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goldfish Effects

Catalog of available effects

Alvaro Uzaheta and Siwei Zhang

This vignette summarizes effects that are implemented in goldfish and are thus available to be used with actor-oriented DyNAM models and tie-oriented Relational Event Models.

Effect arguments

Effect functions have the following 5 arguments:

The following table summaries whether the corresponding arguments can be used for the effects or not.

isTwoMode ignoreRep weighted window transformFun aggregateFun type
Node/actor statistics Structural indeg ×
outdeg ×
nodeTrans × ×
Attribute ego × × × × × ×
alter × × × × × ×
Structural + attribute tertius
Dyadic statistics Structural tie × ×
inertia × ×
recip × ×
Attribute same × × × × ×
diff × × × × ×
sim × × × × ×
egoAlterInt × × × × ×
Structural + attribute tertiusDiff ×
Closure effects trans × × × ×
cycle × × × ×
commonSender × × ×
commonReceiver × × ×
four × × ×
mixedTrans × × × ×
mixedCycle × × × ×
mixedCommonSender × × × ×
mixedCommonReceiver × × × ×

Note that the use of some effects (combinations) are ill-advised. For example, using tie(network, ignoreRep = FALSE), where the network refers to the dependent network, will always result in a change statistic of zero, and thus cannot be used.

Notation

Node or actor statistics

Structural Effects

effect rate choice choice coordination
indeg
outdeg ×
nodeTrans

Here we refer to ego type when type = "ego" and alter type when type = "alter".

Indegree (indeg())

indeg(network,
  isTwoMode = FALSE, weighted = FALSE, window = Inf,
  ignoreRep = FALSE, type = c("alter", "ego"), transformFun = identity
)

\[\begin{align} r(i, t, x) &= \begin{cases} \sum_{j}{I(x(t)_{ji}>0)} & \text{unweighted}\\ \sum_{j}{x(t)_{ji}} & \text{weighted} \end{cases} && \text{ego type}\\ s(i,j, t, x) &= \begin{cases} \sum_{l}{I(x(t)_{lj}>0)} & \text{unweighted}\\ \sum_{l}{x(t)_{lj}} & \text{weighted} \end{cases} && \text{alter type} \end{align}\]

  • DyNAM-Rate model: tendency of actor \(i\) to create an event when \(i\) has a high incoming degree in a covariate network (‘ego’ type).
  • DyNAM-choice model: tendency to create an event \(i\rightarrow j\) when \(j\) has a high incoming degree in a covariate network (‘alter’ type).
  • REM model: tendency to create an event \(i\rightarrow j\) when either \(i\) or \(j\) has a high incoming degree in a covariate network. The argument type allows to choose whether to use the indegree effect for sender \(i\) (type = "ego") or for receiver j (type = "alter").
  • DyNAM-choice_coordination: the effects compute the total degree in a covariate network (‘alter’ type). Note: The effect statistic correspond to the total degree for an undirected network as define when using defineNetwork(net, directed = FALSE).

The degree can be transform with .

Outdegree (outdeg())

outdeg(network,
  isTwoMode = FALSE, weighted = FALSE, window = Inf,
  ignoreRep = FALSE, type = c("alter", "ego"), transformFun = identity
)

\[\begin{align} r(i, t, x) &= \begin{cases} \sum_{j}{I(x(t)_{ij}>0)} & \text{unweighted}\\ \sum_{j}{x(t)_{ij}} & \text{weighted} \end{cases} && \text{ego type}\\ s(i,j, t, x) &= \begin{cases} \sum_{l}{I(x(t)_{jl}>0)} & \text{unweighted}\\ \sum_{l}{x(t)_{jl}} & \text{weighted} \end{cases} && \text{alter type} \end{align}\]

  • DyNAM-Rate model: tendency of actor \(i\) to create an event when \(i\) has a high outgoing degree in a covariate network (‘ego’ type).
  • DyNAM-choice model: tendency to create an event \(i\rightarrow j\) when \(j\) has a high outgoing degree in a covariate network (‘alter’ type).
  • REM model: tendency to create an event \(i\rightarrow j\) when either \(i\) or \(j\) has a high outgoing degree in a covariate network. The argument type allows to choose whether to use the outdegree effect for sender \(i\) (type = "ego") or for receiver \(j\) (type = "alter").

