Quantitative Biology authors titles recent submissions
wTO: an R package for computing weighted topological overlap and
arXiv:1711.04702v1 [q-bio.MN] 13 Nov 2017
consensus networks with an integrated visualization tool
Deisy Morselli Gysi1,2,* , Andre Voigt3 , Tiago Miranda Fragoso4 , Eivind Almaas3,5 , and
Katja Nowick6
1
Department of Computer Science, Interdisciplinary Center of Bioinformatics, University of
Leipzig, Leipzig, D-04109, Leipzig.
2
Swarm Intelligence and Complex Systems Group, Faculty of Mathematics and Computer
Science, University of Leipzig, Leipzig, D-04109, Leipzig.
3
Department of Biotechnology, NTNU - Norwegian University of Science and Technology,
Trondheim, Norway.
4
5
Fundação Cesgranrio, Rio de Janeiro, 20261-903, Brazil.
K.G. Jebsen Center for Genetic Epidemiology, NTNU - Norwegian University of Science
and Technology, Trondheim, Norway.
6
Human Biology Group, Institute for Biology, Department of Biology, Chemistry,
Pharmacy, Free University of Berlin, Konigin-Luise-Str. 1-3, D-14195 Berlin, Germany.
*
To whom correspondence should be addressed. deisy@bioinf.uni-leipzig.de
November 15, 2017
Abstract
Background Gene co-expression network analyses have become a central approach for the systems-level
analysis of biological data. Several software packages exist for generating and analyzing such networks,
either from correlation scores or the absolute value of a transformed score called weighted topological overlap (wT O). However, since some genes are able to up- or down-regulate other genes, it is important to
explicitly consider both positive and negative correlations when constructing a gene co-expression network.
Additionally, there has been a growing interest in the systematic comparison of multiple networks to identify
deferentially changed links. Typically, such analyses are focused on the comparison of networks or data from
two different conditions. Results Here, we present an R package for calculating the weighted topological
overlap (wT O), that explicitly addresses the sign of wTO values. The package includes the calculation of
p-values (raw and adjusted) for each pairwise gene score. Our package also allows the calculation of networks from time series, without replicates. Additionally, our R package incorporates a novel method for
calculating a consensus network (CN ) from two or more networks. To visualize the resulting networks,
1
the R package contains a visualization tool which allows for the direct network manipulation and access
of node and link information. When testing the package on a standard laptop computer, we can conduct
all calculations for systems of 20, 000 genes in under two hours. Conclusion In this work, we developed
an R package that allows the computation of wT O networks, CN s and a visualization tool in the R statistical environment. It is publicly available on CRAN repositories under the GPL−2 Open Source License
(https://cran.r-project.org/web/packages/wTO/).
Keywords: Co-expression network, Network, Expression, R package, Software, Consensus Network, wTO.
Background
Recent applications of complex network analysis methods have provided important new knowledge of
the functioning and interactions of genes at the systems level 1;2;3;4 . Within the area of biological network
analyses, co-expression networks have received much attention 5;6 . For the co-expression networks, a pair of
nodes are typically connected by a link if the genes they represent show a significantly correlated expression
pattern. In the network, this link may be represented as a binary relationship, where 1 = “presence” and
0 = “absence” of the link, or alternatively, the link may have a numeric value (often called weight). The
magnitude of the weight is often interpreted as representing the strength of a gene-pair relationship, and the
sign as indicative of the type of associated gene interaction: positive if the genes are co-regulated, negative
if they are oppositely controlled 7 .
In many implementations of network analyses, we may primarily be interested in an a priori defined
subset of genes with a specific set of properties. Examples include transcription factors (TFs), genes with
known orthologs in a set of organisms of interest, or disease genes
8;9
. When the interest is focused on
a subset of all available genes, sometimes the choice is made to only take into account genes of interest
in the network instead of including the full set of correlations, since the interest lies in a subset of genes.
A major drawback with such an approach, is that relevant information contained in interaction patterns
among excluded genes that would affect network topology and link strength values, is not incorporated in
the network. The loss of such information is not only undesirable, but may also lead to biased results.
The approach of weighted topological (wT O) network analysis is a method that transforms the link
weights for pairs of genes by averaging over all interactions the pair of nodes share. Thus, wTO is a method
that implicitly includes correlations among nodes that are going to be exempt from further analysis. The
wT O method 10;11;12 can be used to determine the overlap among classes of transcripts, for example TFs
and non coding RNAs (ncRNAs). The resulting wTO network provides a more robust representation of
the connections and interactions among the node-set of interest than a simple correlation network analysis
focused only on the node-set of interest 13 . The package WGCNA 14;15 is widely used for weighted gene coexpression network analysis (WGCNA) studies. It provides functions for the calculation of the adjacency
matrix for all pairs of genes as the nth power of absolute correlations, resulting in an unsigned network.
