Test whether or not a network estimates can be considered structurally dependent, i.e. a function of the network structure. By rewiring the graph and calculating a particular statistic \(t\), the test compares the observed mean of \(t\) against the empirical distribution of it obtained from rewiring the network.
n_rewires(graph, p = c(20L, rep(0.1, nslices(graph) - 1)))struct_test(graph, statistic, R, rewire.args = list(), ...)
# S3 method for diffnet_struct_test
c(..., recursive = FALSE)
# S3 method for diffnet_struct_test
print(x, ...)
# S3 method for diffnet_struct_test
hist(
x,
main = "Empirical Distribution of Statistic",
xlab = expression(Values ~ of ~ t),
breaks = 20,
annotated = TRUE,
b0 = expression(atop(plain("") %up% plain("")), t[0]),
b = expression(atop(plain("") %up% plain("")), t[]),
ask = TRUE,
...
)
struct_test_asymp(graph, Y, statistic_name = "distance", p = 2, ...)
A list of class diffnet_struct_test containing the following:
The graph passed to struct_test.
The resulting p-value of the test (see details).
The observed value of the statistic.
The average value of the statistic applied to the simulated networks.
Number of simulations.
The function statistic passed to struct_test.
A boot class object as return from the call to boot.
The list rewire.args passed to struct_test.
A diffnet graph.
Either a Numeric scalar or vector of length nslices(graph)-1
with the number of rewires per links.
A function that returns either a scalar or a vector.
Integer scalar. Number of repetitions.
List. Arguments to be passed to rewire_graph
Further arguments passed to the method (see details).
Ignored
A diffnet_struct_test class object.
Character scalar. Title of the histogram.
Character scalar. x-axis label.
Passed to hist.
Logical scalar. When TRUE marks the observed data average and the simulated data average.
Character scalar. When annotated=TRUE, label for the value of b0.
Character scalar. When annotated=TRUE, label for the value of b.
Logical scalar. When TRUE, asks the user to type <Enter> to see each plot (as
many as statistics where computed).
Numeric vector of length \(n\).
Character scalar. Name of the metric to compute. Currently
this can be either "distance",">","<","==", ">=",
or "<=".
George G. Vega Yon
struct_test computes the test by generating the null distribution using
Monte Carlo simulations (rewiring). struct_test_asymp computes the
test using an asymptotic approximation. While available, we do not recommend
using the asymptotic approximation since it has not shown good results when
compared to the MC approximation. Furthermore, the asymptotic version has only
been implemented for graph as static graph.
The output from the hist method is the same as hist.default.
struct_test is a wrapper for the function boot from the
boot package. Instead of resampling data--vertices or edges--in each iteration the function
rewires the original graph using rewire_graph and applies
the function defined by the user in statistic.
The default values to rewire_graph via rewire.args are:
p | Number or Integer with default n_rewires(graph). |
undirected | Logical scalar with default getOption("diffnet.undirected", FALSE). |
copy.first | Logical scalar with TRUE. |
algorithm | Character scalar with default "swap". |
In struct_test ... are passed to boot, otherwise are passed
to the corresponding method (hist for instance).
From the print method, p-value for the null of the statistic been
equal between graph and its rewired versions is computed as follows
$$% p(\tau)=2\times\min\left(\mbox{Pr}(t\leq\tau), \mbox{Pr}(t\geq\tau)\right) % $$
Where \(\mbox{Pr}\{\cdot\}\) is approximated using the Empirical Distribution Function retrieved from the simulations.
For the case of the asymptotic approximation, under the null we have
$$% \sqrt{n}\left(\hat\beta(Y,G)-\mu_\beta\right)\sim^d\mbox{N}\left(0,\sigma_\beta^2\right) $$
The test is actually on development by Vega Yon and Valente. A copy of the working paper can be distributed upon request to g.vegayon@gmail.com.
The function n_rewires proposes a vector of number of rewirings that
are performed in each iteration.
Vega Yon, George G. and Valente, Thomas W. (On development).
Davidson, R., & MacKinnon, J. G. (2004). Econometric Theory and Methods. New York: Oxford University Press.
Other Functions for inference:
bootnet(),
moran()
# Creating a random graph
set.seed(881)
diffnet <- rdiffnet(100, 5, seed.graph="small-world")
# Testing structure-dependency of threshold
res <- struct_test(
diffnet,
function(g) mean(threshold(g), na.rm=TRUE),
R=100
)
res
hist(res)
# Adding a legend
legend("topright", bty="n",
legend=c(
expression(t[0]:~Baseline),
expression(t:~Rewired~average)
)
)
# Concatenating results
c(res, res)
# Running in parallel fashion
res <- struct_test(
diffnet, function(g) mean(threshold(g), na.rm=TRUE),
R=100, ncpus=2, parallel="multicore"
)
res
hist(res)
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