NetRep

Introduction

The NetRep package provides functions for assessing the preservation of network modules across datasets.

This type of analysis is suitable where networks can be meaningfully inferred from multiple datasets. These include gene coexpression networks, protein-protein interaction networks, and microbial co-occurence networks. Modules within these networks consist of groups of nodes that are particularly interesting: for example a group of tightly connected genes associated with a disease, groups of genes annotated with the same term in the Gene Ontology database, or groups of interacting microbial species, i.e. communities.

Application of this method can answer questions such as:

Do the relationships between genes in a module replicate in an independent cohort?
Are these gene coexpression modules preserved across tissues or are they tissue specific?
Are these modules conserved across species?
Are microbial communities preseved across multiple spatial locations?

A typical workflow for a NetRep analysis will usually contain the following steps, usually as separate scripts.

Calculate the correlation structure and network edges in each dataset using some network inference algorithm.
Load these matrices into R and set up the input lists for NetRep’s functions.
Run the permutation test procedure to determine which modules are preserved in your test dataset(s).
Visualise your modules of interest.
Calculate the network properties in your modules of interest for downstream analyses.

System requirements and installation troubleshooting

NetRep and its dependencies require several third party libraries to be installed. If not found, installation of the package will fail.

NetRep requires:

A compiler with C++11 support for the <thread> libary.
A fortran compiler.
BLAS and LAPACK libraries.

The following sections provide operating system specific advice for getting NetRep working if installation through R fails.

OSX

The necessary fortran and C++11 compilers are provided with the Xcode application and subsequent installation of Command line tools. The most recent version of OSX should prompt you to install these tools when installing the devtools package from RStudio. Those with older versions of OSX should be able to install these tools by typing the following command into their Terminal application: xcode-select --install.

Some users on OSX Mavericks have reported that even after this step they receive errors relating to -lgfortran or -lquadmath. This is reportedly solved by installing the version of gfortran used to compile the R binary for OSX: gfortran-4.8.2. This can be done using the following commands in your Terminal application:

curl -O http://r.research.att.com/libs/gfortran-4.8.2-darwin13.tar.bz2
sudo tar fvxz gfortran-4.8.2-darwin13.tar.bz2 -C /

Windows

For Windows users NetRep requires R version 3.3.0 or later. The necessary fortran and C++11 compilers are provided with the Rtools program. We recommend installation of NetRep through RStudio, which should prompt the user and install these tools when running devtools::install_github("InouyeLab/NetRep"). You may need to run this command again after Rtools finishes installing.

Linux

If installation fails when compiling NetRep at permutations.cpp with an error about namespace thread, you will need to install a newer version of your compiler that supports this C++11 feature. We have found that this works on versions of gcc as old as gcc-4.6.3.

If installation fails prior to this step it is likely that you will need to install the necessary compilers and libraries, then reinstall R. For C++ and fortran compilers we recommend installing g++ and gfortran from the appropriate package manager for your operating system (e.g. apt-get for Ubuntu). BLAS and LAPACK libraries can be installed by installing libblas-dev and liblapack-dev. Note that these libraries must be installed prior to installation of R.

Data required for a NetRep analysis

Any NetRep analysis requires the following data to be provided and pre-computed for each dataset:

An adjacency matrix whose entries indicate the strength of the relationship between nodes.
A matrix whose entries contain the correlation coefficient between each pair of nodes in the network.
a vector containing the module/group label for each node in the network for each discovery dataset.
Optionally, a “data matrix”, which contains the data used to calculate the correlation structure and infer the network, e.g. gene expression data.

There are many different approaches to network inference and module detection. For gene expression data, we recommend using Weighted Gene Coexpression Network Analysis through the WGCNA package. For microbial abundance data we recommend the Python program SparCC. Microbial communities (modules) can then be defined as any group of significantly co-occuring microbes.

Tutorial data

For this vignette, we will use gene expression data simulated for two independent cohorts. The discovery dataset was simulated to contain four modules of varying size, two of which (Modules 1 and 4) replicate in the test dataset.

Details of the simulation are provided in the documentation for the package data (see help("NetRep-data")).

This data is provided with the NetRep package:

library("NetRep")
data("NetRep")

This command loads seven objects into the R session:

discovery_data: a matrix with 150 columns (genes) and 30 rows (samples) whose entries correspond to the expression level of each gene in each sample in the discovery dataset.
discovery_correlation: a matrix with 150 columns and 150 rows containing the correlation-coefficients between each pair of genes calculated from the discovery_data matrix.
discovery_network: a matrix with 150 columns and 150 rows containing the network edge weights encoding the interaction strength between each pair of genes in the discovery dataset.
module_labels: a named vector with 150 entries containing the module assignment for each gene as identified in the discovery dataset. Here, we’ve given genes that are not part of any module/group the label “0”.
test_data: a matrix with 150 columns (genes) and 30 rows (samples) whose entries correspond to the expression level of each gene in each sample in the test dataset.
test_correlation: a matrix with 150 columns and 150 rows containing the correlation-coefficients between each pair of genes calculated from the test_data matrix.
test_network: a matrix with 150 columns and 150 rows containing the network edge weights encoding the interaction strength between each pair of genes in the test dataset.

Setting up the input lists

Next, we will combine these objects into list structures. All functions in the NetRep package take the following arguments:

network: a list of interaction networks, one for each dataset.
data: a list of data matrices used to infer those networks, one for each dataset.
correlation: a list of matrices containing the pairwise correlation coefficients between variables/nodes in each dataset.
moduleAssignments: a list of vectors, one for each discovery dataset, containing the module assignments for each node in that dataset.
modules: a list of vectors, one vector for each discovery dataset, containing the names of the modules from that dataset to run the function on.
discovery: a vector indicating the names or indices to use as the discovery datasets in the network, data, correlation, moduleAssignments, and modules arguments.
test: a list of vectors, one vector for each discovery dataset, containing the names or indices of the network, data, and correlation argument lists to use as the test dataset(s) for the analysis of each discovery dataset.

Each of these lists may contain any number of datasets. The names provided to each list are used by the discovery and test arguments to determine which datasets to compare. More than one dataset can be specified in each of these arguments, for example when performing a pairwise analysis of gene coexpression modules identified in multiple tissues.

