From the dependency network of all CRAN packages, we have seen that reverse dependencies of all CRAN packages seem to follow the power law. This echoes the phenomenon in other software dependencies observed by, for example, LaBelle and Wallingford, 2004, Baxter et al., 2006, Jenkins and Kirk, 2007, Wu et al. 2007, Louridas et al., 2008, Zheng et al., 2008, Kohring, 2009, Li et al., 2013, Bavota et al. 2015, and Cox et al., 2015. In this vignette, we will fit the discrete power law to model this number of reverse dependencies, using functions with the suffix `upp`

, namely `dupp()`

, `Supp()`

and `mcmc_upp()`

. While we shall focus on “Imports”, which is one of the serveral kinds of dependencies in R, the same analysis can be carried out for all other types.

In the vignette on all CRAN pacakges, we looked at the dependency type `Depends`

. Here, we look at the dependency type `Imports`

, using the same workflow. As we are using the forward dependencies to construct the network, the number of reverse dependencies is equivalent to the in-degree of a package, which can be obtained via the function `igraph::degree()`

. We then construct a data frame to hold this in-degree information.

```
g0.imports <- get_graph_all_packages(type = "imports")
d0.imports <- g0.imports |> igraph::degree(mode = "in")
df0.imports <-
data.frame(name = names(d0.imports), degree = as.integer(d0.imports)) |>
dplyr::arrange(dplyr::desc(degree), name)
head(df0.imports, 10)
#> name degree
#> 1 dplyr 3444
#> 2 ggplot2 3273
#> 3 Rcpp 2600
#> 4 rlang 2315
#> 5 magrittr 1991
#> 6 stringr 1763
#> 7 tidyr 1727
#> 8 tibble 1677
#> 9 MASS 1588
#> 10 purrr 1559
```

For the purpose of verification, we use the reverse dependencies to construct the network, then look at the out-degrees of the packages this time.

```
g0.rev_imports <- get_graph_all_packages(type = "reverse imports")
d0.rev_imports <- g0.rev_imports |> igraph::degree(mode = "out") # note the difference to above
df0.rev_imports <-
data.frame(name = names(d0.rev_imports), degree = as.integer(d0.rev_imports)) |>
dplyr::arrange(dplyr::desc(degree), name)
head(df0.rev_imports, 10)
#> name degree
#> 1 dplyr 3444
#> 2 ggplot2 3273
#> 3 Rcpp 2600
#> 4 rlang 2315
#> 5 magrittr 1991
#> 6 stringr 1763
#> 7 tidyr 1727
#> 8 tibble 1677
#> 9 MASS 1588
#> 10 purrr 1559
```

Theoretically, the two data frames are identical. Any possible (but small) difference is due to the CRAN pages being updated while scraping.

We construct a data frame for the empirical frequencies and survival function at the whole range of data.

```
df1.imports <- df0.imports |>
dplyr::filter(degree > 0L) |> # to prevent warning when plotting on log-log scale
dplyr::count(degree, name = "frequency") |>
dplyr::arrange(degree) |>
dplyr::mutate(survival = 1.0 - cumsum(frequency) / sum(frequency))
```

Before fitting the discrete power law, to be determined first is a threshold above which it is appropriate. We will visualise the degree distribution to determine such threshold.

```
plot_log_log <- function(df) {
## useful function for below
df |>
ggplot2::ggplot() +
ggplot2::scale_x_log10() +
ggplot2::scale_y_log10() +
ggplot2::theme_bw(12)
}
gg0 <- df1.imports |>
plot_log_log() +
ggplot2::geom_point(aes(degree, frequency), size = 0.75) +
ggplot2::coord_cartesian(ylim = c(1L, 2e+3L))
gg0
```

The power law seems appropriate for the whole range of data, and so, for illustration purposes, the threshold will be set at 1 inclusive. 0’s will be excluded anyway because, as we will see, the probability mass function (PMF) is not well-defined at 0.

While it is straightforward to determine the threshold here, such linearity over the whole range might not be seen for other data. The package poweRlaw provides functions for and references to more systematic/objective procedures of selecting the threshold.

We use the function `mcmc_pol()`

to fit the discrete power law, of which the PMF is proportional to \(x^{-\alpha}\), where \(\alpha\) is the lone scalar parameter.

