribute list. If this is specified then each node's degree is tabulated only with other nodes having the same value of the by attribute. This term can only be used with undirected bipartite networks.

b2factor(attrname, base=1, levels=NULL) (binary) (bipartite) (undirected) (dyad-independent) (categorical nodal attribute) (frequently-used), b2factor(attrname, base=1, levels=NULL, form="sum") (valued) (bipartite) (undirected) (dyad-independent) (categorical nodal attribute) (frequently-used)

Factor attribute effect for the second mode in a bipartite (aka two-mode) network : The attrname argument is a character string giving the name of a categorical attribute in the network's vertex attribute list. This term adds multiple network statistics to the model, one for each of (a subset of) the unique values of the attrname attribute. Each of these statistics gives the number of times a node with that attribute in the second mode of the network appears in an edge. The second mode of a bipartite network object is sometimes known as the "event" mode. To include all attribute values is usually not a good idea, because the sum of all such statistics equals the number of edges and hence a linear dependency would arise in any model also including edges. Thus, the base argument tells which value(s) (numbered in order according to the sort function) should be omitted. The default value, base=1, means that the smallest (i.e., first in sorted order) attribute value is omitted. For example, if the “fruit” factor has levels “orange”, “apple”, “banana”, and “pear”, then to add just two terms, one for “apple” and one for “pear”, then set “banana” and “orange” to the base (remember to sort the values first) by using nodefactor("fruit", base=2:3). This term can only be used with undirected bipartite networks.

b2mindegree(d) (binary) (bipartite) (undirected)

Minimum degree for the second mode in a bipartite (aka two-mode) network: The d argument is a vector of distinct integers. This term adds one network statistic to the model for each element in d; the $$i$$th such statistic equals the number of nodes in the second mode of a bipartite network with at least degree d[i]. The second mode of a bipartite network object is sometimes known as the "event" mode. This term can only be used with undirected bipartite networks.

b2nodematch(attrname, diff=FALSE, keep=NULL, by=NULL, alpha=1, beta=1, byb1attr=NULL) (binary) (bipartite) (undirected) (dyad-independent) (categorical nodal attribute) (frequently-used)

Nodal attribute-based homophily effect for the second mode in a bipartite (aka two-mode) network: This term is introduced in Bomiriya et al (2014). The attrname argument is a character string giving the name of a categorical attribute in the network's vertex attribute list. Out of the two arguments (discount parameters) alpha and beta, both which takes values from [0,1], only one should be set at a time. If none is set to a value other than 1, this term will simply be a homophily based two-star statistic. This term adds one statistic to the model unless diff is set to TRUE, in which case the term adds multiple network statistics to the model, one for each of (a subset of) the unique values of the attrname attribute. To include only the attribute values you wish, use the keep argument. If an alpha discount parameter is used, each of these statistics gives the sum of the number of common first-mode nodes raised to the power alpha for each pair of second-mode nodes with that attribute. If a beta discount parameter is used, each of these statistics gives half the sum of the number of two-paths with two second-mode nodes with that attribute as the two ends of the two path raised to the power beta for each edge in the network. The byb1attr argument is a character string giving the name of a first mode categorical attribute in the network's attribute list. Setting this argument will separate the orginal statistics based on the values of the set first mode attribute--- i.e. for example, if diff is FALSE, then the sum of all the statistics for each level of this first-mode attribute will be equal to the original b2nodematch statistic where byb1attr set to NULL. This term can only be used with undirected bipartite networks.

