# Custom Network Statistics

#### Donny Williams

#### 5/19/2020

# Background

This vignette describes a new feature to BGGM (`2.0.0`) that allows for computing custom network statistics (e.g., centrality). The new function is called `roll_your_own` and it was suggested by a user of BGGM (see feature request here).

# Basic Idea

The basic idea is to compute the chosen network statistic for each of the sampled partial correlation matrices, resulting in a distribution. All that is required is to define a function that takes either a partial correlation matrix or a weighted adjacency matrix (the partial correlation matrix with values set to zero) as the first argument. Several examples are provided below.

### R packages

    # need the developmental version
    if (!requireNamespace("remotes")) {
      install.packages("remotes")
    }

    # install from github
    remotes::install_github("donaldRwilliams/BGGM")

### Data

In all examples, a subset of `ptsd` data is used. The subset includes two of the “communities” of symptoms (details for these data can be found in Armour et al. 2017). The data are ordinal (5-level Likert).

    # need these packages
    library(BGGM)
    library(ggplot2)
    library(assortnet)
    library(networktools)

    # data
    Y <- ptsd[,1:7]

### Fit Model

For these data, the GGM is estimated with a semi-parametric copula (Hoff 2007). In BGGM, this implemented with `type = mixed` which is kind of a misnomer because the data do not have to be “mixed” (consisting of continuous and discrete variables). Note that the model is fitted only once which highlights that only the posterior samples are needed to compute any network statistic.

    library(BGGM)

    # copula ggm
    fit <- estimate(Y, type = "mixed", iter = 1000)

# Examples

## Expected Influence

The first example computes expected influence (Robinaugh, Millner, and McNally 2016). The first step is to define a function

    # define function
    f <- function(x,...){
      networktools::expectedInf(x,...)$step1
    }

Note that `x` takes the matrix which is then passed to `expectedInf`. The `...` allows for passing additional arguments to the `expectedInf` function. An example is provided below. With the function defined, the next step is to compute the network statistic.

    # iter = 250 for demonstrative purposes
    # (but note even 1000 iters takes less than 1 second)
    # compute
    net_stat <- roll_your_own(object = fit,
                              FUN = f,
                              select = FALSE,
                              iter = 250)
    # print
    net_stat

    #> BGGM: Bayesian Gaussian Graphical Models
    #> ---
    #> Network Stats: Roll Your Own
    #> Posterior Samples: 250
    #> ---
    #> Estimates:
    #>
    #>  Node Post.mean Post.sd Cred.lb Cred.ub
    #>     1     0.701   0.099   0.508   0.871
    #>     2     0.912   0.113   0.722   1.179
    #>     3     0.985   0.112   0.742   1.199
    #>     4     1.056   0.105   0.851   1.247
    #>     5     1.056   0.116   0.862   1.288
    #>     6     0.491   0.092   0.329   0.679
    #>     7     0.698   0.098   0.521   0.878
    #> ---

The option `select = FALSE` indicates to compute the statistics from the partial correlation matrices (nothing set to zero). This can be changed with `select = TRUE`. Internally, each of the sampled partial correlation matrices is multiplied by the adjacency matrix.

    net_stat <- roll_your_own(object = fit,
                              FUN = f,
                              select = TRUE,
                              iter = 250)

    # print
    net_stat

    #> BGGM: Bayesian Gaussian Graphical Models
    #> ---
    #> Network Stats: Roll Your Own
    #> Posterior Samples: 250
    #> ---
    #> Estimates:
    #>
    #>  Node Post.mean Post.sd Cred.lb Cred.ub
    #>     1     0.636   0.136   0.386   0.874
    #>     2     0.792   0.113   0.580   0.996
    #>     3     0.777   0.122   0.544   1.001
    #>     4     0.910   0.121   0.667   1.143
    #>     5     0.525   0.104   0.331   0.727
    #>     6     0.484   0.110   0.270   0.686
    #>     7     0.247   0.081   0.088   0.412
    #> ---

The results are then plotted with

    plot(net_stat)

## Bridge Strength

The next example computes bridge strength (Jones, Ma, and McNally 2019). This requires the user to define clusters or “communities”.