Network modules can be defined with this package by unsupervised clustering. Previously, Nowick and
collaborators 13 developed a mathematical method to calculate the wT O for a set of nodes that explicitly
takes into account both positive and negative correlations. This version of the wT O is especially valuable
for investigating gene regulatory networks and other networks in which positive and negative correlations
explicitly want to be distinguished since one indicates activating and the other one inhibiting/repressing
relationships. This methodology is implemented in the R package presented here.
2
When constructing several wT O networks from independent data sets, the resulting networks’ topologies
may be different 16 . These differences may arise from several sources: (A) technical differences, such as the
platform on which the expression data was measured, the facility where data was collected and prepared,
or how data was processed. (B) Another cause may be biological differences from confounding factors, such
as sex, age, and geographic origin of the individuals measured. It is thus desirable to obtain an integrated
network that considers all independently derived networks as biological replicates and systematically identifies their commonalities. We present here a novel method to compute the network that captures all this
information; we call this the consensus network (CN ).
Here, we present wTO, an R package capable of computing both signed and unsigned wTO networks as
well as the CN , that comes with an integrated tool to visualize the resulting networks. The workflow of the
package is shown in Fig. 1.
Implementation
Weighted Topological Overlap calculation
The representation of interactions between a set of nodes by the wT O method
10;11;12
takes into account
the overall commonality of all the links a node has, instead of basing the analysis only on calculating raw
correlations among the nodes. It thus provides a more comprehensive understanding of how two nodes are
related. The use of the wT O method further reduces false effects coming from incorrectly assigned linked
genes (false positives). Therefore, it is expected that a wT O network contains more robust information about
the connections among nodes than what would result from simply taking direct correlations into account 11;13 .
The wT O can be computed based on a similarity matrix, where the link weights are calculated using Pearson’s
product moment correlation coefficient or the Spearman Rank correlation. The first one measures the linear
relationship between two genes. Note that, the Pearson’s correlation coefficient is sensitive to extreme
values, and therefore it can exaggerate or reduce the strength of a relationship. While, the Spearman Rank
Correlation is recommended when data is monotonically correlated, skewed or ordinal, it is less sensitive to
extreme outliers than the Pearson coefficient 17;18;19;20 .
To calculate the wT O
13
, let i, j = 1, . . . , n be nodes that belong to the set of nodes of interest, u =
1, . . . , m all the nodes, including the ones in the set of interest, and the adjacency matrix A = [ai,j ], defined
as
ai,j =
ρi,j
0
i 6= j
(1)
i = j.
with ρ being a correlation measure. Consequently, the wTO is defined as follows: Ω = wTO = [ωi,j ], where
ωi,j =
Pn
u=1 ai,u au,j + ai,j
,
min(|ki |, |kj |) + 1 − |ai,j |
3
(2)
with
ki =
X
(3)
ai,j .
j
Note that, this expression explicitly includes both positive and negative correlations, and thus allows for
positive and negative wT Oi,j . This is not allowed using other implemented definitions of the wT O method,
making this new version especially valuable for gene regulatory networks.
Since Eq. (2) explicitly allows ai,j 6 0, we need to be aware of the limits of this expression: Consider
three nodes i, j and u and aij 6 0. All the terms in the denominator of Eq. (2) will be negative if also
aiu auj 6 0 for all nodes u. However, if aiu auj > 0, then contributions to the sum will cancel out. Similarly
for the case of aij ≥ 0. To systematically assess the potential effect of term cancellation in Eq. (2), we
calculate the absolute weighted topological overlap, |wT O| which uses the absolute value of the correlations
(ai,j = |ai,j |) as input for Eq. (2) . In this case, the sign of the correlation is excluded from the analysis and
only the magnitude of the link-strength is taken into account. Consequently, by generating a scatter plot of
the signed (wT O) and unsigned (|wT O|) versions of the weighted topological overlap, it is possible to assess
at which wT Oi,j term cancellations start affecting the results. Thus, the closer the plot of wT O vs. |wT O|
is to y = |x|, the better.
Unfortunately, just by computing the wT O network we do not get rid of all spurious correlations. A way
to avoid them is to compute a probability of each one of the links being random or being zero. This can be
easily computed by using 1. Bootstrap 21 or 2. Reshuffle 13 .
1. Bootstrapping the individuals, we allow the dependency inside the individuals to be constant, as a
consequence we can calculate the empirical distribution of each one of the links, and based on that,
estimate the probability that the observed wT Oij is approximately zero. The bootstrap is performed
to test the hypothesis that the wT O found in the real data is different than a zero, and thus, that the
observed values carry reasonably small random noise. The hypotheses being tested with the Bootstrap
are
H0 : wT Oij = 0
H : wT O 6= 0
a
ij
.