Typically we would put the code that reads in our data and sets up the input lists in its own script. This loading script can then be called from our scripts where we calculate the module preservation, visualise our networks, and calculate the network properties:

# Read in the data:
data("NetRep")

# Set up the input data structures for NetRep. We will call these datasets 
# "cohort1" and "cohort2" to avoid confusion with the "discovery" and "test"
# arguments in NetRep's functions:
data_list <- list(cohort1=discovery_data, cohort2=test_data)
correlation_list <- list(cohort1=discovery_correlation, cohort2=test_correlation)
network_list <- list(cohort1=discovery_network, cohort2=test_network)

# We do not need to set up a list for the 'moduleAssignments', 'modules', or 
# 'test' arguments because there is only one "discovery" dataset.

We will call these “cohort1” and “cohort2” to avoid confusion with the arguments “discovery” and “test” common to NetRep’s functions.

Running the permutation procedure to test module preservation

Now we will use NetRep to permutation test whether the network topology of each module is preserved in our test dataset using the modulePreservation function. This function calculates seven module preservation statistics for each module (more on these later), then performs a permutation procedure in the test dataset to determine whether these statistics are significant.

We will run 10,000 permutations, and split calculation across 2 threads so that calculations are run in parallel. By default, modulePreservaton will test the preservation of all modules, excluding the network background which is assumed to have the label “0”. This of course can be changed: there are many more arguments than shown here which control how modulePreservation runs. See help("modulePreservation") for a full list of arguments.

# Assess the preservation of modules in the test dataset.
preservation <- modulePreservation(
 network=network_list, data=data_list, correlation=correlation_list, 
 moduleAssignments=module_labels, discovery="cohort1", test="cohort2", 
 nPerm=10000, nThreads=2
)

## [2023-08-19 15:45:15 BST] Validating user input...
## [2023-08-19 15:45:15 BST]   Checking matrices for problems...
## [2023-08-19 15:45:15 BST] Input ok!
## [2023-08-19 15:45:15 BST] Calculating preservation of network subsets from dataset "cohort1" in
##                           dataset "cohort2".
## [2023-08-19 15:45:15 BST]   Pre-computing network properties in dataset "cohort1"...
## [2023-08-19 15:45:15 BST]   Calculating observed test statistics...
## [2023-08-19 15:45:15 BST]   Generating null distributions from 10000 permutations using 2
##                             threads...
## 
## 
    0% completed.
   69% completed.
  100% completed.
## 
## [2023-08-19 15:45:17 BST]   Calculating P-values...
## [2023-08-19 15:45:17 BST]   Collating results...
## [2023-08-19 15:45:17 BST] Done!

The results returned by modulePreservation for each dataset comparison are a list containing seven elements:

nulls the null distribution for each statistic and module generated by the permutation procedure.
observed the observed value of each module preservation statistic for each module.
p.values the p-values for each module preservation statistic for each module.
nVarsPresent the number of variables in the discovery dataset that had corresponding measurements in the test dataset.
propVarsPresent the proportion of nodes in each module that had corresponding measurements in the test dataset.
totalSize the total number of nodes in the discovery network.
alternative the alternate hypothesis used in the test (e.g. “the module preservation statistics are higher than expected by chance”).

If the test dataset has also had module discovery performed in it, a contigency table tabulating the overlap in module content between the two datasets is returned.

Let’s take a look at our results:

preservation$observed

##    avg.weight coherence    cor.cor  cor.degree cor.contrib      avg.cor avg.contrib
## 1 0.161069393 0.6187688 0.78448573  0.90843993   0.8795006  0.550004272  0.76084777
## 2 0.001872928 0.1359063 0.17270312 -0.03542772   0.5390504  0.034040922  0.23124826
## 3 0.001957475 0.1263280 0.01121223 -0.17179855  -0.1074944 -0.007631867  0.05412794
## 4 0.046291489 0.4871179 0.32610667  0.68122446   0.5251965  0.442614173  0.68239136

preservation$p.value

##   avg.weight  coherence    cor.cor cor.degree cor.contrib    avg.cor avg.contrib
## 1 0.00009999 0.00009999 0.00009999 0.00009999  0.00009999 0.00009999  0.00009999
## 2 0.97960204 0.96560344 0.00799920 0.56514349  0.00249975 0.01609839  0.00549945
## 3 0.98940106 0.98560144 0.42195780 0.81091891  0.71872813 0.99330067  0.87881212
## 4 0.00009999 0.00009999 0.00009999 0.00009999  0.00079992 0.00009999  0.00009999

For now, we will consider all statistics equally important, so we will consider a module to be preserved in “cohort2” if all the statistics have a permutation test P-value < 0.01:

# Get the maximum permutation test p-value
max_pval <- apply(preservation$p.value, 1, max)
max_pval

##          1          2          3          4 
## 0.00009999 0.97960204 0.99330067 0.00079992

Only modules 1 and 4 are reproducible at this significance threshold.

The module preservation statistics

So what do these statistics measure? Let’s take a look at the network topology of Module 1 in the discovery dataset, “cohort1”:

Network topology of Module 1 in the discovery dataset (“cohort1”).

From top to bottom, the plot shows:

A heatmap of the correlation coefficients between nodes in the module.
A heatmap of the network edge weights between nodes in the module.
The scaled weighted degree for each node: this is the sum of each node’s connections to all other nodes in the module, normalised to the most connected node. This is a relative measure of how connected each node is within the module.
The contribution of each node to the module: this is the correlation between each node and the module’s summary profile.
A heatmap of the measurements of each node in the module across samples in the dataset (y-axis)
To the left, the module’s summary profile: a set of observations that best summarise the measurements across all nodes for each sample. This is calculated as the first eigenvector of a principal component analysis: i.e. the linear combination of nodes that explains the greatest portion of the variance in the module’s data.

Now, let’s take a look at the topology of Module 1 in the discovery and the test datasets side by side along with the module preservation statistics:

Network topology of Module 1 in both the discovery (“cohort1”) and test (“cohort2”) datasets.

There are seven module preservation statistics:

‘cor.cor’ measures the concordance of the correlation structure: or, how similar the correlation heatmaps are between the two datasets.
‘avg.cor’ measures the average magnitude of the correlation coefficients of the module in the test dataset: or, how tightly correlated the module is on average in the test dataset. This score is penalised where the correlation coefficients change in sign between the two datasets.
‘avg.weight’ measures the average magnitude of edge weights in the test dataset: or how connected nodes in the module are to each other on average.
‘cor.degree’ measures the concordance of the weighted degree of nodes between the two datasets: or, whether the nodes that are most strongly connected in the discovery dataset remain the most strongly connected in the test dataset.
‘cor.contrib’ measures the concordance of the node contribution between the two datasets: this measures whether the module’s summary profile summarises the data in the same way in both datasets.
‘avg.contrib’ measures the average magnitude of the node contribution in the test dataset: this is a measure of how coherent the data is in the test dataset. This score is penalised where the node contribution changes in sign between the two datasets: for example, where a gene is differentially expressed between the two datasets.
‘coherence’ measures the proportion of variance in the module data explained by the module’s summary profile vector in the test dataset.

