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 functions have the following 5 arguments:

**network(s)/attribute(s)**: Objects upon which the effect should be calculated. For network effects, this would be either an explanatory network or the dependent network (the default). For attribute effects, this should be the attribute name with the indication of the data frame containing the initial values of nodes attributes (`data.frame$attribute`

), see documentation of`defineNodes()`

for details.**isTwoMode**: Identifies whether the effect assumes the network is originated from a two-mode network or not. The default value is`FALSE`

, it means that updates involving self-ties are ignore during the statistics update.**ignoreRep**: Identifies whether the effect recognizes actors to send additional ties beyond the first to receivers (`FALSE`

means additional ties are taken into account). The default value is`FALSE`

. Only available for structural effects.**weighted**: Identifies whether the effect relies on the presence (dummy value) or number of ties (`TRUE`

means it relies also on the number of ties). The default value is`FALSE`

.**window**: Identifies a window length within which changes should apply to events. The default value is`Inf`

meaning that no windows are applied to the effect. Window size can be specified as a number in seconds (i.e. an hour is 3600 seconds), or as a string stating the number of time units in the format “number unit”, for example “30 seconds”, “1 minute”, “2 hours”, “3 days”, “4 weeks”, “5 months”, or “6 years”.*Note:*The time units are converted to seconds using their most common lengths in seconds.**transformFun**: Use this parameter to obtain transformed statistics. The function used as argument is applied to the change statistic.*Note:*Most of the time a transformation function is applied when an effect counts the number of ties`weighted == TRUE`

.**aggregateFun**: Use this parameter to obtain a transformed aggregated statistics for indirect effects like`tertius`

and`tertiusDiff`

. The function used as argument is applied to aggregated statistic.**type**: Applies only to`indegree`

,`outdegree`

and`nodeTrans`

in the case of a REM model`model = "REM"`

. In the default case of`type = "alter"`

, the effect returns change statistics for the potential receivers. In the case where`type = "ego"`

, the effect returns change statistics for the potential sender. This argument does not apply in the case of the DyNAM models: the DyNAM-choice submodel only considers change statistics for potential receivers`type = "alter"`

, while the DyNAM-rate submodel only considers change statistics for potential senders`type = "ego"`

.

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.

- \(x(t)_{ij}\) denotes the value of network \(x\) at time \(t\) between actor \(i\) and actor \(j\).
- \(z(t)_i\) denotes the value of an actor attribute \(z\) at time \(t\) for actor \(i\).
- \(I(y)\) denotes the indicator
function. It takes the value 1 when variable \(y\) is
`TRUE`

and 0 in other case. - We refer as the weighted statistics to the outcome of the effect
when
`weighted = TRUE`

, similarly we refer as the unweighted statistics when`weighted = FALSE`

. - We omit the difference in the computation of the statistics when the
argument
`isTwoMode`

is used. In the default case when`isTwoMode = FALSE`

self ties are excluded in the calculation of the change statistics.

effect | rate | choice | choice coordination |
---|---|---|---|

indeg | √ | √ | √ |

outdeg | √ | √ | × |

nodeTrans | √ | √ | √ |

Here we refer to ego type when `type = "ego"`

and alter
type when `type = "alter"`

.

`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 .

`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"`

).

`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.

effect | rate | choice | choice coordination |
---|---|---|---|

ego | √ | × | × |

alter | × | √ | × |

`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()`

)`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.

effect | rate | choice | choice coordination |
---|---|---|---|

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.

effect | rate | choice | choice coordination |
---|---|---|---|

tie | × | √ | √ |

inertia | × | √ | √ |

recip | × | √ | × |

`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(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.

`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.

effect | rate | choice | choice coordination |
---|---|---|---|

same | × | √ | √ |

diff | × | √ | √ |

sim | × | √ | √ |

egoAlterInt | × | √ | √ |

`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.

`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.

`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.

`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`

.

effect | rate | choice | choice coordination |
---|---|---|---|

tertiusDiff | × | √ | √ |

`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.

effect | rate | choice | choice coordination |
---|---|---|---|

trans | × | √ | √ |

cycle | × | √ | × |

commonSender | × | √ | × |

commonReceiver | × | √ | × |

four | × | √ | √ |

mixedTrans | × | √ | √ |

mixedCycle | × | √ | × |

mixedCommonSender | × | √ | × |

mixedCommonReceiver | × | √ | × |

`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(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.

`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.

`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()`

)```
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.

`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.

`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.

`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.

`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.