(4)
One might also be interested in the use of blocked bootstrap resampling 21 , that can be used for
temporal data without sample replicates for each time point. This kind of resampling is necessary,
once there are two correlation components in those samples: The correlation inside the factors of each
sample and the correlation across the time of different samples. In this case, a time block, often also
called lag, is required, this lag can be chosen using a partial correlation of the time per sample. Then
the wT O is computed for a time series, where the observations are not independent of each other.
2. Reshuffling the expression values inside the individuals, we grant the dependency of the genes to be
removed. By doing that, we can compute the probability of the link being 0, and assume that each
gene is independent of the others. The hypotheses tested with the Reshuffle are the same as in the
bootstrap.
4
A method for determining a Consensus Network
It is well known that co-expression networks vary much among different datasets (even if most conditions
for obtaining the samples and data were the same). Thus, it is necessary for researchers to evaluate variability
between such networks and to focus on their commonalities. This is an important step in co-expression
network analysis that often is overlooked.
Berto and collaborators
16
described a Consensus Network from primates’ frontal lobes by applying a
Wilcoxon test on the links. This approach assumes that there are enough networks to be compared in
two different groups. The usage our proposed methodology allows the minimum of two different wT O,
containing only the significant links, to be combined into one CN . This methodology also has the advantage
of penalizing links with opposite signs. In the same direction, links with the same sign will have it’s CNi,j
value closer to the highest wT Oi,j of a link in the n networks. Our first step is to remove nodes that doesn’t
exist in all networks, meaning that, if a node is absent in at least one network, we are not able to compute
a consensus of the links that belong to that node.
In order to obtain one integrated wT O derived from multiple independent network replicates, we calculate
a CN using the function as follows:
Let k = 1, . . . , w be the number of replicated networks, then Ω∗i,j = wTOCN is defined as
Ω∗ij =
w
X
αij,k ωij,k ,
(5)
|ωij,k |
.
αij,k = Pw
k=1 |ωij,k |
(6)
k=1
where
A threshold can be used to remove links with values close to zero, that are the links in non-consensus.
Results and discussion
Package functions
The function wTO calculates the weights for all links according to Eq. (2) between a set of nodes for a
given input data set. If the user is not interested in the resampling, he can simply run this wT O function.
To test whether the calculated wT O is different from random expectation and to decide on a suitable
threshold value for including link weights, we implemented the function wTO.Complete. Here, the wT O
is calculated a number of times n specified by the user, by either 1) Bootstrapping (method resampling
= ‘‘Bootstrap’’) the individuals, or for time series data (method resampling = ‘‘BlockBootstrap’’)
or 2) Permuting the expression values for each individual (method resampling = ‘‘Reshuffle’’) 13 . The
default value of ∆ is set to 0.2. The user can specify the kind of correlation that the function should use,
Pearson correlation is the default.
Because bootstrapping and permutation tests can be computationally expensive, the wTO.Complete can
also run in parallel, to reduce the wall clock time. For running in parallel, the user may specify a given
number of k computer threads to be used in the calculations. To implement the parallel function, we used
5
th R package parallel 22 .
The output of the wTO.Complete function has two outputs, a diagnosis set of plots and a list with the
following results:
• $Correlation is a data.table containing the Pearson or Spearman correlations between all the nodes,
not only the set of interest. The wT O links for the set of nodes of interest are based on these correlations. The default of this output is set to FALSE.
• $wTO is a data.table containing the nodes, the wT O values (signed and unsigned), the p-values and the
adjusted p-values computed using both signed and unsigned correlations.
• $Quantile is a table containing the empirical and “real” quantiles for: 0.1%, 2.5%, 10%, 90%, 97.5%
and 99.9%. Those empirical values can be used as a threshold for the wT O values, when it is not
desired to visualize low wT O values.
The set of plots shows the quality of the resample, the closer the density of the resampled data is to the
real data, the better. Another produced plot is the scatter plot of the wT O vs |wT O| values, previously
described. The scatter plot of p-values against the wT O and |wT O| values is also plotted along with the
suggested cut-offs using the empirical quantiles.
Computing the CN can be done using the function wTO.Consensus, this function allows the user to give
a list of networks in the format of data.frames with: Node 1, Node 2 and the link weight. The output here
is a data.table containing the node’s names and the consensus weight.