A permutation procedure is necessary to determine whether the value of each statistic is significant: e.g. whether they are higher than expected by chance, i.e. when measuring the statistics between the module in the discovery dataset, and random sets of nodes in the test dataset.

By default, the permutation procedure will sample from only nodes that are present in both datasets. This is appropriate where the assumption is that any nodes that are present in the test dataset but not the discovery dataset are unobserved in the discovery dataset: i.e. they may very well fall in one of your modules of interest. This is appropriate for microarray data. Alternatively, you may set null="all", in which case the permutation procedure will sample from all variables in the test dataset. This is appropriate where the variable can be assumed not present in the discovery dataset: for example microbial abundance or RNA-seq data.

You can also test whether these statistics are smaller than expected by chance by changing the alternative hypothesis in the modulePreservation function (e.g. alternative="lower").

Choosing the right statistics

The module preservation statistics that NetRep calculates were designed for weighted gene coexpression networks. These are complete networks: every gene is connected to every other gene with an edge weight of varying strength. Modules within these networks are groups of genes that are tightly connected or coexpressed.

For other types of networks, some statistics may be more suitable than others when assessing module preservation. Here, we provide some guidelines and pitfalls to be aware of when interpreting the network properties and module preservation statistics in other types of networks.

Sparse networks

Sparse networks are networks where many edges have a “0” value: that is, networks where many nodes have no connection to each other. Typically these are networks where edges are defined as present if the relationship between nodes passes some pre-defined cut-off value, for example where genes are significantly correlated, or where the correlation between microbe presence and absence is significant. In these networks, edges may simply indicate presence or absence, or they may also carry a weight indicating the strength of the relationship.

For networks with unweighted edges, the average edge weight (‘avg.weight’) measures the proportion of nodes that are connected to each other. The weighted degree simply becomes the node degree: the number of connections each node has to any other node in the module.

If the network is sparse the permutation tests for the correlation of weighted degree may be underpowered. Entries in the null distribution will be NA where there were no edges between any nodes in the permuted module. This is because the weighted degree will be 0 for all nodes, and the correlation coefficient cannot be calculated between two vectors if all entries are the same in either vector. This reduces the effective number of permutations for that test: the permutation P-values will be calculated ignoring the NA entries, and the modulePreservation function will generate a warning.

You may wish to consider NA entries where there were no edges as 0 when calculating the permutation test P-values. Note that an NA entry does not necessarily mean that all edges in the permuted module were 0: it can also mean that all edges are present and have identical weights. To distinguish between these cases you should check whether the avg.weight is also 0.

The following code snippet shows how to identify these entries in the null distribution, replace them with zeros, and recalculate the permutation test P-values:

# Handling NA entries in the 'cor.degree' null distribution for sparse networks

# Get the entries in the null distribution where there were no edges in the 
# permuted module
na.entries <- which(is.na(preservation$nulls[,'cor.degree',]))
no.edges <- which(preservation$nulls[,'avg.weight',][na.entries] == 0)

# Set those entries to 0
preservation$nulls[,'cor.degree',][no.edges] <- 0

# Recalculate the permutation test p-values
preservation$p.values <- permutationTest(
  preservation$nulls, preservation$observed, preservation$nVarsPresent,
  preservation$totalSize, preservation$alternative
)

Directed networks

For networks where the edges are directed, the user should be aware that the weighted degree is calculated as the column sum of the module within the supplied network matrix. This usually means that the result will be the in-degree: the number and combined weight of edges ending in each node. To calculate the out-degree you will need to transpose the matrix supplied to the network argument (i.e. using the t() function).

Note that directed networks are typically sparse, and have the same pitfalls as sparse networks described above.

Sparse data

Sparse data is data where many entries are zero. Examples include microbial abundance data: where most microbes are present in only a few samples.

Users should be aware that the average node contribution (‘avg.contrib’), concordance of node contribution (‘cor.contrib’), and the module coherence (‘coherence’) will be systematically underestimated. They are all calculated from the node contribution, which measures the Pearson correlation coefficient between each node and the module summary. Pearson correlation coefficinets are inappropriate when data is sparse: their value will be underestimated when calculated between two vectors where many observations in either vector are equal to 0. However, this should not affect the permutation test P-values since observations in their null distributions will be similarly underestimated.

The biggest problem with sparse data is how to handle variables where all observations are zero in either dataset. These will result in NA values for their node contribution to a module (or permuted module). These will be ignored by the average node contribution (‘avg.contrib’), concordance of node contribution (‘cor.contrib’), and module coherence (‘coherence’) statistics: which only take complete cases. This is problematic if many nodes have NA values, since observations in their null distributions will be for permuted modules of different sizes.

Their are two approaches to dealing with this issue:

Filtering both datasets to contain only variables which are present in both datasets. For examples, microbes that are abundant in at least one sample in both datasets.
Setting observations that are zero to a very small randomly generated number. The goal is for node contribution values to be close to 0 where they would otherwise be set to NA. For microbial abundance data we recommend generating numbers between 0 and 1/the number of samples: the noise values should be small enough that the do not change the node contribution for microbes which are present in one or more samples.

For the latter, code to generate noise would look something like:

not.present <- which(discovery_data == 0)
nSamples <- nrow(discovery_data)
discovery_data[not.present] <- runif(length(not.present), min=0, max=1/nSamples)

Proportional data

Proportional data is data where the sum of measurements across each sample is equal to 1. Examples of this include RNA-seq data and microbial abundance read data.

Users should be aware that the average node contribution (‘avg.controb’), concordance of node contribution (‘cor.contrib’), and the module coherence (‘coherence’) will be systematically overestimated. They are all calculated from the node contribution, which measures the Pearson correlation coefficient between each node and the module summary. Pearson correlation coefficients are overestimated when calculated on proportional data. This should not affect the permutation test P-values since the null distribution observations will be similarly overestimated.

Users should also be aware of this when calculating the correlation structure between all nodes for the correlation matrix input, and use an appropriate method for calculating these relationships.