b2star(k, attrname=NULL, levels=NULL) (binary) (bipartite) (undirected) (categorical nodal attribute)

k-Stars for the second mode in a bipartite (aka two-mode) network: The k argument is a vector of distinct integers. This term adds one network statistic to the model for each element in k. The $$i$$th such statistic counts the number of distinct k[i]-stars whose center node is in the second mode of the network. The second mode of a bipartite network object is sometimes known as the "event" mode. A $$k$$-star is defined to be a center node $$N$$ and a set of $$k$$ different nodes $$\{O_1, \dots, O_k\}$$ such that the ties $$\{N, O_i\}$$ exist for $$i=1, \dots, k$$. The optional argument attrname is a character string giving the name of an attribute in the network's vertex attribute list. If this is specified then the count is over the number of $$k$$-stars (with center node in the second mode) where all nodes have the same value of the attribute. This term can only be used for undirected bipartite networks. Note that b2star(1) is equal to b1star(1) and to edges.

b2starmix(k, attrname, base=NULL, diff=TRUE, levels=NULL) (binary) (bipartite) (undirected) (categorical nodal attribute)

Mixing matrix for k-stars centered on the second mode of a bipartite network: This term is exactly the same as b1starmix except that the roles of b1 and b2 are reversed.

b2twostar(b1attrname, b2attrname, base=NULL, b1levels=NULL, b2levels=NULL) (binary) (bipartite) (undirected) (categorical nodal attribute)

Two-star census for central nodes centered on the second mode of a bipartite network: This term is exactly the same as b1twostar except that the roles of b1 and b2 are reversed.

balance (binary) (triad-related) (directed) (undirected)

Balanced triads: This term adds one network statistic to the model equal to the number of triads in the network that are balanced. The balanced triads are those of type 102 or 300 in the categorization of Davis and Leinhardt (1972). For details on the 16 possible triad types, see ?triad.classify in the {sna} package. For an undirected network, the balanced triads are those with an even number of ties (i.e., 0 and 2).

coincidence(d=NULL,active=0) (binary) (bipartite) (undirected)

Coincident node count for the second mode in a bipartite (aka two-mode) network: By default this term adds one network statistic to the model for each pair of nodes of mode two. It is equal to the number of (first mode) mutual partners of that pair. The first mode of a bipartite network object is sometimes known as the "actor" mode and the seconds as the "event" mode. So this is the number of actors going to both events in the pair. The optional argument d is a two-column matrix of (row-wise) pairs indices where the first row is less than the second row. The second optional argument, active, selects pairs for which the observed count is at least active. This term can only be used with undirected bipartite networks.

concurrent(by=NULL, levels=NULL) (binary) (undirected) (categorical nodal attribute)

Concurrent node count: This term adds one network statistic to the model, equal to the number of nodes in the network with degree 2 or higher. The optional argument by is a character string giving the name of an attribute in the network's vertex attribute list; it functions just like the by argument of the degree term. This term can only be used with undirected networks.

concurrentties(by=NULL, levels=NULL) (binary) (undirected) (categorical nodal attribute)

Concurrent tie count: This term adds one network statistic to the model, equal to the number of ties incident on each actor beyond the first. The optional argument by is a character string giving the name of an attribute in the network's vertex attribute list; it functions just like the by argument of the degree term. This term can only be used with undirected networks.

ctriple(attrname=NULL, levels=NULL) (binary) (directed) (triad-related) (categorical nodal attribute) , a.k.a. ctriad (binary) (directed) (triad-related) (categorical nodal attribute)

Cyclic triples: This term adds one statistic to the model, equal to the number of cyclic triples in the network, defined as a set of edges of the form $$\{(i{\rightarrow}j), (j{\rightarrow}k), (k{\rightarrow}i)\}$$. Note that for all directed networks, triangle is equal to ttriple+ctriple, so at most two of these three terms can be in a model. The optional argument attrname is a character string giving the name of an attribute in the network's vertex attribute list. If this is specified then the count is over the number of cyclic triples where all three nodes have the same value of the attribute. This term can only be used with directed networks.

cycle(k) (binary) (directed) (undirected)

Cycles: The k argument is a vector of distinct integers. This term adds one network statistic to the model for each element in k; the $$i$$th such statistic equals the number of cycles in the network with length exactly k[i]. The cycle statistic applies to both directed and undirected networks. For directed networks, it counts directed cycles of length $$k$$, as opposed to undirected cycles in the undirected case. The directed cycle terms of lengths 2 and 3 are equivalent to mutual and ctriple (respectively). The undirected cycle term of length 3 is equivalent to triangle, and there is no undirected cycle term of length 2.