Node embeddedness transitivity (nodeTrans())

nodeTrans(network,
  isTwoMode = FALSE, window = Inf, ignoreRep = FALSE,
  type = c("alter", "ego"), transformFun = identity
)

\[\begin{align} r(i, t, x) &= \sum_{jk}{I(x(t)_{ik}>0)I(x(t)_{kj}>0)I(x(t)_{ij}>0)} && \text{ego type}\\ s(i,j, t, x) &= \sum_{kl}{I(x(t)_{jk}>0)I(x(t)_{kl}>0)I(x(t)_{jl}>0)} && \text{alter type} \end{align}\]

Embeddedness in transitive structures as a source node.

  • DyNAM-Rate model: tendency of actor \(i\) to create an event when \(i\) is embeddeded in more transitive structures as the source \(i\rightarrow k \rightarrow j \leftarrow i\) in a covariate network (‘ego’ type).
  • DyNAM-choice model: tendency to create an event \(i\rightarrow j\) when \(j\) is embeddeded in more transitive structures as the source \(j\rightarrow k \rightarrow l \leftarrow j\) in a covariate network (‘alter’ type).
  • REM model: tendency to create an event \(i\rightarrow j\) when either \(i\) or \(j\) are embeddeded in more transitive structures as the source in a covariate network. The argument type allows to choose whether to use the node embeddedness transitivity effect for sender \(i\) (type = "ego") or for receiver \(j\) (type = "alter").

The statistic can be transform with , there is not weighted version for this effect.

Attribute effects

effect rate choice choice coordination
ego × ×
alter × ×

Ego attribute (ego())

ego(attribute, isTwoMode = FALSE)

\[\begin{align} r(i, t, z) &= z(t)_i \end{align}\]

  • DyNAM-Rate model: tendency of actors to be more active when they score high on an attribute.
  • REM model: tendency to create an event \(i\rightarrow j\) when \(i\) has a high score on an attribute.

Alter attribute (alter())

alter(attribute, isTwoMode = FALSE)

\[\begin{align} s(i, j, t, z) &= z(t)_j \end{align}\]

  • DyNAM-choice/DyNAM-choice_coordination model: tendency to create an event \(i\rightarrow j\) when \(j\) score high on an attribute.
  • REM model: tendency to create an event \(i\rightarrow j\) when \(j\) has a high score on an attribute.

Structural \(+\) Attribute effect

effect rate choice choice coordination
tertius

Tertius (tertius())

tertius(network, attribute,
  isTwoMode = FALSE, window = Inf,
  ignoreRep = FALSE, type = c("alter", "ego"), transformFun = identity,
  aggregateFun = function(x) mean(x, na.rm = TRUE)
)

\[\begin{align} r(i, t, x, z) &= \frac{\sum_{j:~x(t)_{ji} > 0}{z(t)_j}}{\sum_{j}{I(x(t)_{ji} > 0)}} && \text{ego type}\\ s(i,j, t, x, z) &= \frac{\sum_{k:~x(t)_{kj} > 0}{z(t)_k}}{\sum_{k}{I(x(t)_{kj} > 0)}} && \text{alter type} \end{align}\]

  • DyNAM-Rate submodel: tendency of actor \(i\) to create an event when \(i\) has a high aggregate (aggregateFun) value of its in-neighbors (\(\forall j:~ x[j, i] > 0\)) in a covariate network (‘ego’ type).
  • DyNAM-choice/choice_coordination submodels: tendency to create an event \(i\rightarrow j\) when \(j\) has a high aggregate (aggregateFun) value of its in-neighbors (\(\forall k:~ x[k, j] > 0\)) in a covariate network (‘alter’ type).
  • REM model: tendency to create an event \(i\rightarrow j\) when either \(i\) or \(j\) has a high aggregate (aggregateFun) value of its in-neighbors in a covariate network. The argument type allows to choose whether to use the tertius effect for sender \(i\) (type = "ego") or for receiver \(j\) (type = "alter").