The Bayesian approach is used here for inference, meaning that a prior has to be set for the parameters. We assume a normal distribution \(N(m_{\alpha}=0, s_{\alpha}=10)\) for \(\alpha\). Markov chain Monte Carlo (MCMC) is used as the inference algorithm.

```
x <- df1.imports$degree
counts <- df1.imports$frequency
alpha0 <- 1.5 # initial value
theta0 <- 1.0 # initial value, but fixed to 1.0 for discrete power law
m_alpha <- 0.0 # prior mean
s_alpha <- 10.0 # prior standard deviation
set.seed(3075L)
mcmc0.imports <-
mcmc_pol(
x = x,
count = counts,
alpha = alpha0,
theta = theta0,
a_alpha = m_alpha,
b_alpha = s_alpha,
a_theta = 1.0,
b_theta = 1.0,
a_pseudo = 10.0,
b_pseudo = 1.0,
pr_power = 1.0,
iter = 2500L,
thin = 1L,
burn = 500L,
freq = 500L,
invt = 1.0,
mc3_or_marg = TRUE,
x_max = 100000
) # takes seconds
```

Now we have the samples representing the posterior distribution of \(\alpha\):

This means the number of reverse “Imports” follows approximately a power law with exponent 1.65. We can also obtain the posterior density via numerical integration using `marg_pol()`

. The returned list contains `log.marginal`

for the marginal log-likelihood, and `posterior`

for the posterior density in a data frame. We verify its alignment with the MCMC results:

```
marg0.imports <-
marg_pow(
data.frame(x = x, count = counts),
lower = 1.001,
upper = 20.0,
m_alpha = m_alpha,
s_alpha = s_alpha
)
ggplot2::last_plot() +
ggplot2::geom_line(ggplot2::aes(alpha, density), marg0.imports$posterior, col = 2) +
ggplot2::coord_cartesian(xlim = range(mcmc0.imports$pars$alpha))
```

We can also calculate the fitted frequencies and survival function, using `dpol()`

and `Spol()`

respectively. Their summaries are however readily available in the MCMC output. We overlay the fitted line (blue, dashed) and credible intervals (red, dotted) on the plot above to check goodness-of-fit:

```
n0 <- sum(df1.imports$frequency) # TOTAL number of data points in x
## or n0 <- length(x)
df0.fitted <- mcmc0.imports$fitted
gg1 <- df1.imports |>
plot_log_log() +
ggplot2::geom_point(aes(degree, frequency), size = 0.75) +
ggplot2::geom_line(aes(x, f_med * n0), data = df0.fitted, col = 4, lty = 2) +
ggplot2::geom_line(aes(x, f_025 * n0), data = df0.fitted, col = 2, lty = 3) +
ggplot2::geom_line(aes(x, f_975 * n0), data = df0.fitted, col = 2, lty = 3) +
ggplot2::coord_cartesian(ylim = c(1L, 2e+3L))
gg1
```

The corresponding survival function according to the power law fit can also be obtained:

```
gg2 <- df1.imports |>
plot_log_log() +
ggplot2::geom_point(aes(degree, survival), size = 0.75) +
ggplot2::geom_line(aes(x, S_med), data = df0.fitted, col = 4, lty = 2) +
ggplot2::geom_line(aes(x, S_025), data = df0.fitted, col = 2, lty = 3) +
ggplot2::geom_line(aes(x, S_975), data = df0.fitted, col = 2, lty = 3)
gg2
#> Warning: Transformation introduced infinite values in continuous y-axis
```

This shows the discrete power law doesn’t fit as good as it seems according to the frequency plot. To improve the fit, we use the discrete extreme value mixture distributions introduced in Lee, Eastoe and Farrell, 2024. Provided in this package for such purpose are `dmix2()`

, `Smix2()`

and `mcmc_mix2()`

. As it takes hours to run the inference algorithm using `mcmc_mix()`

, we hardcode the parameter values based on their component-wise posterior median, and overlay the data by the fitted line.

```
gg3 <- df1.imports |>
filter(degree > 1L) |>
dplyr::mutate(
survival = 1.0 - cumsum(frequency) / sum(frequency),
survival.mix = Smix2(degree, 1606, 1.73, 1.00, 0.237, 4.03, 0.003)
) |>
ggplot2::ggplot() +
ggplot2::geom_point(aes(degree, survival), size = 0.75) +
ggplot2::geom_line(aes(degree, survival.mix), col = 4, lty = 2, lwd = 1.2) +
ggplot2::scale_x_log10() +
ggplot2::scale_y_log10() +
ggplot2::theme_bw(12)
gg3
#> Warning: Transformation introduced infinite values in continuous y-axis
```

This illustrates the mixture distribution’s better fit.