cyclicalties(attrname=NULL, levels=NULL) (binary) (directed), cyclicalties(threshold=0) (valued) (directed) (undirected)

Cyclical ties: This term adds one statistic, equal to the number of ties $$i\rightarrow j$$ such that there exists a two-path from $$i$$ to $$j$$. (Related to the ttriple term.) The binary version takes a nodal attribute attrname, and, if given, all three nodes involved ($$i$$, $$j$$, and the node on the two-path) must match on this attribute in order for $$i\rightarrow j$$ to be counted. The binary version of this term can only be used with directed networks. The valued version can be used with both directed and undirected.

cyclicalweights(twopath="min",combine="max",affect="min") (valued) (directed) (undirected)

Cyclical weights: This statistic implements the cyclical weights statistic, like that defined by Krivitsky (2012), Equation 13, but with the focus dyad being $$y_{j,i}$$ rather than $$y_{i,j}$$. The currently implemented options for twopath is the minimum of the constituent dyads ("min") or their geometric mean ("geomean"); for combine, the maximum of the 2-path strengths ("max") or their sum ("sum"); and for affect, the minimum of the focus dyad and the combined strength of the two paths ("min") or their geometric mean ("geomean"). For each of these options, the first (and the default) is more stable but also more conservative, while the second is more sensitive but more likely to induce a multimodal distribution of networks.

ddsp(d, type="OTP") (binary) (directed)

Directed dyadwise shared partners: This term adds one network statistic to the model for each element in d where the $$i$$th such statistic equals the number of dyads in the network with exactly d[i] shared partners. This term can only be used with directed networks. Multiple shared partner definitions are possible; the type argument may be used to select the type of shared partner to be counted (see below for type codes). By default, outgoing two-paths are employed.

While there is only one shared partner configuration in the undirected case, nine distinct configurations are possible for directed graphs. Currently, edgewise shared partner terms may be defined with respect to five of these configurations; they are defined here as follows (using terminology from Butts (2008) and the relevent package):

Outgoing Two-path (OTP)

vertex $$k$$ is an OTP shared partner of ordered pair $$(i,j)$$ iff $$i \to k \to j$$. Also known as "transitive shared partner".

Incoming Two-path (ITP)

vertex $$k$$ is an ITP shared partner of ordered pair $$(i,j)$$ iff $$j \to k \to i$$. Also known as "cyclical shared partner"

Outgoing Shared Partner (OSP)

vertex $$k$$ is an OSP shared partner of ordered pair $$(i,j)$$ iff $$i \to k, j \to k$$.

Incoming Shared Partner (ISP)

vertex $$k$$ is an ISP shared partner of ordered pair $$(i,j)$$ iff $$k \to i, k \to j$$.

Note that Robins et al. (2009) define closely related statistics to several of the above, using slightly different terminology.

degrange(from, to=+Inf, by=NULL, homophily=FALSE, levels=NULL) (binary) (undirected) (categorical nodal attribute)

Degree range: The from and to arguments are vectors of distinct integers (or +Inf, for to (its default)). If one of the vectors has length 1, it is recycled to the length of the other. Otherwise, they must have the same length. This term adds one network statistic to the model for each element of from (or to); the $$i$$th such statistic equals the number of nodes in the network of degree greater than or equal to from[i] but strictly less than to[i], i.e. with edges in semiopen interval [from,to). The optional argument by is a character string giving the name of an attribute in the network's vertex attribute list. If this is specified and homophily is TRUE, then degrees are calculated using the subnetwork consisting of only edges whose endpoints have the same value of the by attribute. If by is specified and homophily is FALSE (the default), then separate degree range statistics are calculated for nodes having each separate value of the attribute.