Homogenous modules

Homogenous modules are modules where all nodes are similarly correlated or similarly connected: differences in edge weights, correlation coefficients, and node contributions are due to noise.

For these modules, the concordance of correlation (‘cor.cor’), concordance of node contribution (‘cor.contrib’), and correlation of weighted degree (‘cor.degree’) may be small, with large permutation test P-values, even where a module is preserved, due to irrelevant changes in node rank for each property between the discovery and test datasets.

These statistics should be considered in the context of their “average” counterparts: the average correlation coefficient (‘avg.cor’), average node contribution (‘avg.contrib’) and average edge weight (‘avg.weight’). If these are high, with significant permutation test P-values, and the module coherence is high, then the module should be investigated further.

Module homogeneity can be investigated through plotting their network topology in both datasets (see next section). In our experience, the smaller the module, the more likely it is to be topologically homogenous.

Small network modules

The module preservation statistics break down for modules with less than four nodes. The number of nodes is effectively the sample size when calculating the value of a module preservation statistic. If you wish to use NetRep to analyse these modules, you should use only the average edge weight (‘avg.weight’), module coherence (‘coherence’), average node contribution (‘avg.contrib’), and average correlation coefficient (‘avg.cor’) statistics.

Visualising network modules

We can visualise the network topology of our modules using the plotModule function. It takes the same input data as the modulePreservation function:

network: a list of network adjacency matrices, one for each dataset.
correlation: a list of matrices containing the correlation coefficients between nodes.
data: a list of data matrices used to infer the network and correlation matrices.
moduleAssignments: a list of vectors, one for each discovery dataset, containing the module labels for each node.
modules: the modules we want to plot.
discovery: the dataset the modules were identified in.
test: the dataset we want to plot the modules in.

First, let’s look at the four modules in the discovery dataset:

plotModule(
  data=data_list, correlation=correlation_list, network=network_list, 
  moduleAssignments=module_labels, modules=c(1,2,3,4),
  discovery="cohort1", test="cohort1"
)

## [2023-08-19 15:45:17 BST] Validating user input...
## [2023-08-19 15:45:17 BST]   Checking matrices for problems...
## [2023-08-19 15:45:17 BST] User input ok!
## [2023-08-19 15:45:17 BST] Calculating network properties of network subsets from dataset "cohort1"
##                           in dataset "cohort1"...
## [2023-08-19 15:45:17 BST] Ordering nodes...
## [2023-08-19 15:45:17 BST] Ordering samples...
## [2023-08-19 15:45:17 BST] Ordering samples...
## [2023-08-19 15:45:17 BST] rendering plot components...
## [2023-08-19 15:45:18 BST] Done!

By default, nodes are ordered from left to right in decreasing order of weighted degree: the sum of edge weights within each module, i.e. how strongly connected each node is within its module. For visualisation, the weighted degree is normalised within each module by the maximum value since the weighted degree of nodes can be dramatically different for modules of different sizes.

Samples are ordered from top to bottom in descending order of the module summary profile of the left-most shown module.

When we plot the four modules in the test dataset, the nodes remain in the same order: that is, in decreasing order of weighted degree in the discovery dataset. This allows you to directly compare topology plots in each dataset of interest:

plotModule(
  data=data_list, correlation=correlation_list, network=network_list, 
  moduleAssignments=module_labels, modules=c(1,2,3,4),
  discovery="cohort1", test="cohort2"
)

## [2023-08-19 15:45:18 BST] Validating user input...
## [2023-08-19 15:45:18 BST]   Checking matrices for problems...
## [2023-08-19 15:45:18 BST] User input ok!
## [2023-08-19 15:45:18 BST] Calculating network properties of network subsets from dataset "cohort1"
##                           in dataset "cohort1"...
## [2023-08-19 15:45:18 BST] Calculating network properties of network subsets from dataset "cohort1"
##                           in dataset "cohort2"...
## [2023-08-19 15:45:18 BST] Ordering nodes...
## [2023-08-19 15:45:18 BST] Ordering samples...
## [2023-08-19 15:45:18 BST] Ordering samples...
## [2023-08-19 15:45:18 BST] rendering plot components...
## [2023-08-19 15:45:19 BST] Done!

Here we can clearly see from the correlation structure and network edge weight heatmaps that Modules 1 and 4 replicate.

By default, samples in this new plot are orderded in descending order of the left most module’s summary profile, as calculated in the test dataset. If we’re analysing module preservation across datasets drawn from the same samples, e.g. different tissues, we can change the plot so that samples are ordered as per the discovery dataset by setting orderSamplesBy = "cohort1". We won’t do this here, since our two datasets have different samples.

We can change the order of nodes on the plot by setting orderNodesBy. If we want to order nodes instead by our test dataset, we can set orderNodesBy = "cohort2". However, a more informative setting is to tell plotModule to order the nodes by the average weighted degree across our datasets. For preserved modules, this provides a more robust estimate of the weighted degree and a more robust ordering of nodes by relative importance to their module, so we will plot just Modules 1 and 4.

plotModule(
  data=data_list, correlation=correlation_list, network=network_list, 
  moduleAssignments=module_labels, modules=c(1,4), # only the preserved modules
  discovery="cohort1", test="cohort2",
  orderNodesBy=c("cohort1", "cohort2") # this can be any number of datasets
)

## [2023-08-19 15:45:19 BST] Validating user input...
## [2023-08-19 15:45:19 BST]   Checking matrices for problems...
## [2023-08-19 15:45:19 BST] User input ok!
## [2023-08-19 15:45:19 BST] Calculating network properties of network subsets from dataset "cohort1"
##                           in dataset "cohort1"...
## [2023-08-19 15:45:19 BST] Calculating network properties of network subsets from dataset "cohort1"
##                           in dataset "cohort2"...
## [2023-08-19 15:45:19 BST] Ordering nodes...
## [2023-08-19 15:45:19 BST] Ordering samples...
## [2023-08-19 15:45:19 BST] Ordering samples...
## [2023-08-19 15:45:19 BST] rendering plot components...
## [2023-08-19 15:45:19 BST] Done!

Tweaking the plot appearance

When drawing these plots yourself, you may need to tweak the appearance and placement of the axis labels and legends, which may change depending on the size of the device you are drawing the plot on. There is an extensive set of options for modifying the size and placement of the axes, legends, and their individual elements. A list and description of these can be found in the “plot layout and device size” section of the help file for plotModule.

