How to model the GWESP Effect (and more) in Goldfish?

[SamALIENWARE]

In Stadtfeld et al.'s (2017) introduction paper of DyNAM, They stated the following:

"Higher order effects and effects that involve more than three nodes can be specified. These extend relational algebra to implicate further ties in dependencies. Examples include geometrically weighted edgewise shared partners’ effects (for a detailed discussion, see Snijders et al.2006) or attractiveness mechanisms such as preferential attachment."

However, I have not seen the realization of gwesp or other higher-order effects in the goldfish package, on the rdrr.io page or the github page. I do appreciate that integrating every function into the package is massive work. However, I wonder if it is possible to estimate gwesp or other higher order effects with this version of Goldfish? Alternatively, is it possible to share some related codes so that I could learn how to write a customized effect estimation function myself, please?

[auzaheta]

Hi Sam,

I think this is a relevant question. I'm splitting my comments about it into two parts:

GWESP

The geometrically weighted edgewise partners effects family is base on triadic effect as far I understand it. According to the Rsiena documentation (page 131), you can have five versions of the effect, which depends on the direction that you are interested in. From there, I think you can have a GWESP effect with a fixed α. For all value using the transformFun argument of some of the triadic effects implemented in goldfish. For example, the transitivity effect is the main component of what is used for the GWESPFF on formula 18a (Rsiena manual). The following code should do the trick:

#' GWESP transformation of goldfish closure effects
#' A transformation function generator to compute GWESP as describe in RSiena manual
#'
#' Using closure idea from Advance R Wickham book,
#' it implements formula 19 (pg. 133) RSiena manual
#' @param alpha tuning parameter that may range from 0 to \infty, RSiena manual recommend a default value of log(2) 
#' @param eventNetStat event network statistic that compute value $\sum_h x_{ih}{hj}$ in the case of GWESPFF
#'
#' @return transform statistic value
#' @noRd
#'
#' @examples
#' GWESPlog2 <- funTransformationGWESP(log(2))
#' GWESPlog2(0) # 0
#' GWESPlog2(1) # 1
#' GWESPlog2(6) # maximum slightly less than exp(log(2)), i.e., less than 2
funTransformationGWESP <- function(alpha) {
  function(eventNetStat)
    exp(alpha) * (1 - (1 - exp(-1 * alpha)) ^ eventNetStat)
}

# example of use
data("Social_Evolution")
callNetwork <- defineNetwork(nodes = actors, directed = TRUE)
callNetwork <- linkEvents(x = callNetwork, changeEvent = calls, nodes = actors)
callsDependent <- defineDependentEvents(events = calls, nodes = actors,
                                        defaultNetwork = callNetwork)
GWESPFFlog2 <- funTransformationGWESP(log(2))
mod01 <- estimate(callsDependent ~ inertia + recip + trans(callNetwork, transformFun = funTransformationGWESP(log(2))),
                  model = "DyNAM", subModel = "choice")
summary(mod01)

I hope @stadtfeldethz or @jhollway could correct me if I'm wrong with this approach.

According to the Rsiena documentation, you can have five versions of the GWESP effects. Right now, goldfish has four of the five triadic base effects you would need to have the corresponding GWESP effect, three of them are right now undocumented, and in the develop branch, the following table provides the equivalence and the code to use with a fix α value of log(2).

GWESP effect	goldfish base effect	Use with α= log(2)
gwespFF	transitivity	r trans(network, transformFun = funTransformationGWESP(log(2)))
gwespBB	cycle	r cycle(network, transformFun = funTransformationGWESP(log(2)))
gwespFB	sender closure	r clSender(network, transformFun = funTransformationGWESP(log(2)))
gwespBF	receiver closure	r clReceiver(network, transformFun = funTransformationGWESP(log(2)))
gwespRR	not available

The remaining problem is tunning the alpha parameter. For this, you can do a grid search and keep the model with better information criteria (a similar approach has been made for the size of the window for windowed effects).

Higher-order effects

If you have a specific effect in mind, I'm happy to help with the process. I'll prepare a vignette with a tutorial on developing new effects. In the meantime, you can take a look at the transitivity effect and the four effect as examples of effects involving three and four actors.

Let me know if you need more info to proceed. I think I could put together an initial short raw version of the tutorial in a new discussion thread.

[SamALIENWARE]

Thank you for your explanation.

What if the networks are undirected (strategic alliance network with firms collaborating/coordinating with each other)? Would the same effects mentioned here be available, except for the direction of sender /receiver of course?

[auzaheta]

In the undirected case, it is not possible to distinguish between the different closure effects. All of them would report the same information. In that case, you could only use one of them.