This term can only be used with undirected networks; for directed networks see idegrange and odegrange. This term can be used with bipartite networks, and will count nodes of both first and second mode in the specified degree range. To count only nodes of the first mode ("actors"), use b1degrange and to count only those fo the second mode ("events"), use b2degrange.

degree(d, by=NULL, homophily=FALSE, levels=NULL) (binary) (undirected) (categorical nodal attribute) (frequently-used)

Degree: The d argument is a vector of distinct integers. This term adds one network statistic to the model for each element in d; the $$i$$th such statistic equals the number of nodes in the network of degree d[i], i.e. with exactly d[i] edges. The optional argument by is a character string giving the name of an attribute in the network's vertex attribute list. If this is specified and homophily is TRUE, then degrees are calculated using the subnetwork consisting of only edges whose endpoints have the same value of the by attribute. If by is specified and homophily is FALSE (the default), then separate degree statistics are calculated for nodes having each separate value of the attribute. This term can only be used with undirected networks; for directed networks see idegree and odegree.

degree1.5 (binary) (undirected)

Degree to the 3/2 power: This term adds one network statistic to the model equaling the sum over the actors of each actor's degree taken to the 3/2 power (or, equivalently, multiplied by its square root). This term is an undirected analog to the terms of Snijders et al. (2010), equations (11) and (12). This term can only be used with undirected networks.

degreepopularity (binary) (undirected) (deprecated)

Degree popularity (deprecated): see degree1.5.

degcrossprod (binary) (undirected)

Degree Cross-Product: This term adds one network statistic equal to the mean of the cross-products of the degrees of all pairs of nodes in the network which are tied. Only coded for undirected networks.

degcor (binary) (undirected)

Degree Correlation: This term adds one network statistic equal to the correlation of the degrees of all pairs of nodes in the network which are tied. Only coded for undirected networks.

density (binary) (dyad-independent) (directed) (undirected)

Density: This term adds one network statistic equal to the density of the network. For undirected networks, density equals kstar(1) or edges divided by $$n(n-1)/2$$; for directed networks, density equals edges or istar(1) or ostar(1) divided by $$n(n-1)$$.

diff(attrname, pow=1, dir="t-h", sign.action="identity") (binary) (dyad-independent) (frequently-used) (directed) (undirected) (quantitative nodal attribute), diff(attrname, pow=1, dir="t-h", sign.action="identity", form ="sum") (valued) (dyad-independent) (directed) (undirected) (quantitative nodal attribute)

Difference: The attrname argument is a character string giving the name of a quantitative attribute in the network's vertex attribute list. For values of pow other than 0, this term adds one network statistic to the model, equaling the sum, over directed edges $$(i,j)$$, of sign.action(attrname[i]-attrname[j])^pow if dir is "t-h" (the default), "tail-head", or "b1-b2" and of sign.action(attrname[j]-attrname[i])^pow if "h-t", "head-tail", or "b2-b1". That is, the argument dir determines which vertex's attribute is subtracted from which, with tail being the origin of a directed edge and head being its destination, and bipartite networks' edges being treated as going from the first part (b1) to the second (b2).

If pow==0, the exponentiation is replaced by the signum function: +1 if the difference is positive, 0 if there is no difference, and -1 if the difference is negative. Note that this function is applied after the sign.action. The comparison is exact, so when using calculated values of attrname, ensure that values that you want to be considered equal are, in fact, equal.

The following sign.actions are possible:

"identity" (the default)

no transformation of the difference regardless of sign

"abs"

absolute value of the difference: equivalent to the absdiff term

"posonly"

positive differences are kept, negative differences are replaced by 0

"negonly"

negative differences are kept, positive differences are replaced by 0

Note that this term may not be meaningful for unipartite undirected networks unless sign.action=="abs". When used on such a network, it behaves as if all edges were directed, going from the lower-indexed vertex to the higher-indexed vertex.

desp(d, type="OTP") (binary) (directed)