When tweaking these parameters, you should set the dryRun argument to TRUE. When dryRun = TRUE, only the axes and labels will be drawn, avoiding the drawing time for the heatmaps, which may take some time for large modules.

Let’s tweak the previous plot:

plotModule(
  data=data_list, correlation=correlation_list, network=network_list, 
  moduleAssignments=module_labels, modules=c(1,4),
  discovery="cohort1", test="cohort2",
  orderNodesBy=c("cohort1", "cohort2"),
  dryRun=TRUE
)

## [2023-08-19 15:45:19 BST] Validating user input...
## [2023-08-19 15:45:19 BST]   Checking matrices for problems...
## [2023-08-19 15:45:20 BST] User input ok!
## [2023-08-19 15:45:20 BST] Calculating network properties of network subsets from dataset "cohort1"
##                           in dataset "cohort1"...
## [2023-08-19 15:45:20 BST] Calculating network properties of network subsets from dataset "cohort1"
##                           in dataset "cohort2"...
## [2023-08-19 15:45:20 BST] Ordering nodes...
## [2023-08-19 15:45:20 BST] Ordering samples...
## [2023-08-19 15:45:20 BST] Ordering samples...
## [2023-08-19 15:45:20 BST] rendering plot components...
## [2023-08-19 15:45:20 BST] Done!

Now we can quickly iterate over parameters until we’re happy with the plot:

# Change the margins so the plot is more compressed. Alternatively we could 
# change the device window.
par(mar=c(3,10,3,10)) # bottom, left, top, right margin sizes
plotModule(
  data=data_list, correlation=correlation_list, network=network_list, 
  moduleAssignments=module_labels, modules=c(1,4),
  discovery="cohort1", test="cohort2",
  orderNodesBy=c("cohort1", "cohort2"),
  dryRun=TRUE, 
  # Title of the plot
  main = "Preserved modules", 
  # Use the maximum edge weight as the highest value instead of 1 in the
  # network heatmap
  netRange=NA,
  # Turn off the node and sample labels:
  plotNodeNames=FALSE, plotSampleNames=FALSE,
  # The distance from the bottom axis should the module labels be drawn:
  maxt.line=0,
  # The distance from the legend the legend titles should be drawn:
  legend.main.line=2
)

## [2023-08-19 15:45:20 BST] Validating user input...
## [2023-08-19 15:45:20 BST]   Checking matrices for problems...
## [2023-08-19 15:45:20 BST] User input ok!
## [2023-08-19 15:45:20 BST] Calculating network properties of network subsets from dataset "cohort1"
##                           in dataset "cohort1"...
## [2023-08-19 15:45:20 BST] Calculating network properties of network subsets from dataset "cohort1"
##                           in dataset "cohort2"...
## [2023-08-19 15:45:20 BST] Ordering nodes...
## [2023-08-19 15:45:20 BST] Ordering samples...
## [2023-08-19 15:45:20 BST] Ordering samples...
## [2023-08-19 15:45:20 BST] rendering plot components...
## [2023-08-19 15:45:20 BST] Done!

Once we’re happy, we can turn off the dryRun parameter:

par(mar=c(3,10,3,10)) 
plotModule(
  data=data_list, correlation=correlation_list, network=network_list, 
  moduleAssignments=module_labels, modules=c(1,4),
  discovery="cohort1", test="cohort2",
  orderNodesBy=c("cohort1", "cohort2"), main = "Preserved modules", 
  netRange=NA, plotNodeNames=FALSE, plotSampleNames=FALSE,
  maxt.line=0, legend.main.line=2
)

## [2023-08-19 15:45:20 BST] Validating user input...
## [2023-08-19 15:45:20 BST]   Checking matrices for problems...
## [2023-08-19 15:45:20 BST] User input ok!
## [2023-08-19 15:45:20 BST] Calculating network properties of network subsets from dataset "cohort1"
##                           in dataset "cohort1"...
## [2023-08-19 15:45:20 BST] Calculating network properties of network subsets from dataset "cohort1"
##                           in dataset "cohort2"...
## [2023-08-19 15:45:20 BST] Ordering nodes...
## [2023-08-19 15:45:20 BST] Ordering samples...
## [2023-08-19 15:45:20 BST] Ordering samples...
## [2023-08-19 15:45:20 BST] rendering plot components...
## [2023-08-19 15:45:20 BST] Done!

Plotting the individual components

We can also plot individual components of the plot separately. For example, a heatmap of the correlation structure:

par(mar=c(5,5,4,4)) 
plotCorrelation(
  data=data_list, correlation=correlation_list, network=network_list, 
  moduleAssignments=module_labels, modules=0:4, discovery="cohort1",
  test="cohort1", symmetric=TRUE, orderModules=FALSE
)

## [2023-08-19 15:45:21 BST] Validating user input...
## [2023-08-19 15:45:21 BST]   Checking matrices for problems...
## [2023-08-19 15:45:21 BST] User input ok!
## [2023-08-19 15:45:21 BST] Calculating network properties of network subsets from dataset "cohort1"
##                           in dataset "cohort1"...
## [2023-08-19 15:45:21 BST] Ordering nodes...
## [2023-08-19 15:45:21 BST] rendering plot components...
## [2023-08-19 15:45:21 BST] Done!

A full list of function and arguments for these individual plots can be found at help("plotTopology").

Calculating the network properties of a module

Finally, we can calculate the topological properties of the network modules for use in other downstream analyses. Possible downstream analyses include:

Assessing the association between a module and a phenotype of interest using the module summary profile.
Ranking nodes by relative importance using the weighted node degree

To do this, we use the networkProperties function, which has the same arguments as the modulePreservation function. We will calculate the network properties of modules 1 and 4, which were preserved in “cohort2”, in both datasets:

properties <- networkProperties(
  data=data_list, correlation=correlation_list, network=network_list, 
  moduleAssignments=module_labels, 
  # Only calculate for the reproducible modules
  modules=c(1,4),
  # what dataset were the modules identified in?
  discovery="cohort1", 
  # which datasets do we want to calculate their properties in?
  test=c("cohort1", "cohort2")
)

## [2023-08-19 15:45:22 BST] Validating user input...
## [2023-08-19 15:45:22 BST]   Checking matrices for problems...
## [2023-08-19 15:45:22 BST] User input ok!
## [2023-08-19 15:45:22 BST] Calculating network properties of network subsets from dataset "cohort1"
##                           in dataset "cohort1"...
## [2023-08-19 15:45:22 BST] Calculating network properties of network subsets from dataset "cohort1"
##                           in dataset "cohort2"...
## [2023-08-19 15:45:22 BST] Done!

