Structure dependence test

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Description

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.

Usage

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n_rewires(graph, p = c(20L, rep(0.1, nslices(graph) - 1)))

struct_test(graph, statistic, R, rewire.args = list(p = n_rewires(graph),
  undirected = getOption("diffnet.undirected", FALSE), copy.first = TRUE,
  algorithm = "swap"), ...)

## S3 method for class 'diffnet_struct_test'
c(..., recursive = FALSE)

## S3 method for class 'diffnet_struct_test'
print(x, ...)

## S3 method for class '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[]), ...)

struct_test_asymp(graph, Y, statistic_name = "distance", p = 2, ...)

Arguments

graph

A diffnet graph.

p

Either a Numeric scalar or vector of length nslices(graph)-1 with the number of rewires per links.

statistic

A function that returns either a scalar or a vector.

R

Integer scalar. Number of repetitions.

rewire.args

List. Arguments to be passed to rewire_graph

...

Further arguments passed to the method (see details).

recursive

Ignored

x

A diffnet_boot class object.

main

Character scalar. Title of the histogram.

xlab

Character scalar. x-axis label.

breaks

Passed to hist.

annotated

Logical scalar. When TRUE marks the observed data average and the simulated data average.

b0

Character scalar. When annotated=TRUE, label for the value of b0.

b

Character scalar. When annotated=TRUE, label for the value of b.

Y

Numeric vector of length n.

statistic_name

Character scalar. Name of the metric to compute. Currently this can be either "distance",">","<","==", ">=", or "<=".

Details

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. In particular, the "swap" algorithm is used in order to preserve the degree sequence of the graph, in other words, each rewired version of the original graph has the same degree sequence.

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*min[Pr(t<=tau), Pr(t>=tau)]

Where Pr(.) 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)*[mean(beta) - E[beta]]/Var[beta] ~ N(0,1)

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.

Value

A list of class diffnet_bot containing the following:

graph

The graph passed to struct_test.

p.value

The resulting p-value of the test (see details).

t0

The observed value of the statistic.

mean_t

The average value of the statistic applied to the simulated networks.

R

Number of simulations.

statistic

The function statistic passed to struct_test.

boot

A boot class object as return from the call to boot.

rewire.args

The list rewire.args passed to struct_test.

Author(s)

George G. Vega Yon

References

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.

Examples

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# 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
## Not run: 
res <- struct_test(diffnet, function(g) mean(threshold(g), na.rm=TRUE), R=100,
ncpus=4, parallel="multicore")
res
hist(res)

## End(Not run)

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