Directed edgewise shared partners: This term adds one network statistic to the model for each element in d where the $$i$$th such statistic equals the number of edges in the network with exactly d[i] shared partners. This term can only be used with directed networks. Multiple shared partner definitions are possible; the type argument may be used to select the type of shared partner to be counted (see ddsp for type codes). By default, outgoing two-paths are employed.

dgwdsp(decay=0, fixed=FALSE, cutoff=30, type="OTP") (binary) (directed)

Geometrically weighted dyadwise shared partner distribution: This term adds one network statistic to the model equal to the geometrically weighted dyadwise shared partner distribution with decay parameter decay parameter, which should be non-negative. (this parameter was called alpha prior to ergm 3.7). The value supplied for this parameter may be fixed (if fixed=TRUE), or it may be used instead as the starting value for the estimation of decay in a curved exponential family model (when fixed=FALSE, the default) (see Hunter and Handcock, 2006). Note that the GWDSP statistic is equal to the sum of GWNSP plus GWESP. For a directed network, multiple shared partner definitions are possible; the type argument may be used to select the type of shared partner to employ (see ddsp for definitions). By default, outgoing two-paths are employed. The optional argument cutoff sets the number of underlying DSP terms to use in computing the statistics to reduce the computational burden.

dgwesp(decay=0, fixed=FALSE, cutoff=30, type="OTP") (binary) (directed)

Geometrically weighted edgewise shared partner distribution: This term adds a statistic equal to the geometrically weighted edgewise (not dyadwise) shared partner distribution with decay parameter decay parameter, which should be non-negative. (this parameter was called alpha prior to ergm 3.7). The value supplied for this parameter may be fixed (if fixed=TRUE), or it may be used instead as the starting value for the estimation of decay in a curved exponential family model (when fixed=FALSE, the default) (see Hunter and Handcock, 2006). For a directed network, multiple shared partner definitions are possible; the type argument may be used to select the type of shared partner to employ (see ddsp for definitions). By default, outgoing two-paths are employed. The optional argument cutoff sets the number of underlying ESP terms to use in computing the statistics to reduce the computational burden.

dgwnsp(decay=0, fixed=FALSE, cutoff=30, type="OTP") (binary) (directed)

Geometrically weighted non-edgewise shared partner distribution: This term is just like gwesp and gwdsp except it adds a statistic equal to the geometrically weighted nonedgewise (that is, over dyads that do not have an edge) shared partner distribution with decay parameter decay parameter, which should be non-negative. (this parameter was called alpha prior to ergm 3.7). The value supplied for this parameter may be fixed (if fixed=TRUE), or it may be used instead as the starting value for the estimation of decay in a curved exponential family model (when fixed=FALSE, the default) (see Hunter and Handcock, 2006). For a directed network, multiple shared partner definitions are possible; the type argument may be used to select the type of shared partner to employ (see ddsp for definitions). By default, outgoing two-paths are employed. The optional argument cutoff sets the number of underlying NSP terms to use in computing the statistics to reduce the computational burden.

dnsp(d, type="OTP") (binary) (directed)

Directed non-edgewise shared partners: This term adds one network statistic to the model for each element in d where the $$i$$th such statistic equals the number of non-edges in the network with exactly d[i] shared partners. This term can only be used with directed networks. Multiple shared partner definitions are possible; the type argument may be used to select the type of shared partner to be counted (see ddsp for type codes). By default, outgoing two-paths are employed.

dsp(d) (binary) (directed) (undirected)

Dyadwise shared partners: The d argument is a vector of distinct integers. This term adds one network statistic to the model for each element in d; the $$i$$th such statistic equals the number of dyads in the network with exactly d[i] shared partners. This term can be used with directed and undirected networks. For directed networks the count is over homogeneous shared partners only (i.e., only partners on a directed two-path connecting the nodes in the dyad).

dyadcov(x, attrname=NULL) (binary) (dyad-independent) (directed) (undirected) (categorical nodal attribute)