Our R package also includes options to visualize the resulting networks. We have used a combination of the
packages network 23;24 , igraph 25 and visNetwork 26 to build the visualization capability. Here, the function
NetVis generates a graph using as input a list of links and their corresponding weights. The analysis functions
wTO.Complete and Consensus both generate network data-structures (edge list) that can be visualized with
this function. The user needs to choose a relevant wT O-threshold, or p-value cut-off, to select the set of links
to be plotted. Additionally, the user may choose a layout for the network visualization from those available
in the igraph 25 package. By default, the wT O-threshold value is set to 0.5, and the network layout-style
is set to layout nicely. The size of the nodes is relative to the degree of its node. An option allowed
on this package is also to make clusters from the nodes. The MakeGroups parameter allows the user to
choose clustering algorithms from: “walktrap” 27 , “optimal” 28 , “spinglass” 29;30;31 , “edge.betweenness’ 32;33 ,
‘fast greedy” 34 , “infomap’ 35;36 , “louvain” 37 , “label prop’ 38 and “leading eigen” 39 . All those algorithms are
implemented in the igraph package 25 . Nodes are colored according to the cluster they belong to. The width
of a link is relative to the wT Oi,j , and its color is relative to its sign (if a signed network was calculated).
Nodes can have different shapes, allowing for labeling of nodes of different classes, for example target genes
or protein coding and non-protein coding genes.
The package provides a example dataset and an example of nodes of interest as well.
Consensus example
A visual representation of the Consensus Network methodology is shown in Fig. 3. The thicker the link
between two nodes is, the stronger the correlation between them. The signs are represented by the colours,
blue and orange. If a link has different signs in the networks, the strength of the link in the CN is close to
6
zero. When all links agree to the same or close value, the strength of the CN value is closer to the maximum
value.
Stress Test
Normally, when running the wT O, the interest lies on a subset of nodes of interest. In Fig 2 we show
the runtime for different network sizes, and different proportions of nodes of interest. Normally, running the
wT O for TF using the expressed genes, we have around 14% of nodes of interest. In a normal notebook, it’s
possible to compute the wTO for a full network with 20,000 nodes in 20 miliseconds per link. It shows that
is possible to compute the wTO easily for a full set of real gene expression networks.
Conclusion
The new wTO package allows the wT O network calculation for both positive and negative correlation,
which is not provided in any other published R package. Our package computes p-values for each link based
on its empirical distribution, which is another novel feature. This improvement to the wT O method permits
the reduction of false positives. Our package also allows the computation of a CN , which allows integrating
networks derived from different studies or datasets and focusing on links that consistently show in these
networks. We also provide an interactive visualization tool that can be used to visualize both, wT O network
and CN .
Availability and requirements
R relies on the following packages: som 40 , plyr 41 , stringr 42 , network 23;24 , igraph 25 , visNetwork 26 ,
data.table 43 and the standard packages stats and parallel 22 . It is publicly available on CRAN repositories under the GPL-2 Open Source License https://cran.r-project.org/web/packages/wTO/. It is
platform independent.
Declarations
Abbreviations
wTO: Weighted topological overlap; CN: Consensus Network; TF: Transcription Factor; lncRNA: long
non coding RNA; miRNA: micro RNA.
Acknowledgements
We thank Daniel Gerighausen for discussions. We thank Martina Hall for discussions and suggestions on
the package.
7
Author’s contributions
DG implemented the code in R. DG and TM conceived the idea of p-values for the edges. KN and
EA generalized the wT O for signed values. DG wrote the draft of manuscript. All authors discussed the
manuscript, read and approved the final version of the manuscript.
Competing interests
The authors declare that they have no competing interests.
Consent for publication
Not applicable.
Ethics approval and consent to participate
Not applicable.
Funding
This work was supported partially by a doctoral grant from the Brazilian government’s Science without
Borders program (GDE 204111/2014-5).
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Figures
Figure 1: The wTO package workflow. The wTO.Complete function takes as input a data set, in the
data.frame format, with columns being the individuals, and the rows the measured variables (for example
gene expression) and a vector of overlapping nodes (for example a set of transcription factor genes). The
user has to set the resampling and the correlation methods. The resampling will be done for as many times
as desired. The output of the function can be used to plot the network and to create the CN , using the
Net.Vis or wTO.Consensus functions, respectively. The output from the wTO.Consensus can be plotted as
well, using the same visualization function.
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Figure 2: Running time for the computation of wTO for each link for different sizes of networks
and proportions of sets of nodes of interest: The running time of the wTO increases with the increase
of the proportion of nodes of interest. If we are interested, for example, in having a TF-TF wTO network,
using as background the full set of expressed genes, we have around 3K TFs and 22K genes, i.e. 14% of
nodes of interest.
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B
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A
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Figure 3: Consensus Network: A schematic example of how the CN works. Links that have multiple signs
has its weight dropped to zero. Links that have a consistency of signs are kept, and the weight is closer to
the highest value.