Note: When a node does not have in-neighbors, the tertius effect is impute as the average of the aggregate values of nodes with in-neighbors.

Dyadic statistics

Structural effects

effect rate choice choice coordination
tie ×
inertia ×
recip × ×

Tie (tie())

tie(network,
  weighted = FALSE, window = Inf, ignoreRep = FALSE,
  transformFun = identity
)

\[\begin{equation} s(i,j, t, x) = \begin{cases} I(x(t)_{ij}>0) & \text{unweighted}\\ x(t)_{ij} & \text{weighted} \end{cases} \end{equation}\]

Tendency to create an event \(i\rightarrow j\) if the tie \(i\rightarrow j\) exists in a covariate network. Parameter weighted can be set to TRUE if the value in the covariate network for the dyad \(i\rightarrow j\) is to be taken as a statistic. It can be transformed by using transformFun (This might make sense when weighted = TRUE).

Inertia (inertia())

inertia(network, weighted = FALSE, window = Inf, transformFun = identity)

\[\begin{equation} s(i,j, t, x) = \begin{cases} I(x(t)_{ij}>0) & \text{unweighted}\\ x(t)_{ij} & \text{weighted} \end{cases} \end{equation}\]

Usually used as the “intercept” for the choice submodel, inertia is the tendency to create an event \(i\rightarrow j\) if the event \(i\rightarrow j\) happened before. It can be interpreted as the differential tendency to update existing ties rather than creating new ones. Thus, inertia is similar to tie, but defined on the network to which the dependent events relate. Parameter weighted can be set to TRUE if the count of past events \(i\rightarrow j\) is to be taken as a statistic. It can be transformed by using transformFun (this might make sense when weighted = TRUE). Note: inertia can never be used in combination with a ignoreRep = TRUE parameter as this would replace all positive statistics with zeros.

Tie reciprocation (recip())

recip(network,
  weighted = FALSE, window = Inf, ignoreRep = FALSE,
  transformFun = identity
)

\[\begin{equation} s(i,j, t, x) = \begin{cases} I(x(t)_{ji}>0) & \text{unweighted}\\ x(t)_{ji} & \text{weighted} \end{cases} \end{equation}\]

Effect of a tie j->i on event \(i\rightarrow j\). Recip cannot be used with binary = T in undirected dependent network

Tendency to create an event \(i\rightarrow j\) if one or several \(j\rightarrow i\) happened before. Parameter weighted can be set to TRUE if the count/weight of \(j\rightarrow i\) events/ties is to be taken as a statistic. It can be transformed by using transformFun (this might make sense when weighted = TRUE). This effect cannot be used for two-mode networks and for DyNAM-choice_coordination submodel.

Attribute effects

effect rate choice choice coordination
same ×
diff ×
sim ×
egoAlterInt ×

Same value in attribute (same())

same(attribute)

\[\begin{equation} s(i,j, t, z) = I(z(t)_i = z(t)_j) \end{equation}\]

Homophily (same value). The tendency of an event \(i\rightarrow j\) to happen if actors i and j have the same attribute value. This effect cannot be used for two-mode networks and for the DyNAM-rate submodel.

Differences in attribute (diff())

diff(attribute, transformFun = abs)

\[\begin{equation} s(i,j, t, z) = |z(t)_i - z(t)_j| \end{equation}\]

Heterophily. The tendency of an event \(i\rightarrow j\) to happen if actors i and j have different attribute values (high absolute differences regarding attribute if transformFun = abs). This effect cannot be used for two-mode networks.