Dyadic covariate: The x argument is either a square matrix of covariates, one for each possible edge in the network, the name of a network attribute of covariates, or a network; if the latter, optional argument attrname provides the name of the quantitative edge attribute to use for covariate values (in this case, missing edges in x are assigned a covariate value of zero). This term adds three statistics to the model, each equal to the sum of the covariate values for all dyads occupying one of the three possible non-empty dyad states (mutual, upper-triangular asymmetric, and lower-triangular asymmetric dyads, respectively), with the empty or null state serving as a reference category. If the network is undirected, x is either a matrix of edgewise covariates, or a network; if the latter, optional argument attrname provides the name of the edge attribute to use for edge values. This term adds one statistic to the model, equal to the sum of the covariate values for each edge appearing in the network. The edgecov and dyadcov terms are equivalent for undirected networks.

edgecov(x, attrname=NULL) (binary) (dyad-independent) (directed) (undirected) (frequently-used) , edgecov(x, attrname=NULL, form="sum") (valued) (directed) (undirected) (dyad-independent)

Edge covariate: The x argument is either a square matrix of covariates, one for each possible edge in the network, the name of a network attribute of covariates, or a network; if the latter, optional argument attrname provides the name of the quantitative edge attribute to use for covariate values (in this case, missing edges in x are assigned a covariate value of zero). This term adds one statistic to the model, equal to the sum of the covariate values for each edge appearing in the network. The edgecov term applies to both directed and undirected networks. For undirected networks the covariates are also assumed to be undirected. The edgecov and dyadcov terms are equivalent for undirected networks.

edges (binary) (valued) (dyad-independent) (directed) (undirected) (frequently-used) , a.k.a nonzero (valued) (directed) (undirected) (dyad-independent)

Edges: This term adds one network statistic equal to the number of edges (i.e. nonzero values) in the network. For undirected networks, edges is equal to kstar(1); for directed networks, edges is equal to both ostar(1) and istar(1).

esp(d) (binary) (directed) (undirected)

Edgewise shared partners: This is just like the dsp term, except this term adds one network statistic to the model for each element in d where the $$i$$th such statistic equals the number of edges (rather than dyads) in the network with exactly d[i] shared partners. This term can be used with directed and undirected networks. For directed networks the count is over homogeneous shared partners only (i.e., only partners on a directed two-path connecting the nodes in the edge and in the same direction).

equalto(value=0, tolerance=0) (valued) (directed) (undirected) (dyad-independent)

Number of dyads with values equal to a specific value (within tolerance): Adds one statistic equal to the number of dyads whose values are within tolerance of value, i.e., between value-tolerance and value+tolerance, inclusive.

greaterthan(threshold=0) (valued) (directed) (undirected) (dyad-independent)

Number of dyads with values strictly greater than a threshold: Adds one statistic equal to the number of dyads whose values exceed threshold.

gwb1degree(decay, fixed=FALSE, attrname=NULL, cutoff=30, levels=NULL) (binary) (bipartite) (undirected) (curved)

Geometrically weighted degree distribution for the first mode in a bipartite (aka two-mode) network: This term adds one network statistic to the model equal to the weighted degree distribution with decay controlled by the decay parameter, which should be non-negative, for nodes in the first mode of a bipartite network. The first mode of a bipartite network object is sometimes known as the "actor" mode. The decay parameter is the same as theta_s in equation (14) in Hunter (2007). The value supplied for this parameter may be fixed (if fixed=TRUE), or it may be used as merely the starting value for the estimation in a curved exponential family model (the default). The optional argument cutoff is only relevant if fixed=FALSE. In that case it only uses this number of terms in computing the statistics to reduce the computational burden. If attrname is specified then separate degree statistics are calculated for nodes having each separate value of the attribute. This term can only be used with undirected bipartite networks.

gwb2degree(decay, fixed=FALSE, attrname=NULL, cutoff=30, levels=NULL) (binary) (bipartite) (undirected) (curved)

