Introduction to Patterns: a modeling tool dedicated to biological network modeling

Frédéric Bertrand and Myriam Maumy-Bertrand

Université de Strasbourg et CNRS,IRMA, labex IRMIA
frederic.bertrand@math.unistra.fr

2021-03-21

Patterns: a modeling tool dedicated to biological network modeling

It allows for single or joint modeling of, for instance, genes and proteins. It is design to work with patterned data. Famous examples of problems related to patterned data are: * recovering signals in networks after a stimulation (cascade network reverse engineering), * analysing periodic signals.

The weights are viewed as a penalty factors in the penalized regression model: it is a number that multiplies the lambda value in the minimization problem to allow differential shrinkage, Friedman et al. 2010, equation 1 page 3. If equal to 0, it implies no shrinkage, and that variable is always included in the model. Default is 1 for all variables. Infinity means that the variable is excluded from the model. Note that the weights are rescaled to sum to the number of variables.

Due to maximum size requirement for CRAN packages, most of the graphics of the vignette will be created when the code is run.

A word for those that have been using our seminal work, the Cascade package that we created several years ago and that was a very efficient network reverse engineering tool for cascade networks (Jung, N., Bertrand, F., Bahram, S., Vallat, L., and Maumy-Bertrand, M. (2014), https://doi.org/10.1093/bioinformatics/btt705, https://cran.r-project.org/package=Cascade, https://github.com/fbertran/Cascade and https://fbertran.github.io/Cascade/).

The Patterns package is more than (at least) a threeway major extension of the Cascade package :

Hence the Patterns package should be viewed more as a completely new modelling tools than as an extension of the Cascade package.

This website and these examples were created by F. Bertrand and M. Maumy-Bertrand.

Installation

You can install the released version of Patterns from CRAN with:

install.packages("Patterns")

You can install the development version of Patterns from github with:

devtools::install_github("fbertran/Patterns")

Examples

Data management

Import Cascade Data (repeated measurements on several subjects) from the CascadeData package and turn them into a micro array object. The second line makes sure the CascadeData package is installed.

library(Patterns)
if(!require(CascadeData)){install.packages("CascadeData")}
data(micro_US)
micro_US<-as.micro_array(micro_US[1:100,],time=c(60,90,210,390),subject=6)
str(micro_US)
#> Formal class 'micro_array' [package "Patterns"] with 7 slots
#>   ..@ microarray: num [1:100, 1:24] 103.2 26 70.7 213.7 13.7 ...
#>   .. ..- attr(*, "dimnames")=List of 2
#>   .. .. ..$ : chr [1:100] "1007_s_at" "1053_at" "117_at" "121_at" ...
#>   .. .. ..$ : chr [1:24] "N1_US_T60" "N1_US_T90" "N1_US_T210" "N1_US_T390" ...
#>   ..@ name      : chr [1:100] "1007_s_at" "1053_at" "117_at" "121_at" ...
#>   ..@ gene_ID   : num 0
#>   ..@ group     : num 0
#>   ..@ start_time: num 0
#>   ..@ time      : num [1:4] 60 90 210 390
#>   ..@ subject   : num 6

Get a summay and plots of the data:

summary(micro_US)
#>    N1_US_T60        N1_US_T90        N1_US_T210       N1_US_T390    
#>  Min.   :  12.2   Min.   :  12.9   Min.   :   1.5   Min.   :  10.1  
#>  1st Qu.: 177.7   1st Qu.: 198.7   1st Qu.: 189.0   1st Qu.: 196.7  
#>  Median : 513.0   Median : 499.4   Median : 608.5   Median : 541.2  
#>  Mean   :1386.6   Mean   :1357.7   Mean   :1450.4   Mean   :1331.2  
#>  3rd Qu.:1912.3   3rd Qu.:1883.4   3rd Qu.:2050.2   3rd Qu.:1646.2  
#>  Max.   :6348.4   Max.   :6507.3   Max.   :6438.5   Max.   :6351.4  
#>    N2_US_T60        N2_US_T90        N2_US_T210       N2_US_T390    
#>  Min.   :  16.7   Min.   :   3.4   Min.   :   5.5   Min.   :   6.1  
#>  1st Qu.: 212.4   1st Qu.: 185.7   1st Qu.: 214.7   1st Qu.: 230.1  
#>  Median : 584.1   Median : 501.5   Median : 596.0   Median : 601.8  
#>  Mean   :1381.9   Mean   :1345.4   Mean   :1410.5   Mean   :1403.7  
#>  3rd Qu.:1616.2   3rd Qu.:1830.5   3rd Qu.:2005.8   3rd Qu.:1901.7  
#>  Max.   :6149.3   Max.   :6090.8   Max.   :6160.6   Max.   :6143.1  
#>    N3_US_T60        N3_US_T90        N3_US_T210       N3_US_T390    
#>  Min.   :   1.9   Min.   :  10.3   Min.   :   3.3   Min.   :   6.6  
#>  1st Qu.: 187.4   1st Qu.: 194.6   1st Qu.: 177.8   1st Qu.: 222.6  
#>  Median : 611.4   Median : 576.2   Median : 552.2   Median : 593.7  
#>  Mean   :1365.4   Mean   :1381.2   Mean   :1310.1   Mean   :1427.1  
#>  3rd Qu.:1855.2   3rd Qu.:2040.2   3rd Qu.:1784.5   3rd Qu.:2131.7  
#>  Max.   :6636.6   Max.   :6515.5   Max.   :6530.4   Max.   :6177.2  
#>    N4_US_T60        N4_US_T90        N4_US_T210       N4_US_T390    
#>  Min.   :  20.2   Min.   :  15.6   Min.   :  19.8   Min.   :   9.3  
#>  1st Qu.: 199.3   1st Qu.: 215.4   1st Qu.: 207.0   1st Qu.: 197.8  
#>  Median : 610.8   Median : 614.0   Median : 544.9   Median : 590.7  
#>  Mean   :1505.1   Mean   :1526.7   Mean   :1401.6   Mean   :1458.8  
#>  3rd Qu.:2198.1   3rd Qu.:2168.9   3rd Qu.:1831.2   3rd Qu.:1984.8  
#>  Max.   :6986.2   Max.   :7148.0   Max.   :6820.0   Max.   :6762.3  
#>    N5_US_T60        N5_US_T90        N5_US_T210       N5_US_T390    
#>  Min.   :   3.4   Min.   :  10.0   Min.   :  10.7   Min.   :  16.5  
#>  1st Qu.: 213.2   1st Qu.: 209.8   1st Qu.: 202.0   1st Qu.: 208.2  
#>  Median : 609.4   Median : 561.3   Median : 555.6   Median : 570.5  
#>  Mean   :1498.2   Mean   :1424.8   Mean   :1394.1   Mean   :1435.3  
#>  3rd Qu.:2008.7   3rd Qu.:1906.5   3rd Qu.:1923.9   3rd Qu.:1867.8  
#>  Max.   :7268.2   Max.   :6857.8   Max.   :6574.0   Max.   :6896.6  
#>    N6_US_T60        N6_US_T90        N6_US_T210       N6_US_T390    
#>  Min.   :  13.0   Min.   :   6.6   Min.   :   3.8   Min.   :  14.4  
#>  1st Qu.: 207.5   1st Qu.: 198.6   1st Qu.: 203.9   1st Qu.: 195.8  
#>  Median : 516.2   Median : 530.6   Median : 578.0   Median : 580.0  
#>  Mean   :1412.9   Mean   :1388.3   Mean   :1416.5   Mean   :1360.8  
#>  3rd Qu.:2037.4   3rd Qu.:1889.8   3rd Qu.:2030.8   3rd Qu.:1872.6  
#>  Max.   :6898.1   Max.   :6749.4   Max.   :6490.0   Max.   :6780.2

plot(micro_US)

Gene selection

There are several functions to carry out gene selection before the inference. They are detailed in the vignette of the package.

Data simulation

Let’s simulate some cascade data and then do some reverse engineering.

We first design the F matrix for \(T_i=4\) times and \(Ngrp=4\) groups. The Fmatobject is an array of sizes \((T_i,T-i,Ngrp^2)=(4,4,16)\).