Similarity in attribute (sim())

sim(attribute, transformFun = abs)

\[\begin{equation} s(i,j, t, z) = -|z(t)_i - z(t)_j| \end{equation}\]

Homophily (similar value). The tendency of an event \(i\rightarrow j\) to happen if actors i and j have similar attribute values (low absolute differences regarding attribute if transformFun = abs). This effect cannot be used for two-mode networks.

Ego Alter interaction (egoAlterInt())

egoAlterInt(attribute = list(attribute1, attribute2))

\[\begin{equation} s(i,j, t, z^{(1)}, z^{(2)}) = z(t)_i^{(1)} * z(t)_j^{(2)} \end{equation}\]

In a model that includes an alter effect using attribute2, the ego alter interaction helps to study the tendency to create an event \(i \rightarrow j\) when \(j\) score high on attribute2 moderated by the score of ego on attribute1.

Structural \(+\) Attribute effect

effect rate choice choice coordination
tertiusDiff ×

Tertius difference (tertiusDiff())

tertiusDiff(network, attribute,
  isTwoMode = FALSE, weighted = FALSE,
  window = Inf, ignoreRep = FALSE, transformFun = abs,
  aggregateFun = function(x) mean(x, na.rm = TRUE)
)

\[\begin{equation} s(i,j, t, x, z) = \left|\frac{z(t)_i - \sum_{k:~x(t)_{kj} > 0}{z(t)_k}}{ \sum_{k}{I(x(t)_{kj} > 0)}}\right| \end{equation}\]

The tendency to create an event \(i\rightarrow j\) when \(i\) has a similar value as \(j\) aggregate (aggregateFun) value of its in-neighbors (\(\forall j:~ x[j, i] > 0\)). Note: When the node \(j\) does not have in-neighbors, its value is imputed by the average of the similarities computed for the pairs \(i\), \(k\) for all \(k\) that has at least one in-neighbor.

Closure effect statistics (Dyadic effects)

effect rate choice choice coordination
trans ×
cycle × ×
commonSender × ×
commonReceiver × ×
four ×
mixedTrans ×
mixedCycle × ×
mixedCommonSender × ×
mixedCommonReceiver × ×

Transitivity (trans())

trans(network, window = Inf, ignoreRep = FALSE, transformFun = identity)

\[\begin{equation} s(i,j, t, x) = \sum_{k}{I(x(t)_{ik}>0)I(x(t)_{kj}>0)} \end{equation}\]

It is the tendency to create an event \(i\rightarrow j\) when it closes more two-paths (\(i\rightarrow k\rightarrow j\)) observed in the past events in a covariate network. It can be transformed by using transformFun. This effect cannot be used for two-mode networks.

Cycle (cycle())

cycle(network, window = Inf, ignoreRep = FALSE, transformFun = identity)

\[\begin{equation} s(i,j, t, x) = \sum_{k}{I(x(t)_{jk}>0)I(x(t)_{ki}>0)} \end{equation}\]

It is the tendency to create an event \(i\rightarrow j\) when it closes more two-paths (\(j\rightarrow k\rightarrow i\)) observed in the past events in a covariate network. It can be transformed by using transformFun. This effect cannot be used for two-mode networks and DyNAM-choice_coordination.

Common sender closure (commonSender())

commonSender(network, window = Inf, ignoreRep = FALSE, transformFun = identity)

\[\begin{equation} s(i,j, t, x) = \sum_{k}{I(x(t)_{ki}>0)I(x(t)_{kj}>0)} \end{equation}\]

It is the tendency to create an event \(i\rightarrow j\) when it closes more two-paths (\(i\leftarrow k\rightarrow j\)) observed in the past events in a covariate network. It can be transformed by using transformFun. This effect cannot be used for two-mode networks and DyNAM-choice_coordination.