Geometrically weighted degree distribution for the second mode in a bipartite (aka two-mode) network: This term adds one network statistic to the model equal to the weighted degree distribution with decay controlled by the which should be non-negative, for nodes in the second mode of a bipartite network. The second mode of a bipartite network object is sometimes known as the "event" mode. The decay parameter is the same as theta_s in equation (14) in Hunter (2007). The value supplied for this parameter may be fixed (if fixed=TRUE), or it may be used as merely the starting value for the estimation in a curved exponential family model (the default). The optional argument cutoff is only relevant if fixed=FALSE. In that case it only uses this number of terms in computing the statistics to reduce the computational burden. If attrname is specified then separate degree statistics are calculated for nodes having each separate value of the attribute. This term can only be used with undirected bipartite networks.

gwdegree(decay, fixed=FALSE, attrname=NULL, cutoff=30, levels=NULL) (binary) (undirected) (curved) (frequently-used)

Geometrically weighted degree distribution: This term adds one network statistic to the model equal to the weighted degree distribution with decay controlled by the decay parameter. The decay parameter is the same as theta_s in equation (14) in Hunter (2007). The value supplied for this parameter may be fixed (if fixed=TRUE), or it may be used instead as the starting value for the estimation of decay in a curved exponential family model (when fixed=FALSE, the default) (see Hunter and Handcock, 2006). The optional argument cutoff is only relevant if fixed=FALSE. In that case it only uses this number of terms in computing the statistics to reduce the computational burden. If attrname is specified then separate degree statistics are calculated for nodes having each separate value of the attribute. This term can only be used with undirected networks.

gwdsp(decay=0, fixed=FALSE, cutoff=30) (binary) (directed) (undirected) (curved)

Geometrically weighted dyadwise shared partner distribution: This term adds one network statistic to the model equal to the geometrically weighted dyadwise shared partner distribution with decay parameter decay parameter, which should be non-negative. The value supplied for this parameter may be fixed (if fixed=TRUE), or it may be used instead as the starting value for the estimation of decay in a curved exponential family model (when fixed=FALSE, the default) (see Hunter and Handcock, 2006). For directed networks the count is over homogeneous shared partners only (i.e., only partners on a directed two-path connecting the nodes in the dyad). The optional argument cutoff is only relevant if fixed=FALSE. In that case it only uses this number of terms in computing the statistics to reduce the computational burden.

gwesp(decay=0, fixed=FALSE, cutoff=30) (binary) (frequently-used) (directed) (undirected) (curved)

Geometrically weighted edgewise shared partner distribution: This term is just like gwdsp except it adds a statistic equal to the geometrically weighted edgewise (not dyadwise) shared partner distribution with decay parameter decay parameter, which should be non-negative. The value supplied for this parameter may be fixed (if fixed=TRUE), or it may be used instead as the starting value for the estimation of decay in a curved exponential family model (when fixed=FALSE, the default) (see Hunter and Handcock, 2006). This term can be used with directed and undirected networks. For directed networks the geometric weighting is over homogeneous shared partners only (i.e., only partners on a directed two-path connecting the nodes in the edge and in the same direction). The optional argument cutoff is only relevant if fixed=FALSE. In that case it only uses this number of terms in computing the statistics to reduce the computational burden.

gwidegree(decay, fixed=FALSE, attrname=NULL, cutoff=30, levels=NULL) (binary) (directed) (curved)

Geometrically weighted in-degree distribution: This term adds one network statistic to the model equal to the weighted in-degree distribution with decay parameter decay parameter, which should be non-negative. (this parameter was called alpha prior to ergm 3.7). The value supplied for this parameter may be fixed (if fixed=TRUE), or it may be used instead as the starting value for the estimation of decay in a curved exponential family model (when fixed=FALSE, the default) (see Hunter and Handcock, 2006). This term can only be used with directed networks. The optional argument cutoff is only relevant if fixed=FALSE. In that case it only uses this number of terms in computing the statistics to reduce the computational burden. If attrname is specified then separate degree statistics are calculated for nodes having each separate value of the attribute.