Ti<-4
Ngrp<-4

Fmat=array(0,dim=c(Ti,Ti,Ngrp^2))

for(i in 1:(Ti^2)){
  if(((i-1) %% Ti) > (i-1) %/% Ti){
    Fmat[,,i][outer(1:Ti,1:Ti,function(x,y){0<(x-y) & (x-y)<2})]<-1
    }
}

The Patterns function CascadeFinit is an utility function to easily define such an F matrix.

Fbis=Patterns::CascadeFinit(Ti,Ngrp,low.trig=FALSE)
str(Fbis)
#>  num [1:4, 1:4, 1:16] 0 0 0 0 0 0 0 0 0 0 ...

Check if the two matrices Fmat and Fbis are identical.

print(all(Fmat==Fbis))
#> [1] TRUE

End of F matrix definition.

Fmat[,,3]<-Fmat[,,3]*0.2
Fmat[3,1,3]<-1
Fmat[4,2,3]<-1
Fmat[,,4]<-Fmat[,,3]*0.3
Fmat[4,1,4]<-1
Fmat[,,8]<-Fmat[,,3]

We set the seed to make the results reproducible and draw a scale free random network.

set.seed(1)
Net<-Patterns::network_random(
  nb=100,
  time_label=rep(1:4,each=25),
  exp=1,
  init=1,
  regul=round(rexp(100,1))+1,
  min_expr=0.1,
  max_expr=2,
  casc.level=0.4
)
Net@F<-Fmat
str(Net)
#> Formal class 'network' [package "Patterns"] with 6 slots
#>   ..@ network: num [1:100, 1:100] 0 0 0 0 0 0 0 0 0 0 ...
#>   ..@ name   : chr [1:100] "gene 1" "gene 2" "gene 3" "gene 4" ...
#>   ..@ F      : num [1:4, 1:4, 1:16] 0 0 0 0 0 0 0 0 0 0 ...
#>   ..@ convF  : num [1, 1] 0
#>   ..@ convO  : num 0
#>   ..@ time_pt: int [1:4] 1 2 3 4

Plot the simulated network.

Patterns::plot(Net, choice="network")

If a gene clustering is known, it can be used as a coloring scheme.

plot(Net, choice="network", gr=rep(1:4,each=25))

Plot the F matrix, for low dimensional F matrices.

plot(Net, choice="F")

Plot the F matrix using the pixmap package, for high dimensional F matrices.

plot(Net, choice="Fpixmap")

We simulate gene expression according to the network that was previously drawn

set.seed(1)
M <- Patterns::gene_expr_simulation(
  network=Net,
  time_label=rep(1:4,each=25),
  subject=5,
  peak_level=200,
  act_time_group=1:4)
str(M)
#> Formal class 'micro_array' [package "Patterns"] with 7 slots
#>   ..@ microarray: num [1:100, 1:20] 86.1 -146.8 228.3 505.1 -36.6 ...
#>   .. ..- attr(*, "dimnames")=List of 2
#>   .. .. ..$ : chr [1:100] "gene 1" "gene 2" "gene 3" "gene 4" ...
#>   .. .. ..$ : chr [1:20] "log(S/US) : P1T1" "log(S/US) : P1T2" "log(S/US) : P1T3" "log(S/US) : P1T4" ...
#>   ..@ name      : chr [1:100] "gene 1" "gene 2" "gene 3" "gene 4" ...
#>   ..@ gene_ID   : num 0
#>   ..@ group     : int [1:100] 1 1 1 1 1 1 1 1 1 1 ...
#>   ..@ start_time: num 0
#>   ..@ time      : int [1:4] 1 2 3 4
#>   ..@ subject   : num 5

Get a summay and plots of the simulated data:

summary(M)
#>  log(S/US) : P1T1   log(S/US) : P1T2    log(S/US) : P1T3   log(S/US) : P1T4  
#>  Min.   :-486.823   Min.   :-1962.641   Min.   :-1923.74   Min.   :-3391.76  
#>  1st Qu.: -54.618   1st Qu.:  -23.300   1st Qu.:  -71.99   1st Qu.:  -58.69  
#>  Median :  -8.319   Median :    0.000   Median :   -3.85   Median :    1.53  
#>  Mean   :   8.799   Mean   :   22.064   Mean   :   20.72   Mean   :   19.82  
#>  3rd Qu.:  69.340   3rd Qu.:    9.707   3rd Qu.:   33.00   3rd Qu.:   69.45  
#>  Max.   : 942.229   Max.   : 1616.469   Max.   : 2899.61   Max.   : 3231.17  
#>  log(S/US) : P2T1  log(S/US) : P2T2   log(S/US) : P2T3   log(S/US) : P2T4  
#>  Min.   :-359.82   Min.   :-1126.13   Min.   :-600.757   Min.   :-797.980  
#>  1st Qu.: -39.27   1st Qu.:  -14.15   1st Qu.: -49.998   1st Qu.: -73.700  
#>  Median :  12.54   Median :    0.00   Median :  -7.749   Median :  -9.760  
#>  Mean   :  21.35   Mean   :   20.02   Mean   :  23.579   Mean   :   8.361  
#>  3rd Qu.:  73.02   3rd Qu.:   28.54   3rd Qu.:  67.216   3rd Qu.:  62.297  
#>  Max.   : 451.61   Max.   :  946.22   Max.   :1869.077   Max.   : 935.321  
#>  log(S/US) : P3T1   log(S/US) : P3T2    log(S/US) : P3T3    log(S/US) : P3T4   
#>  Min.   :-430.696   Min.   :-1003.575   Min.   :-718.3926   Min.   :-1211.592  
#>  1st Qu.: -65.939   1st Qu.:   -3.637   1st Qu.: -43.3783   1st Qu.:  -61.300  
#>  Median :   1.392   Median :    0.000   Median :  -0.3233   Median :   -4.124  
#>  Mean   :   4.032   Mean   :   11.431   Mean   :  19.7553   Mean   :    5.993  
#>  3rd Qu.:  62.391   3rd Qu.:   36.945   3rd Qu.:  69.6650   3rd Qu.:   62.634  
#>  Max.   : 514.510   Max.   :  752.066   Max.   :1497.5790   Max.   : 1296.563  
#>  log(S/US) : P4T1   log(S/US) : P4T2     log(S/US) : P4T3    log(S/US) : P4T4 
#>  Min.   :-451.370   Min.   :-1113.1785   Min.   :-1506.536   Min.   :-446.40  
#>  1st Qu.: -44.107   1st Qu.:   -8.8769   1st Qu.:  -48.512   1st Qu.: -56.95  
#>  Median :   7.193   Median :    0.0000   Median :    4.243   Median :   3.71  
#>  Mean   :  16.034   Mean   :    0.9469   Mean   :  -28.544   Mean   :  28.74  
#>  3rd Qu.:  62.840   3rd Qu.:   33.2306   3rd Qu.:   46.374   3rd Qu.:  64.89  
#>  Max.   : 657.434   Max.   :  844.0070   Max.   :  694.220   Max.   :1368.62  
#>  log(S/US) : P5T1   log(S/US) : P5T2  log(S/US) : P5T3    log(S/US) : P5T4   
#>  Min.   :-747.865   Min.   :-862.59   Min.   :-1493.607   Min.   :-1420.740  
#>  1st Qu.: -78.166   1st Qu.: -15.10   1st Qu.:  -59.853   1st Qu.:  -32.799  
#>  Median :  -8.900   Median :   0.00   Median :   -2.776   Median :    4.008  
#>  Mean   :  -7.693   Mean   :  66.20   Mean   :  -13.865   Mean   :  -12.332  
#>  3rd Qu.:  45.926   3rd Qu.:  31.74   3rd Qu.:   40.324   3rd Qu.:   59.892  
#>  Max.   : 960.419   Max.   :1899.57   Max.   :  770.655   Max.   :  489.916
plot(M)

Network inferrence

We infer the new network using subjectwise leave one out cross-validation (default setting): all measurements from the same subject are removed from the dataset). The inference is carried out with a general Fshape.

