| edges |
 |
\(\sum_{i<j}
y_{ij}\) |
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 |
 |
\(\sum_{i<j} y_{ij}
y_{ji}\) |
Counts the dyads in which both directed ties are
present, capturing the tendency for ties to be returned.
(holland1981) |
| triangle |
 |
\(\sum_{i<j<k} y_{ij}
y_{jk} y_{ik}\) |
Counts the sets of three mutually connected nodes, the
basic measure of local clustering in an undirected network.
(frank1986) |
| gwesp |
 |
\(\exp{(\tau)}
\sum_{i=1}^{n-2} \left[1 - \left(1 - \exp{(-\tau)}\right)^i\right]
EP_i(y)\) |
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 |
 |
\(\exp{(\tau)}
\sum_{i=1}^{n-2} \left[1 - \left(1 - \exp{(-\tau)}\right)^i\right]
DP_i(y)\) |
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 |
 |
\(\exp{(\tau)}
\sum_{i=1}^{n-1} \left[1 - \left(1 - \exp{(-\tau)}\right)^i\right]
D_i(y)\) |
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 |
 |
\(\sum_{k=2}^{n-1} (-1)^k
\frac{S_k(y)}{\lambda^{k-2}}\) |
Alternating sum of the k-star counts, an equivalent
parameterization of the geometrically weighted degree distribution used
to model degree heterogeneity. (snijders2006; hunter2007) |
| nodematch |
 |
\(\sum_{i<j} y_{ij}
\mathbf{1}(x_i = x_j)\) |
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 |
 |
\(\sum_{i<j} y_{ij}
\left[\mathbf{1}(x_i = k) + \mathbf{1}(x_j = k)\right]\) |
Counts the tie endpoints belonging to each level of a
categorical attribute, measuring how active nodes with that value are in
forming ties. |
| nodemix |
 |
\(\sum_{i<j} y_{ij}
\mathbf{1}(\{x_i, x_j\} = \{k, l\})\) |
Counts the ties for every pairing of attribute values,
reproducing the full mixing matrix of a categorical attribute.
(wasserman1996; morris1991) |
| nodecov |
 |
\(\sum_{i<j} y_{ij} (x_i +
x_j)\) |
Sums a quantitative attribute over both ends of each
tie, measuring how strongly that attribute drives tie formation. |
| absdiff |
|
\(\sum_{i<j} y_{ij}
\left\lvert{}x_i - x_j\right\rvert{}\) |
Sums the absolute difference in a quantitative
attribute across tied nodes. Negative estimates indicate homophily,
since similar nodes contribute less. |
| edgecov |
 |
\(\sum_{i<j} y_{ij}
x_{ij}\) |
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 |
 |
\(\sum_{i \neq j} y_{ij}
\mathbf{1}\left(\exists k : y_{ik} y_{kj} = 1\right)\) |
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 |
 |
\(\sum_{i \neq j} y_{ij}
\mathbf{1}\left(\exists k : y_{jk} y_{ki} = 1\right)\) |
Counts the ties that take part in at least one cycle,
capturing generalized exchange rather than hierarchy. |
| nodeicov |
 |
\(\sum_{i \neq j} y_{ij}
x_j\) |
Sums the receiving node’s attribute value over all
ties, measuring how a quantitative attribute drives incoming ties
(popularity). |
| nodeocov |
 |
\(\sum_{i \neq j} y_{ij}
x_i\) |
Sums the sending node’s attribute value over all ties,
measuring how a quantitative attribute drives outgoing ties
(activity). |
| gwidegree |
 |
\(\exp{(\tau)}
\sum_{i=1}^{n-1} \left[1 - \left(1 - \exp{(-\tau)}\right)^i\right]
D^{\mathrm{in}}_i(y)\) |
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 |
 |
\(\exp{(\tau)}
\sum_{i=1}^{n-1} \left[1 - \left(1 - \exp{(-\tau)}\right)^i\right]
D^{\mathrm{out}}_i(y)\) |
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 |
|
\(\sum_{i \neq j} y_{ij}
\mathbf{1}(x_j = k)\) |