Common receiver closure (commonReceiver())

commonReceiver(network,
  window = Inf, ignoreRep = FALSE,
  transformFun = identity
)

\[\begin{equation} s(i,j, t, x) = \sum_{k}{I(x(t)_{ik}>0)I(x(t)_{jk}>0)} \end{equation}\]

It is the tendency to create an event \(i\rightarrow j\) when it closes more two-paths (\(i\rightarrow k\leftarrow j\)) observed in the past events in a covariate network. It can be transformed by using transformFun. This effect cannot be used for two-mode networks and DyNAM-choice_coordination.

Four cycle (four())

four(network,
  isTwoMode = FALSE, window = Inf, ignoreRep = FALSE,
  transformFun = identity
)

\[\begin{equation} s(i,j, t, x) = \sum_{kl}{I(x(t)_{ik}>0)I(x(t)_{lk}>0)I(x(t)_{lj}>0)} \end{equation}\]

Closure of three-paths. It is the tendency to create an event \(i\rightarrow j\) when it closes more three-paths (\(i\rightarrow k\leftarrow l\rightarrow j\)) observed in the past events in a covariate network. It can be transformed by using transformFun. This effect cannot be used for two-mode networks and DyNAM-choice_coordination.

Mixed Transitivity (mixedTrans())

mixedTrans(
  network = list(network1, network2), window = Inf,
  ignoreRep = FALSE, transformFun = identity
)

\[\begin{equation} s(i,j, t, x^{(1)}, x^{(2)}) = \sum_{k}{I(x(t)_{ik}^{(1)}>0)I(x(t)_{kj}^{(2)}>0)} \end{equation}\]

Transitivity within 2 networks. It is the tendency to create an event \(i\rightarrow j\) when it closes more two-paths with events (\(i\rightarrow k\)) in network1 and (\(k\rightarrow j\)) in network2 observed in the past events in the covariate networks. It can be transformed by using transformFun. This effect cannot be used for two-mode networks.

Mixed Cycle (mixedCycle())

mixedCycle(
  network = list(network1, network2), window = Inf,
  ignoreRep = FALSE, transformFun = identity
)

\[\begin{equation} s(i,j, t, x^{(1)}, x^{(2)}) = \sum_{k}{I(x(t)_{ki}^{(1)}>0)I(x(t)_{jk}^{(2)}>0)} \end{equation}\]

Cycle within 2 networks. It is the tendency to create an event \(i\rightarrow j\) when it closes more two-paths with events (\(k\rightarrow i\)) in network1 and (\(j\rightarrow k\)) in network2 observed in the past events in the covariate networks. It can be transformed by using transformFun. This effect cannot be used for two-mode networks and DyNAM-choice_coordination.

Mixed common sender closure (mixedCommonSender())

mixedCommonSender(
  network = list(network1, network2), window = Inf,
  ignoreRep = FALSE, transformFun = identity
)

\[\begin{equation} s(i,j, t, x^{(1)}, x^{(2)}) = \sum_{k}{I(x(t)_{ki}^{(1)}>0)I(x(t)_{kj}^{(2)}>0)} \end{equation}\]

Closure common sender within 2 networks. It is the tendency to create an event \(i\rightarrow j\) when it closes more two-paths with events (\(k\rightarrow i\)) in network1 and (\(k\rightarrow j\)) in network2 observed in the past events in the covariate networks. It can be transformed by using transformFun. This effect cannot be used for two-mode networks and DyNAM-choice_coordination.

Mixed common receiver closure (mixedCommonReceiver())

mixedCommonReceiver(
  network = list(network1, network2), window = Inf,
  ignoreRep = FALSE, transformFun = identity
)

\[\begin{equation} s(i,j, t, x^{(1)}, x^{(2)}) = \sum_{k}{I(x(t)_{ik}^{(1)}>0)I(x(t)_{jk}^{(2)}>0)} \end{equation}\]

Closure receiver within 2 networks. It is the tendency to create an event \(i\rightarrow j\) when it closes more two-paths with events (\(i\rightarrow k\)) in network1 and (\(j\rightarrow k\)) in network2 observed in the past events in the covariate networks. It can be transformed by using transformFun. This effect cannot be used for two-mode networks and DyNAM-choice_coordination.

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