13
arXiv:1711.04702v1 [q-bio.MN] 13 Nov 2017
consensus networks with an integrated visualization tool
Deisy Morselli Gysi1,2,* , Andre Voigt3 , Tiago Miranda Fragoso4 , Eivind Almaas3,5 , and
Katja Nowick6
1
Department of Computer Science, Interdisciplinary Center of Bioinformatics, University of
Leipzig, Leipzig, D-04109, Leipzig.
2
Swarm Intelligence and Complex Systems Group, Faculty of Mathematics and Computer
Science, University of Leipzig, Leipzig, D-04109, Leipzig.
3
Department of Biotechnology, NTNU - Norwegian University of Science and Technology,
Trondheim, Norway.
4
5
Fundação Cesgranrio, Rio de Janeiro, 20261-903, Brazil.
K.G. Jebsen Center for Genetic Epidemiology, NTNU - Norwegian University of Science
and Technology, Trondheim, Norway.
6
Human Biology Group, Institute for Biology, Department of Biology, Chemistry,
Pharmacy, Free University of Berlin, Konigin-Luise-Str. 1-3, D-14195 Berlin, Germany.
*
To whom correspondence should be addressed. deisy@bioinf.uni-leipzig.de
November 15, 2017
Abstract
Background Gene co-expression network analyses have become a central approach for the systems-level
analysis of biological data. Several software packages exist for generating and analyzing such networks,
either from correlation scores or the absolute value of a transformed score called weighted topological overlap (wT O). However, since some genes are able to up- or down-regulate other genes, it is important to
explicitly consider both positive and negative correlations when constructing a gene co-expression network.
Additionally, there has been a growing interest in the systematic comparison of multiple networks to identify
deferentially changed links. Typically, such analyses are focused on the comparison of networks or data from
two different conditions. Results Here, we present an R package for calculating the weighted topological
overlap (wT O), that explicitly addresses the sign of wTO values. The package includes the calculation of
p-values (raw and adjusted) for each pairwise gene score. Our package also allows the calculation of networks from time series, without replicates. Additionally, our R package incorporates a novel method for
calculating a consensus network (CN ) from two or more networks. To visualize the resulting networks,
1
the R package contains a visualization tool which allows for the direct network manipulation and access
of node and link information. When testing the package on a standard laptop computer, we can conduct
all calculations for systems of 20, 000 genes in under two hours. Conclusion In this work, we developed
an R package that allows the computation of wT O networks, CN s and a visualization tool in the R statistical environment. It is publicly available on CRAN repositories under the GPL−2 Open Source License
(https://cran.r-project.org/web/packages/wTO/).
Keywords: Co-expression network, Network, Expression, R package, Software, Consensus Network, wTO.
Background
Recent applications of complex network analysis methods have provided important new knowledge of
the functioning and interactions of genes at the systems level 1;2;3;4 . Within the area of biological network
analyses, co-expression networks have received much attention 5;6 . For the co-expression networks, a pair of
nodes are typically connected by a link if the genes they represent show a significantly correlated expression
pattern. In the network, this link may be represented as a binary relationship, where 1 = “presence” and
0 = “absence” of the link, or alternatively, the link may have a numeric value (often called weight). The
magnitude of the weight is often interpreted as representing the strength of a gene-pair relationship, and the
sign as indicative of the type of associated gene interaction: positive if the genes are co-regulated, negative
if they are oppositely controlled 7 .
In many implementations of network analyses, we may primarily be interested in an a priori defined
subset of genes with a specific set of properties. Examples include transcription factors (TFs), genes with
known orthologs in a set of organisms of interest, or disease genes
8;9
. When the interest is focused on
a subset of all available genes, sometimes the choice is made to only take into account genes of interest
in the network instead of including the full set of correlations, since the interest lies in a subset of genes.
A major drawback with such an approach, is that relevant information contained in interaction patterns
among excluded genes that would affect network topology and link strength values, is not incorporated in
the network. The loss of such information is not only undesirable, but may also lead to biased results.
The approach of weighted topological (wT O) network analysis is a method that transforms the link
weights for pairs of genes by averaging over all interactions the pair of nodes share. Thus, wTO is a method
that implicitly includes correlations among nodes that are going to be exempt from further analysis. The
wT O method 10;11;12 can be used to determine the overlap among classes of transcripts, for example TFs
and non coding RNAs (ncRNAs). The resulting wTO network provides a more robust representation of
the connections and interactions among the node-set of interest than a simple correlation network analysis
focused only on the node-set of interest 13 . The package WGCNA 14;15 is widely used for weighted gene coexpression network analysis (WGCNA) studies. It provides functions for the calculation of the adjacency
matrix for all pairs of genes as the nth power of absolute correlations, resulting in an unsigned network.