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 |
|
\(\sum_{i \neq j} y_{ij}
\mathbf{1}(x_i = k)\) |
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 |
 |
\(\sum_{i} \binom{\sum_{j
\neq i} y_{ij}}{k}\) |
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 |
 |
\(\sum_{j} \binom{\sum_{i
\neq j} y_{ij}}{k}\) |
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 |
 |
\(\sum_{i} \binom{\sum_{j
\neq i} y_{ij}}{k}\) |
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 |
 |
\(\sum_{i}
\mathbf{1}\left(\sum_{j \neq i} y_{ij} = 0\right)\) |
Counts the nodes with no ties, capturing an excess (or
shortage) of isolated nodes relative to the rest of the degree
distribution. |
| degree |
 |
\(\sum_{i}
\mathbf{1}\left(\sum_{j \neq i} y_{ij} = d\right)\) |
Counts the nodes with exactly d ties. Pass several
values of d to include one statistic per degree, e.g. to model
low-degree nodes explicitly. |
| concurrent |
|
\(\sum_{i}
\mathbf{1}\left(\sum_{j \neq i} y_{ij} \geq 2\right)\) |
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 |
 |
\(\exp{(\tau)}
\sum_{i=1}^{n-2} \left[1 - \left(1 - \exp{(-\tau)}\right)^i\right]
EP^{\mathrm{OTP}}_i(y)\) |
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 |
 |
\(\exp{(\tau)}
\sum_{i=1}^{n-2} \left[1 - \left(1 - \exp{(-\tau)}\right)^i\right]
DP^{\mathrm{OTP}}_i(y)\) |
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 |
 |
\(\exp{(\tau)}
\sum_{i=1}^{n_{B_2}} \left[1 - \left(1 - \exp{(-\tau)}\right)^i\right]
DP_i(y)\) |
Summarizes how many second-mode nodes each pair of
first-mode nodes has in common, weighting additional shared partners
geometrically less. (wang2009; hunter2007) |
| gwb2dsp |
 |
\(\exp{(\tau)}
\sum_{i=1}^{n_{B_1}} \left[1 - \left(1 - \exp{(-\tau)}\right)^i\right]
DP_i(y)\) |
Summarizes how many first-mode nodes each pair of
second-mode nodes has in common, weighting additional shared partners
geometrically less. (wang2009; hunter2007) |
| gwb1degree |
 |
\(\exp{(\tau)}
\sum_{i=1}^{n_{B_2}} \left[1 - \left(1 - \exp{(-\tau)}\right)^i\right]
D^{B_1}_i(y)\) |
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 |
 |
\(\exp{(\tau)}
\sum_{i=1}^{n_{B_1}} \left[1 - \left(1 - \exp{(-\tau)}\right)^i\right]
D^{B_2}_i(y)\) |
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 |
 |
\(\sum_{i \in B_1} \sum_{j
\in B_2} y_{ij} \mathbf{1}(x_i = k)\) |
Counts the ties incident on first-mode nodes at each
level of a categorical attribute, measuring how active those nodes
are. |
| b2factor |
 |
\(\sum_{i \in B_1} \sum_{j
\in B_2} y_{ij} \mathbf{1}(x_j = k)\) |
Counts the ties incident on second-mode nodes at each
level of a categorical attribute, measuring how active those nodes
are. |
| b1nodematch |
 |
\(\sum_{k\in B_2}
\sum_{i<j \in B_1} \mathbf{1}(x_i = x_j) y_{ik} y_{jk}\) |
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 |
 |
\(\sum_{k\in B_1}
\sum_{i<j \in B_2} \mathbf{1}(x_i = x_j) y_{ik} y_{jk}\) |
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 |
 |
\(\sum_{i \in B_1}
\mathbf{1}(x_i = p) \sum_{j_1 < \cdots < j_k \in B_2}
\prod_{l=1}^{k} y_{i j_l} \mathbf{1}(x_{j_l} = q)\) |
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 |
 |
\(\sum_{j \in B_2}
\mathbf{1}(x_j = p) \sum_{i_1 < \cdots < i_k \in B_1}
\prod_{l=1}^{k} y_{i_l j} \mathbf{1}(x_{i_l} = q)\) |
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. |