# The summary profile of module 1 in the discovery dataset:
properties[["cohort1"]][["1"]][["summary"]]

##  Discovery_1  Discovery_2  Discovery_3  Discovery_4  Discovery_5  Discovery_6  Discovery_7 
##  -0.15173019  -0.09817810  -0.10356266  -0.21351111  -0.06424053  -0.25787365  -0.06191222 
##  Discovery_8  Discovery_9 Discovery_10 Discovery_11 Discovery_12 Discovery_13 Discovery_14 
##  -0.05886898   0.04544493   0.16790065  -0.16163254  -0.07158769  -0.16775343   0.39457572 
## Discovery_15 Discovery_16 Discovery_17 Discovery_18 Discovery_19 Discovery_20 Discovery_21 
##   0.10762551   0.25872801   0.01187731   0.57266243   0.15737963   0.02368060  -0.07088476 
## Discovery_22 Discovery_23 Discovery_24 Discovery_25 Discovery_26 Discovery_27 Discovery_28 
##   0.03726126  -0.13770047  -0.01978039  -0.06336512  -0.06360727  -0.30044215   0.14682841 
## Discovery_29 Discovery_30 
##   0.07036710   0.07229971

# Along with the proportion of variance in the module data explained by the 
# summary profile:
properties[["cohort1"]][["1"]][["coherence"]]

## [1] 0.585781

# The same information in the test dataset:
properties[["cohort2"]][["1"]][["summary"]]

##       Test_1       Test_2       Test_3       Test_4       Test_5       Test_6       Test_7 
## -0.099957918  0.061501299  0.043541623  0.051055323  0.056572949  0.136605203  0.116491092 
##       Test_8       Test_9      Test_10      Test_11      Test_12      Test_13      Test_14 
## -0.395294200 -0.099564626  0.092715774 -0.005526985  0.256963062  0.028746029 -0.076793357 
##      Test_15      Test_16      Test_17      Test_18      Test_19      Test_20      Test_21 
## -0.435677499  0.100475978 -0.339161521 -0.195830382 -0.104643904  0.050046780  0.238180614 
##      Test_22      Test_23      Test_24      Test_25      Test_26      Test_27      Test_28 
##  0.144114251  0.211841029  0.228291634 -0.171340087 -0.188991911 -0.093239829  0.063972325 
##      Test_29      Test_30 
##  0.278339356  0.046567899

properties[["cohort2"]][["1"]][["coherence"]]

## [1] 0.6187688

Managing memory with large datasets

When analysing large datasets, e.g. transcriptome-wide gene coexpression networks, it may not be possible to fit all matrices for both datasets in memory. NetRep provides an additional class, disk.matrix, which stores a filepath to a matrix on disk, along with meta-data on how to read that file. This allows NetRep’s functions to load matrices into RAM only when required, so that only one dataset is kept in memory at any point in time.

The disk.matrix class recognises two types of files: matrix data saved in table format (i.e. a file that is normally read in by read.table or read.csv), and serialized R objects saved through saveRDS. Serialized R objects are much faster to load into R than files in table format, but cannot be read by other programs. We recommend storing your files in both formats unless you are low on disk space.

First, we need to make sure our matrices are saved to disk. Matrices can be converted to disk.matrix objects directly through the as.disk.matrix function:

# serialize=TRUE will save the data using 'saveRDS'. 
# serialize=FALSE will save the data as a tab-separated file ('sep="\t"').
discovery_data <- as.disk.matrix(
  x=discovery_data, 
  file="discovery_data.rds", 
  serialize=TRUE)
discovery_correlation <- as.disk.matrix(
  x=discovery_correlation, 
  file="discovery_correlation.rds", 
  serialize=TRUE)
discovery_network <- as.disk.matrix(
  x=discovery_network, 
  file="discovery_network.rds",
  serialize=TRUE)
test_data <- as.disk.matrix(
  x=test_data, 
  file="test_data.rds", 
  serialize=TRUE)
test_correlation <- as.disk.matrix(
  x=test_correlation, 
  file="test_correlation.rds", 
  serialize=TRUE)
test_network <- as.disk.matrix(
  x=test_network, 
  file="test_network.rds",
  serialize=TRUE)

Now, these matrices are stored simply as file paths:

test_network

## Pointer to matrix stored at test_network.rds

To load the matrix into R we can convert it back to a matrix:

as.matrix(test_network)[1:5, 1:5]

##            Node_1       Node_2     Node_3     Node_4       Node_5
## Node_1 1.00000000 0.0284607734 0.29433703 0.27292044 0.0774910679
## Node_2 0.02846077 1.0000000000 0.04594941 0.04747009 0.0001403167
## Node_3 0.29433703 0.0459494090 1.00000000 0.48140887 0.1392228920
## Node_4 0.27292044 0.0474700916 0.48140887 1.00000000 0.0996614770
## Node_5 0.07749107 0.0001403167 0.13922289 0.09966148 1.0000000000

Once our matrices are saved to disk, we can load them as disk.matrix objects in new R sessions using attach.disk.matrix. Typically, we would save our matrices to disk after running our network inference pipeline, then use attach.disk.matrix in our new R session when we run NetRep at some point in the future.

# If files are saved as tables, set 'serialized=FALSE' and specify arguments 
# that would normally be provided to 'read.table'. Note: this function doesnt
# check whether the file can actually be read in as a matrix!
discovery_data <- attach.disk.matrix("discovery_data.rds")
discovery_correlation <- attach.disk.matrix("discovery_correlation.rds")
discovery_network <- attach.disk.matrix("discovery_network.rds")
test_data <- attach.disk.matrix("test_data.rds")
test_correlation <- attach.disk.matrix("test_correlation.rds")
test_network <- attach.disk.matrix("test_network.rds")

And we need to set up our input lists for NetRep:

data_list <- list(cohort1=discovery_data, cohort2=test_data)
correlation_list <- list(cohort1=discovery_correlation, cohort2=test_correlation)
network_list <- list(cohort1=discovery_network, cohort2=test_network)

Now we can run our analyses as previously described in the tutorial:

# Assess the preservation of modules in the test dataset.
preservation <- modulePreservation(
 network=network_list, data=data_list, correlation=correlation_list, 
 moduleAssignments=module_labels, discovery="cohort1", test="cohort2", 
 nPerm=10000, nThreads=2
)