Network modules can be defined with this package by unsupervised clustering. Previously, Nowick and
collaborators 13 developed a mathematical method to calculate the wT O for a set of nodes that explicitly
takes into account both positive and negative correlations. This version of the wT O is especially valuable
for investigating gene regulatory networks and other networks in which positive and negative correlations
explicitly want to be distinguished since one indicates activating and the other one inhibiting/repressing
relationships. This methodology is implemented in the R package presented here.
2
When constructing several wT O networks from independent data sets, the resulting networks’ topologies
may be different 16 . These differences may arise from several sources: (A) technical differences, such as the
platform on which the expression data was measured, the facility where data was collected and prepared,
or how data was processed. (B) Another cause may be biological differences from confounding factors, such
as sex, age, and geographic origin of the individuals measured. It is thus desirable to obtain an integrated
network that considers all independently derived networks as biological replicates and systematically identifies their commonalities. We present here a novel method to compute the network that captures all this
information; we call this the consensus network (CN ).
Here, we present wTO, an R package capable of computing both signed and unsigned wTO networks as
well as the CN , that comes with an integrated tool to visualize the resulting networks. The workflow of the
package is shown in Fig. 1.
Implementation
Weighted Topological Overlap calculation
The representation of interactions between a set of nodes by the wT O method
10;11;12
takes into account
the overall commonality of all the links a node has, instead of basing the analysis only on calculating raw
correlations among the nodes. It thus provides a more comprehensive understanding of how two nodes are
related. The use of the wT O method further reduces false effects coming from incorrectly assigned linked
genes (false positives). Therefore, it is expected that a wT O network contains more robust information about
the connections among nodes than what would result from simply taking direct correlations into account 11;13 .
The wT O can be computed based on a similarity matrix, where the link weights are calculated using Pearson’s
product moment correlation coefficient or the Spearman Rank correlation. The first one measures the linear
relationship between two genes. Note that, the Pearson’s correlation coefficient is sensitive to extreme
values, and therefore it can exaggerate or reduce the strength of a relationship. While, the Spearman Rank
Correlation is recommended when data is monotonically correlated, skewed or ordinal, it is less sensitive to
extreme outliers than the Pearson coefficient 17;18;19;20 .
To calculate the wT O
13
, let i, j = 1, . . . , n be nodes that belong to the set of nodes of interest, u =
1, . . . , m all the nodes, including the ones in the set of interest, and the adjacency matrix A = [ai,j ], defined
as
ai,j =
ρi,j
0
i 6= j
(1)
i = j.
with ρ being a correlation measure. Consequently, the wTO is defined as follows: Ω = wTO = [ωi,j ], where
ωi,j =
Pn
u=1 ai,u au,j + ai,j
,
min(|ki |, |kj |) + 1 − |ai,j |
3
(2)
with
ki =
X
(3)
ai,j .
j
Note that, this expression explicitly includes both positive and negative correlations, and thus allows for
positive and negative wT Oi,j . This is not allowed using other implemented definitions of the wT O method,
making this new version especially valuable for gene regulatory networks.
Since Eq. (2) explicitly allows ai,j 6 0, we need to be aware of the limits of this expression: Consider
three nodes i, j and u and aij 6 0. All the terms in the denominator of Eq. (2) will be negative if also
aiu auj 6 0 for all nodes u. However, if aiu auj > 0, then contributions to the sum will cancel out. Similarly
for the case of aij ≥ 0. To systematically assess the potential effect of term cancellation in Eq. (2), we
calculate the absolute weighted topological overlap, |wT O| which uses the absolute value of the correlations
(ai,j = |ai,j |) as input for Eq. (2) . In this case, the sign of the correlation is excluded from the analysis and
only the magnitude of the link-strength is taken into account. Consequently, by generating a scatter plot of
the signed (wT O) and unsigned (|wT O|) versions of the weighted topological overlap, it is possible to assess
at which wT Oi,j term cancellations start affecting the results. Thus, the closer the plot of wT O vs. |wT O|
is to y = |x|, the better.
Unfortunately, just by computing the wT O network we do not get rid of all spurious correlations. A way
to avoid them is to compute a probability of each one of the links being random or being zero. This can be
easily computed by using 1. Bootstrap 21 or 2. Reshuffle 13 .
1. Bootstrapping the individuals, we allow the dependency inside the individuals to be constant, as a
consequence we can calculate the empirical distribution of each one of the links, and based on that,
estimate the probability that the observed wT Oij is approximately zero. The bootstrap is performed
to test the hypothesis that the wT O found in the real data is different than a zero, and thus, that the
observed values carry reasonably small random noise. The hypotheses being tested with the Bootstrap
are
H0 : wT Oij = 0
H : wT O 6= 0
a
ij
.