gwnsp(decay=0, fixed=FALSE, cutoff=30) (binary) (directed) (undirected) (curved)

Geometrically weighted nonedgewise shared partner distribution: This term is just like gwesp and gwdsp except it adds a statistic equal to the geometrically weighted nonedgewise (that is, over dyads that do not have an edge) shared partner distribution with weight parameter decay parameter, which should be non-negative. (this parameter was called alpha prior to ergm 3.7). The optional argument fixed indicates whether the decay parameter is fixed at the given value, or is to be fit as a curved exponential-family model (see Hunter and Handcock, 2006). The default is FALSE, which means the scale parameter is not fixed and thus the model is a CEF model. This term can be used with directed and undirected networks. For directed networks the geometric weighting is over homogeneous shared partners only (i.e., only partners on a directed two-path connecting the nodes in the non-edge and in the same direction). The optional argument cutoff is only relevant if fixed=FALSE. In that case it only uses this number of terms in computing the statistics to reduce the computational burden.

gwodegree(decay, fixed=FALSE, attrname=NULL, cutoff=30, levels=NULL) (binary) (directed) (curved)

Geometrically weighted out-degree distribution: This term adds one network statistic to the model equal to the weighted out-degree distribution with decay parameter decay pa

## References

• Bomiriya, R. P, Bansal, S., and Hunter, D. R. (2014). Modeling Homophily in ERGMs for Bipartite Networks. Submitted.

• Butts, CT. (2008). “A Relational Event Framework for Social Action.” Sociological Methodology, 38(1).

• Davis, J.A. and Leinhardt, S. (1972). The Structure of Positive Interpersonal Relations in Small Groups. In J. Berger (Ed.), Sociological Theories in Progress, Volume 2, 218--251. Boston: Houghton Mifflin.

• Holland, P. W. and S. Leinhardt (1981). An exponential family of probability distributions for directed graphs. Journal of the American Statistical Association, 76: 33--50.

• Hunter, D. R. and M. S. Handcock (2006). Inference in curved exponential family models for networks. Journal of Computational and Graphical Statistics, 15: 565--583.

• Hunter, D. R. (2007). Curved exponential family models for social networks. Social Networks, 29: 216--230.

• Krackhardt, D. and Handcock, M. S. (2007). Heider versus Simmel: Emergent Features in Dynamic Structures. Lecture Notes in Computer Science, 4503, 14--27.

• Krivitsky P. N. (2012). Exponential-Family Random Graph Models for Valued Networks. Electronic Journal of Statistics, 2012, 6, 1100-1128. 10.1214/12-EJS696

• Robins, G; Pattison, P; and Wang, P. (2009). “Closure, Connectivity, and Degree Distributions: Exponential Random Graph (p*) Models for Directed Social Networks.” Social Networks, 31:105-117.

• Snijders T. A. B., G. G. van de Bunt, and C. E. G. Steglich. Introduction to Stochastic Actor-Based Models for Network Dynamics. Social Networks, 2010, 32(1), 44-60. 10.1016/j.socnet.2009.02.004

• Morris M, Handcock MS, and Hunter DR. Specification of Exponential-Family Random Graph Models: Terms and Computational Aspects. Journal of Statistical Software, 2008, 24(4), 1-24. http://www.jstatsoft.org/v24/i04

• Snijders, T. A. B., P. E. Pattison, G. L. Robins, and M. S. Handcock (2006). New specifications for exponential random graph models, Sociological Methodology, 36(1): 99-153.

ergm package, search.ergmTerms, ergm, network, %v%, %n%

## Examples

Run this code
# NOT RUN {
ergm(flomarriage ~ kstar(1:2) + absdiff("wealth") + triangle)

ergm(molecule ~ edges + kstar(2:3) + triangle
+ nodematch("atomic type",diff=TRUE)
+ triangle + absdiff("atomic type"))
# }
# NOT RUN {
<!-- % TODO: Write a valued example. -->
# }


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