## [2023-08-19 15:45:22 BST] Validating user input...
## [2023-08-19 15:45:22 BST]   Loading matrices of dataset "cohort2" into RAM...
## [2023-08-19 15:45:22 BST]   Checking matrices for problems...
## [2023-08-19 15:45:22 BST]   Unloading dataset from RAM...
## [2023-08-19 15:45:22 BST]   Loading matrices of dataset "cohort1" into RAM...
## [2023-08-19 15:45:22 BST]   Checking matrices for problems...
## [2023-08-19 15:45:22 BST] Input ok!
## [2023-08-19 15:45:22 BST] Calculating preservation of network subsets from dataset "cohort1" in
##                           dataset "cohort2".
## [2023-08-19 15:45:22 BST]   Pre-computing network properties in dataset "cohort1"...
## [2023-08-19 15:45:22 BST]   Unloading dataset from RAM...
## [2023-08-19 15:45:22 BST]   Loading matrices of dataset "cohort2" into RAM...
## [2023-08-19 15:45:22 BST]   Calculating observed test statistics...
## [2023-08-19 15:45:22 BST]   Generating null distributions from 10000 permutations using 2
##                             threads...
## 
## 
    0% completed.
   69% completed.
  100% completed.
## 
## [2023-08-19 15:45:24 BST]   Calculating P-values...
## [2023-08-19 15:45:24 BST]   Collating results...
## [2023-08-19 15:45:24 BST] Unloading dataset from RAM...
## [2023-08-19 15:45:24 BST] Done!

You can now see that modulePreservation loads and unloads the two datasets as required.

Using `disk.matrix` with the plotting functions

Earlier in the tutorial, we showed you how to use the dryRun argument to quickly set up the plot axes before actually drawing the module(s) of interest. This does not work so well with disk.matrix input since we need to know which nodes and samples are being drawn to display their labels. This means that all datasets used for the plot need to be loaded, which can be quite slow if the datasets are large. There are two solutions: (1) do not use disk.matrix so that all matrices are kept in memory, or (2) use the nodeOrder and sampleOrder functions to determine the nodes and samples that will be on the plot in advance:

# Determine the nodes and samples on a plot in advance:
nodesToPlot <- nodeOrder(
  data=data_list, correlation=correlation_list, network=network_list, 
  moduleAssignments=module_labels, modules=c(1,4), discovery="cohort1", 
  test=c("cohort1", "cohort2"), mean=TRUE
)

## [2023-08-19 15:45:24 BST] Validating user input...
## [2023-08-19 15:45:24 BST]   Loading matrices of dataset "cohort2" into RAM...
## [2023-08-19 15:45:24 BST]   Checking matrices for problems...
## [2023-08-19 15:45:24 BST]   Unloading dataset from RAM...
## [2023-08-19 15:45:25 BST]   Loading matrices of dataset "cohort1" into RAM...
## [2023-08-19 15:45:25 BST]   Checking matrices for problems...
## [2023-08-19 15:45:25 BST] User input ok!
## [2023-08-19 15:45:25 BST] Calculating network properties of network subsets from dataset "cohort1"
##                           in dataset "cohort1"...
## [2023-08-19 15:45:25 BST] Unloading dataset from RAM...
## [2023-08-19 15:45:25 BST] Loading matrices of dataset "cohort2" into RAM...
## [2023-08-19 15:45:25 BST] Calculating network properties of network subsets from dataset "cohort1"
##                           in dataset "cohort2"...
## [2023-08-19 15:45:25 BST] Unloading dataset from RAM...
## [2023-08-19 15:45:25 BST] Ordering nodes...
## [2023-08-19 15:45:25 BST] Done!

# We need to know which module will appear left-most on the plot:
firstModule <- module_labels[nodesToPlot[1]]

samplesToPlot <- sampleOrder(
  data=data_list, correlation=correlation_list, network=network_list, 
  moduleAssignments=module_labels, modules=firstModule, discovery="cohort1",
  test="cohort2"
)

## [2023-08-19 15:45:25 BST] Validating user input...
## [2023-08-19 15:45:25 BST]   Loading matrices of dataset "cohort2" into RAM...
## [2023-08-19 15:45:25 BST]   Checking matrices for problems...
## [2023-08-19 15:45:25 BST] User input ok!
## [2023-08-19 15:45:25 BST] Calculating network properties of network subsets from dataset "cohort1"
##                           in dataset "cohort2"...
## [2023-08-19 15:45:25 BST] Unloading dataset from RAM...
## [2023-08-19 15:45:25 BST] Ordering samples...
## [2023-08-19 15:45:25 BST] Done!

# Load in the dataset we are plotting:
test_data <- as.matrix(test_data)
test_correlation <- as.matrix(test_correlation)
test_network <- as.matrix(test_network)

# Now we can use 'dryRun=TRUE' quickly:
plotModule(
  data=test_data[samplesToPlot, nodesToPlot], 
  correlation=test_correlation[nodesToPlot, nodesToPlot], 
  network=test_network[nodesToPlot, nodesToPlot],
  moduleAssignments=module_labels[nodesToPlot],
  orderNodesBy=NA, orderSamplesBy=NA, dryRun=TRUE
)

## [2023-08-19 15:45:25 BST] Validating user input...
## [2023-08-19 15:45:25 BST]   Checking matrices for problems...
## [2023-08-19 15:45:25 BST] User input ok!
## [2023-08-19 15:45:25 BST] Calculating network properties of network subsets from dataset "Dataset1"
##                           in dataset "Dataset1"...
## [2023-08-19 15:45:25 BST] rendering plot components...
## [2023-08-19 15:45:25 BST] Done!

# And draw the final plot once we determine the plot parameters 
par(mar=c(3,10,3,10)) 
plotModule(
  data=test_data[samplesToPlot, nodesToPlot], 
  correlation=test_correlation[nodesToPlot, nodesToPlot], 
  network=test_network[nodesToPlot, nodesToPlot],
  moduleAssignments=module_labels[nodesToPlot],
  orderNodesBy=NA, orderSamplesBy=NA
)

## [2023-08-19 15:45:25 BST] Validating user input...
## [2023-08-19 15:45:25 BST]   Checking matrices for problems...
## [2023-08-19 15:45:25 BST] User input ok!
## [2023-08-19 15:45:25 BST] Calculating network properties of network subsets from dataset "Dataset1"
##                           in dataset "Dataset1"...
## [2023-08-19 15:45:25 BST] rendering plot components...
## [2023-08-19 15:45:25 BST] Done!