(4)
One might also be interested in the use of blocked bootstrap resampling 21 , that can be used for
temporal data without sample replicates for each time point. This kind of resampling is necessary,
once there are two correlation components in those samples: The correlation inside the factors of each
sample and the correlation across the time of different samples. In this case, a time block, often also
called lag, is required, this lag can be chosen using a partial correlation of the time per sample. Then
the wT O is computed for a time series, where the observations are not independent of each other.
2. Reshuffling the expression values inside the individuals, we grant the dependency of the genes to be
removed. By doing that, we can compute the probability of the link being 0, and assume that each
gene is independent of the others. The hypotheses tested with the Reshuffle are the same as in the
bootstrap.
4
A method for determining a Consensus Network
It is well known that co-expression networks vary much among different datasets (even if most conditions
for obtaining the samples and data were the same). Thus, it is necessary for researchers to evaluate variability
between such networks and to focus on their commonalities. This is an important step in co-expression
network analysis that often is overlooked.
Berto and collaborators
16
described a Consensus Network from primates’ frontal lobes by applying a
Wilcoxon test on the links. This approach assumes that there are enough networks to be compared in
two different groups. The usage our proposed methodology allows the minimum of two different wT O,
containing only the significant links, to be combined into one CN . This methodology also has the advantage
of penalizing links with opposite signs. In the same direction, links with the same sign will have it’s CNi,j
value closer to the highest wT Oi,j of a link in the n networks. Our first step is to remove nodes that doesn’t
exist in all networks, meaning that, if a node is absent in at least one network, we are not able to compute
a consensus of the links that belong to that node.
In order to obtain one integrated wT O derived from multiple independent network replicates, we calculate
a CN using the function as follows:
Let k = 1, . . . , w be the number of replicated networks, then Ω∗i,j = wTOCN is defined as
Ω∗ij =
w
X
αij,k ωij,k ,
(5)
|ωij,k |
.
αij,k = Pw
k=1 |ωij,k |
(6)
k=1
where
A threshold can be used to remove links with values close to zero, that are the links in non-consensus.
Results and discussion
Package functions
The function wTO calculates the weights for all links according to Eq. (2) between a set of nodes for a
given input data set. If the user is not interested in the resampling, he can simply run this wT O function.
To test whether the calculated wT O is different from random expectation and to decide on a suitable
threshold value for including link weights, we implemented the function wTO.Complete. Here, the wT O
is calculated a number of times n specified by the user, by either 1) Bootstrapping (method resampling
= ‘‘Bootstrap’’) the individuals, or for time series data (method resampling = ‘‘BlockBootstrap’’)
or 2) Permuting the expression values for each individual (method resampling = ‘‘Reshuffle’’) 13 . The
default value of ∆ is set to 0.2. The user can specify the kind of correlation that the function should use,
Pearson correlation is the default.
Because bootstrapping and permutation tests can be computationally expensive, the wTO.Complete can
also run in parallel, to reduce the wall clock time. For running in parallel, the user may specify a given
number of k computer threads to be used in the calculations. To implement the parallel function, we used
5
th R package parallel 22 .
The output of the wTO.Complete function has two outputs, a diagnosis set of plots and a list with the
following results:
• $Correlation is a data.table containing the Pearson or Spearman correlations between all the nodes,
not only the set of interest. The wT O links for the set of nodes of interest are based on these correlations. The default of this output is set to FALSE.
• $wTO is a data.table containing the nodes, the wT O values (signed and unsigned), the p-values and the
adjusted p-values computed using both signed and unsigned correlations.
• $Quantile is a table containing the empirical and “real” quantiles for: 0.1%, 2.5%, 10%, 90%, 97.5%
and 99.9%. Those empirical values can be used as a threshold for the wT O values, when it is not
desired to visualize low wT O values.
The set of plots shows the quality of the resample, the closer the density of the resampled data is to the
real data, the better. Another produced plot is the scatter plot of the wT O vs |wT O| values, previously
described. The scatter plot of p-values against the wT O and |wT O| values is also plotted along with the
suggested cut-offs using the empirical quantiles.
Computing the CN can be done using the function wTO.Consensus, this function allows the user to give
a list of networks in the format of data.frames with: Node 1, Node 2 and the link weight. The output here
is a data.table containing the node’s names and the consensus weight.
