---
title: "goldfish Effects"
subtitle: "Catalog of available effects"
author: "Alvaro Uzaheta and Siwei Zhang"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{goldfish Effects}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r setup, include = FALSE}
#| purl = FALSE
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>",
  fig.path = "teaching/",
  eval = FALSE,
  purl = FALSE
)
```

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:

* **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 | âˆš | âˆš | âˆš | âˆš | âˆš | Ã— | âˆš |
|  |  | outdeg | âˆš | âˆš | âˆš | âˆš | âˆš | Ã— | âˆš |
|  |  | nodeTrans | Ã— | âˆš | âˆš | âˆš | âˆš | Ã— | âˆš |
|  | Attribute | ego | âˆš | Ã— | Ã— | Ã— | Ã— | Ã— | Ã— |
|  |  | alter | âˆš | Ã— | Ã— | Ã— | Ã— | Ã— | Ã— |
|  | Structural + attribute | tertius | âˆš | âˆš | âˆš | âˆš | âˆš | âˆš | âˆš |
| Dyadic statistics | Structural | tie | âˆš | âˆš | âˆš | âˆš | âˆš | Ã— | Ã— |
|  |  | inertia | âˆš | âˆš | âˆš | âˆš | âˆš | Ã— | Ã— |
|  |  | recip | âˆš | âˆš | âˆš | âˆš | âˆš | Ã— | Ã— |
|  | Attribute | same | âˆš | Ã— | Ã— | Ã— | âˆš | Ã— | Ã— |
|  |  | diff | âˆš | Ã— | Ã— | Ã— | âˆš | Ã— | Ã— |
|  |  | sim | âˆš | Ã— | Ã— | Ã— | âˆš | Ã— | Ã— |
|  |  | egoAlterInt | âˆš | Ã— | Ã— | Ã— | âˆš | Ã— | Ã— |
|  | Structural + attribute | tertiusDiff | âˆš | âˆš | âˆš | âˆš | âˆš | âˆš | Ã— |
|  | Closure effects | trans | Ã— | âˆš | Ã— | âˆš | âˆš | Ã— | Ã— |
|  |  | cycle | Ã— | âˆš | Ã— | âˆš | âˆš | Ã— | Ã— |
|  |  | commonSender | âˆš | âˆš | Ã— | âˆš | âˆš | Ã— | Ã— |
|  |  | commonReceiver | âˆš | âˆš | Ã— | âˆš | âˆš | Ã— | Ã— |
|  |  | four | âˆš | âˆš | Ã— | âˆš | âˆš | Ã— | Ã— |
|  |  | mixedTrans | Ã— | âˆš | Ã— | âˆš | âˆš | Ã— | Ã— |
|  |  | mixedCycle | Ã— | âˆš | Ã— | âˆš | âˆš | Ã— | Ã— |
|  |  | mixedCommonSender | Ã— | âˆš | Ã— | âˆš | âˆš | Ã— | Ã— |
|  |  | mixedCommonReceiver | Ã— | âˆš | Ã— | âˆš | âˆš | Ã— | Ã— |


_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

* $x(t)_{ij}$ 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. 
  
# Node or actor statistics

## Structural Effects
| effect          |   rate   |  choice  | choice coordination |
|-----------------|:--------:|:--------:|:-------------------:|
|     indeg       |       âˆš  |       âˆš  |            âˆš        |
| outdeg          |       âˆš  |       âˆš  |           Ã—         |
| nodeTrans       |       âˆš  |       âˆš  |            âˆš        |

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

### Indegree (`indeg()`)

```{r}
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 \code{transformFun}.


### Outdegree (`outdeg()`)

```{r}
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()`)

```{r}
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 \code{transformFun},
there is not weighted version for this effect.

## Attribute effects

| effect          |   rate   |  choice  | choice coordination |
|-----------------|:--------:|:--------:|:-------------------:|
|     ego         |       âˆš  |     Ã—    |           Ã—         |
|     alter       |     Ã—    |       âˆš  |           Ã—         |


### Ego attribute (`ego()`)

```{r}
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()`)

```{r}
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         |       âˆš  |       âˆš  |            âˆš        |

### Tertius (`tertius()`)

```{r}
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         |     Ã—    |     âˆš    |            âˆš        |
|     inertia     |     Ã—    |     âˆš    |            âˆš        |
|     recip       |     Ã—    |     âˆš    |           Ã—         |

### Tie (`tie()`)

```{r}
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()`)

```{r}
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()`)

```{r}
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        |     Ã—    |     âˆš    |            âˆš        |
|     diff        |     Ã—    |     âˆš    |            âˆš        |
|     sim         |     Ã—    |     âˆš    |            âˆš        |
| egoAlterInt     |     Ã—    |     âˆš    |            âˆš        |


### Same value in attribute (`same()`)

```{r}
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()`)

```{r}
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()`)

```{r}
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()`)

```{r}
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     |     Ã—    |     âˆš    |            âˆš        |

### Tertius difference (`tertiusDiff()`)

```{r}
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       |     Ã—    |     âˆš    |            âˆš        |
| cycle           |     Ã—    |     âˆš    |           Ã—         |
| commonSender        |     Ã—    |     âˆš    |           Ã—         |
| commonReceiver      |     Ã—    |     âˆš    |           Ã—         |
|     four        |     Ã—    |     âˆš    |            âˆš        |
|     mixedTrans  |     Ã—    |     âˆš    |            âˆš        |
| mixedCycle      |     Ã—    |     âˆš    |           Ã—         |
| mixedCommonSender   |     Ã—    |     âˆš    |           Ã—         |
| mixedCommonReceiver |     Ã—    |     âˆš    |           Ã—         |

### Transitivity (`trans()`)

```{r}
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()`)

```{r}
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()`)

```{r}
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()`)

```{r}
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()`)

```{r}
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()`)

```{r}
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()`)

```{r}
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()`)

```{r}
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()`)

```{r}
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.