Running NetRep on a cluster

The permutation procedure is typically too computationally intense to run interactively on the head node of a cluster. We recommend splitting your analysis into the following scripts:

A script to save your networks, the data, and the correlation structure matrices as disk.matrix format if all your datasets will not fit in memory at once.
A script to load in the matrix data and set up the input lists used by NetRep’s functions.
A script that runs the modulePreservation analysis for your modules of interest.
A script that visualises your modules of interest.
A script that calculates and saves the network properties for your modules of interest.

We recommend writing the visualisation script with the dryRun parameter set to TRUE at first. This can be run interactively to determine whether modifications need to be made to figures. Once you’re happy with the plot size and layout, you should set dryRun to FALSE and run the script as a batch job: the heatmaps for large modules can take a long time to render. Since these heatmaps contain many points, we also recommend saving plots in a rasterised format (png or jpeg) rather than in a vectorised format (pdf).

Setting the number of threads

The permutation procedure in modulePreservation can only be parallelised over CPUs that shared memory. On most clusters, this means that NetRep’s functions can only be parallelised on one physical node when submitting batch jobs. You should not run modulePreservation with more threads than the number of cores you have allocated to your job. Doing so will cause the program to “thrash”: all threads will run very slowly as they compete for resources and R may possibly crash.

To parallelise the permutation procedure in modulePreservation across multiple nodes you can use the combineAnalyses function. In this case, you must submit multiple jobs, and set the nPerm argument to be the total number of permutations you wish to run in total, divided by the number of nodes/jobs you are submitting. The combineAnalyses function will take the output of the modulePreservation function, combine the null distributions, and calculate the permutation test p-values using the combined permutations of each module preservation statistic.

Estimating wall time

The required runtime of the permutation procedure will vary depending on the size of the network, the size of the modules, the number of samples in each dataset, the number of modules, and the number of permutations.

The required Wall time can be estimated by running modulePreservation with a few permutations per core and setting the verbose flag to TRUE. The required Wall time can then be estimated from the time stamps of the output.

For example, consider the following output from our cluster:

## [2016-06-14 17:25:16 AEST] Validating user input...
## [2016-06-14 17:25:16 AEST]   Loading matrices of dataset "liver" into RAM...
## [2016-06-14 17:26:29 AEST]   Checking matrices for problems...
## [2016-06-14 17:26:31 AEST]   Unloading dataset from RAM...
## [2016-06-14 17:26:31 AEST]   Loading matrices of dataset "brain" into RAM...
## [2016-06-14 17:27:45 AEST]   Checking matrices for problems...
## [2016-06-14 17:27:47 AEST] Input ok!
## [2016-06-14 17:27:47 AEST] Calculating preservation of network subsets from
##                            dataset "brain" in dataset "liver".
## [2016-06-14 17:27:47 AEST]   Pre-computing intermediate properties in dataset
##                              "brain"...
## [2016-06-14 17:27:48 AEST]   Unloading dataset from RAM...
## [2016-06-14 17:27:48 AEST]   Loading matrices of dataset "liver" into RAM...
## [2016-06-14 17:29:01 AEST]   Calculating observed test statistics...
## [2016-06-14 17:29:02 AEST]   Generating null distributions from 320
##                              permutations using 32 threads...
## 
##   100% completed.
## 
## [2016-06-14 17:29:24 AEST]   Calculating P-values...
## [2016-06-14 17:29:24 AEST]   Collating results...
## [2016-06-14 17:29:24 AEST] Unloading dataset from RAM...
## [2016-06-14 17:29:25 AEST] Done!

Here, we are running modulePreservation to test whether all gene coexpression network modules discovery in the adiposed tissue are preserved in the liver tissue of the same samples. These datasets consist of roughly 22,000 genes and 300 samples. We have run 320 permutations on 32 cores: i.e. 10 permutations per core.

We can use the timestamps surrounding the progress report (“100% completed”) in the output to estimate the total runtime for an arbitrary number of permutations. It took 22 seconds to run 10 permutations per core, so 2.2 seconds per permutation per core. If we want to run 20,000 permutations, this will take approximately 23 minutes. Adding the time taken to check the input and swap datasets (approximately 4 minutes), we would allocate 30 minutes for the job. It is always better to provide an overly cautious estimate of the job runtime so that the cluster does not cancel the job just as it is finishing.

Estimating memory usage

Memory usage of modulePreservation depends on the total size of the test dataset, the sizes of each module that will be tested, and the number of threads. If disk.matrix objects are supplied as input NetRep will only keep the data, correlation and network matrices of one dataset in memory at any point in time. Each thread requires additional memory to store the network properties of each permuted module at each permutation. The additional memory usage of each thread depends on the sizes of the modules to be tested.

The simplest way to run the permutation procedure is to allocate a full node for your job: that is, set the number of threads to the number of cores on that node, and request all the memory of that node.

If you wish to allocate less memory, you can estimate the memory requirements of NetRep through the same job we used to estimate runtime. You could then allocate the maximum memory used by this job (plus 10%).

Optimising runtime

There are several approaches that can be used to reduce runtime of the permutation procedure.

If your system has sufficient memory, you may see a performance improvement by running multiple instances of NetRep rather than parallelising over multiple threads. The results from these multiple jobs can then be combined using the combineAnalyses function. This is useful if you see a difference in performance between a single threaded instance vs. a multi thread instance.

Performance may also improve by compiling R against different BLAS and LAPACK libraries prior to installation of NetRep. This requires some experimentation as different libraries will work better for different systems. Note however that changing these typically means recompiling all of R from source.

The runtime of the permutation procedure is primarily influenced by the size of the modules and the number of samples in each test dataset. Permutation testing of large modules takes a much longer time than small modules; by a factor of \(n^{2}\) for \(n\) nodes. Excluding large modules, or filtering modules to the top most connected nodes, can thus dramatically reduce runtime. For example, in our ouput above in the section on estimating runtime each permutation took 2.2 seconds to complete. By excluding modules with more than 250 nodes (12 of 37 modules) runtime was reduced to 0.12 seconds: almost a 20-fold speed increase. Performing dimensionality reduction prior to network inference will also have this effect.

The permutation procedure will also take longer the more samples in the test dataset. This is due to the single value decomposition required to calculate the summary profile of each module at each permutation: this is the most computationally complex network property to calculate. Runtime will be dramatically reduced by setting the data argument to NULL, however this will prevent three of the seven statistics from being calculated. Alternatively downsampling may be employed to reduce the sample size in the test dataset.