Our R package also includes options to visualize the resulting networks. We have used a combination of the
packages network 23;24 , igraph 25 and visNetwork 26 to build the visualization capability. Here, the function
NetVis generates a graph using as input a list of links and their corresponding weights. The analysis functions
wTO.Complete and Consensus both generate network data-structures (edge list) that can be visualized with
this function. The user needs to choose a relevant wT O-threshold, or p-value cut-off, to select the set of links
to be plotted. Additionally, the user may choose a layout for the network visualization from those available
in the igraph 25 package. By default, the wT O-threshold value is set to 0.5, and the network layout-style
is set to layout nicely. The size of the nodes is relative to the degree of its node. An option allowed
on this package is also to make clusters from the nodes. The MakeGroups parameter allows the user to
choose clustering algorithms from: “walktrap” 27 , “optimal” 28 , “spinglass” 29;30;31 , “edge.betweenness’ 32;33 ,
‘fast greedy” 34 , “infomap’ 35;36 , “louvain” 37 , “label prop’ 38 and “leading eigen” 39 . All those algorithms are
implemented in the igraph package 25 . Nodes are colored according to the cluster they belong to. The width
of a link is relative to the wT Oi,j , and its color is relative to its sign (if a signed network was calculated).
Nodes can have different shapes, allowing for labeling of nodes of different classes, for example target genes
or protein coding and non-protein coding genes.
The package provides a example dataset and an example of nodes of interest as well.
Consensus example
A visual representation of the Consensus Network methodology is shown in Fig. 3. The thicker the link
between two nodes is, the stronger the correlation between them. The signs are represented by the colours,
blue and orange. If a link has different signs in the networks, the strength of the link in the CN is close to
6
zero. When all links agree to the same or close value, the strength of the CN value is closer to the maximum
value.
Stress Test
Normally, when running the wT O, the interest lies on a subset of nodes of interest. In Fig 2 we show
the runtime for different network sizes, and different proportions of nodes of interest. Normally, running the
wT O for TF using the expressed genes, we have around 14% of nodes of interest. In a normal notebook, it’s
possible to compute the wTO for a full network with 20,000 nodes in 20 miliseconds per link. It shows that
is possible to compute the wTO easily for a full set of real gene expression networks.
Conclusion
The new wTO package allows the wT O network calculation for both positive and negative correlation,
which is not provided in any other published R package. Our package computes p-values for each link based
on its empirical distribution, which is another novel feature. This improvement to the wT O method permits
the reduction of false positives. Our package also allows the computation of a CN , which allows integrating
networks derived from different studies or datasets and focusing on links that consistently show in these
networks. We also provide an interactive visualization tool that can be used to visualize both, wT O network
and CN .
Availability and requirements
R relies on the following packages: som 40 , plyr 41 , stringr 42 , network 23;24 , igraph 25 , visNetwork 26 ,
data.table 43 and the standard packages stats and parallel 22 . It is publicly available on CRAN repositories under the GPL-2 Open Source License https://cran.r-project.org/web/packages/wTO/. It is
platform independent.
Declarations
Abbreviations
wTO: Weighted topological overlap; CN: Consensus Network; TF: Transcription Factor; lncRNA: long
non coding RNA; miRNA: micro RNA.
Acknowledgements
We thank Daniel Gerighausen for discussions. We thank Martina Hall for discussions and suggestions on
the package.
7
Author’s contributions
DG implemented the code in R. DG and TM conceived the idea of p-values for the edges. KN and
EA generalized the wT O for signed values. DG wrote the draft of manuscript. All authors discussed the
manuscript, read and approved the final version of the manuscript.
Competing interests
The authors declare that they have no competing interests.
Consent for publication
Not applicable.
Ethics approval and consent to participate
Not applicable.
Funding
This work was supported partially by a doctoral grant from the Brazilian government’s Science without
Borders program (GDE 204111/2014-5).
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Figures
Figure 1: The wTO package workflow. The wTO.Complete function takes as input a data set, in the
data.frame format, with columns being the individuals, and the rows the measured variables (for example
gene expression) and a vector of overlapping nodes (for example a set of transcription factor genes). The
user has to set the resampling and the correlation methods. The resampling will be done for as many times
as desired. The output of the function can be used to plot the network and to create the CN , using the
Net.Vis or wTO.Consensus functions, respectively. The output from the wTO.Consensus can be plotted as
well, using the same visualization function.
11
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20
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25%
50%
75%
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10000
15000
20000
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Figure 2: Running time for the computation of wTO for each link for different sizes of networks
and proportions of sets of nodes of interest: The running time of the wTO increases with the increase
of the proportion of nodes of interest. If we are interested, for example, in having a TF-TF wTO network,
using as background the full set of expressed genes, we have around 3K TFs and 22K genes, i.e. 14% of
nodes of interest.
12
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A
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E
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F
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C
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B
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G
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G
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F
C
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A
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E
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F
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G
B
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D
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A
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E
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F
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G
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Figure 3: Consensus Network: A schematic example of how the CN works. Links that have multiple signs
has its weight dropped to zero. Links that have a consistency of signs are kept, and the weight is closer to
the highest value.
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