    # clusters
    communities <- substring(colnames(Y), 1, 1)

    # function is slow
    f <- function(x, ...){
    networktools::bridge(x, ...)$`Bridge Strength`
    }

    # compute
    net_stat <- roll_your_own(object = fit,
                              FUN = f,
                              communities = communities,
                              iter = 250)

    # print
    net_stat

    #> BGGM: Bayesian Gaussian Graphical Models
    #> ---
    #> Network Stats: Roll Your Own
    #> Posterior Samples: 250
    #> ---
    #> Estimates:
    #>
    #>  Node Post.mean Post.sd Cred.lb Cred.ub
    #>     1     0.162   0.082   0.035   0.347
    #>     2     0.250   0.113   0.061   0.501
    #>     3     0.180   0.104   0.049   0.480
    #>     4     0.280   0.098   0.090   0.480
    #>     5     0.375   0.093   0.196   0.558
    #>     6     0.617   0.166   0.339   1.002
    #>     7     0.628   0.166   0.400   1.025
    #> ---

Notice `communities`. This is passed to `...` in the function `f`, which, in turn, is passed to the function `bridge`. Any number of arguments can be passed this way. Here are the results

This can then be plotted and further customized (the returned object is a `ggplot`)

    plot(net_stat,
         fill = "lightblue") +
      ggtitle("Bridge Strength") +
      xlab("Score")

## Assortment

The next example computes assortment (Newman 2003).

    # clusters
    communities <- substring(colnames(Y), 1, 1)

    # define function
    f <- function(x,...){
      assortnet::assortment.discrete(x, ...)$r
    }

    net_stat <- roll_your_own(object = fit,
                              FUN = f,
                              types = communities,
                              weighted = TRUE,
                              SE = FALSE, M = 1,
                              iter = 250)

    # print
    net_stat

    #> BGGM: Bayesian Gaussian Graphical Models
    #> ---
    #> Network Stats: Roll Your Own
    #> Posterior Samples: 250
    #> ---
    #> Estimates:
    #>
    #>  Post.mean Post.sd Cred.lb Cred.ub
    #>      0.261   0.124   -0.01   0.469
    #> ---

This example demonstrate that `...` can take several arguments. The results are stored in the `net_stat` object. They can be accessed with

    hist(net_stat$results, main = "Assortment")

# Note

The function `roll_your_own` is expecting the custom function to return either a single number or a number for each node. This ensures all the printing and plotting functions work. However, you could return anything you want and then access the results to plot, summarize, etc.

# References

Armour, Cherie, Eiko I Fried, Marie K Deserno, Jack Tsai, and Robert H Pietrzak. 2017. “A Network Analysis of Dsm-5 Posttraumatic Stress Disorder Symptoms and Correlates in Us Military Veterans.” Journal of Anxiety Disorders 45: 49–59. https://doi.org/10.31234/osf.io/p69m7.

Hoff, Peter D. 2007. “Extending the Rank Likelihood for Semiparametric Copula Estimation.” The Annals of Applied Statistics 1 (1): 265–83. https://doi.org/10.1214/07-AOAS107.

Jones, Payton J, Ruofan Ma, and Richard J McNally. 2019. “Bridge Centrality: A Network Approach to Understanding Comorbidity.” Multivariate Behavioral Research, 1–15. https://doi.org/10.1080/00273171.2019.1614898.

Newman, Mark EJ. 2003. “Mixing Patterns in Networks.” Physical Review E 67 (2): 026126.

Robinaugh, Donald J, Alexander J Millner, and Richard J McNally. 2016. “Identifying Highly Influential Nodes in the Complicated Grief Network.” Journal of Abnormal Psychology 125 (6): 747.