Net_inf_P <- Patterns::inference(M, cv.subjects=TRUE)
#> We are at step :  1
#> Computing Group (out of 4) : 
#>  1.........................
#>  2.........................
#>  3.........................
#>  4.........................
#> The convergence of the network is (L1 norm) : 0.01
#> We are at step :  2
#> Computing Group (out of 4) : 
#>  1.........................
#>  2.........................
#>  3.........................
#>  4.........................
#> The convergence of the network is (L1 norm) : 0.00522
#> We are at step :  3
#> Computing Group (out of 4) : 
#>  1.........................
#>  2.........................
#>  3.........................
#>  4.........................
#> The convergence of the network is (L1 norm) : 0.0034
#> We are at step :  4
#> Computing Group (out of 4) : 
#>  1.........................
#>  2.........................
#>  3.........................
#>  4.........................
#> The convergence of the network is (L1 norm) : 0.00235
#> We are at step :  5
#> Computing Group (out of 4) : 
#>  1.........................
#>  2.........................
#>  3.........................
#>  4.........................
#> The convergence of the network is (L1 norm) : 0.00181
#> We are at step :  6
#> Computing Group (out of 4) : 
#>  1.........................
#>  2.........................
#>  3.........................
#>  4.........................
#> The convergence of the network is (L1 norm) : 0.00142
#> We are at step :  7
#> Computing Group (out of 4) : 
#>  1.........................
#>  2.........................
#>  3.........................
#>  4.........................
#> The convergence of the network is (L1 norm) : 0.00117
#> We are at step :  8
#> Computing Group (out of 4) : 
#>  1.........................
#>  2.........................
#>  3.........................
#>  4.........................
#> The convergence of the network is (L1 norm) : 0.00098

Plot of the inferred F matrix

plot(Net_inf_P, choice="F")

Heatmap of the inferred coefficients of the Omega matrix

stats::heatmap(Net_inf_P@network, Rowv = NA, Colv = NA, scale="none", revC=TRUE)

Default values fot the \(F\) matrices. The Finit matrix (starting values for the algorithm). In our case, the Finitobject is an array of sizes \((T_i,T-i,Ngrp^2)=(4,4,16)\).

Ti<-4;
ngrp<-4
nF<-ngrp^2
Finit<-array(0,c(Ti,Ti,nF)) 
              for(ii in 1:nF){    
                if((ii%%(ngrp+1))==1){
                  Finit[,,ii]<-0
                } else {
                  Finit[,,ii]<-cbind(rbind(rep(0,Ti-1),diag(1,Ti-1)),rep(0,Ti))+rbind(cbind(rep(0,Ti-1),diag(1,Ti-1)),rep(0,Ti))
                }
              }

The Fshape matrix (default shape for F matrix the algorithm). Any interaction between groups and times are permitted except the retro-actions (a group on itself, or an action at the same time for an actor on another one).

Fshape<-array("0",c(Ti,Ti,nF)) 
for(ii in 1:nF){  
  if((ii%%(ngrp+1))==1){
    Fshape[,,ii]<-"0"
  } else {
    lchars <- paste("a",1:(2*Ti-1),sep="")
    tempFshape<-matrix("0",Ti,Ti)
    for(bb in (-Ti+1):(Ti-1)){
      tempFshape<-replaceUp(tempFshape,matrix(lchars[bb+Ti],Ti,Ti),-bb)
    }
    tempFshape <- replaceBand(tempFshape,matrix("0",Ti,Ti),0)
    Fshape[,,ii]<-tempFshape
  }
}

Any other form can be used. A “0” coefficient is missing from the model. It allows testing the best structure of an “F” matrix and even performing some significance tests of hypothses on the structure of the \(F\) matrix.

The IndicFshape function allows to design custom F matrix for cascade networks with equally spaced measurements by specifying the zero and non zero \(F_{ij}\) cells of the \(F\) matrix. It is useful for models featuring several clusters of actors that are activated at the time. Let’s define the following indicatrix matrix (action of all groups on each other, which is not a possible real modeling setting and is only used as an example):

TestIndic=matrix(!((1:(Ti^2))%%(ngrp+1)==1),byrow=TRUE,ngrp,ngrp)
TestIndic
#>       [,1]  [,2]  [,3]  [,4]
#> [1,] FALSE  TRUE  TRUE  TRUE
#> [2,]  TRUE FALSE  TRUE  TRUE
#> [3,]  TRUE  TRUE FALSE  TRUE
#> [4,]  TRUE  TRUE  TRUE FALSE

For that choice, we get those init and shape \(F\) matrices.

IndicFinit(Ti,ngrp,TestIndic)
#> , , 1
#> 
#>      [,1] [,2] [,3] [,4]
#> [1,]    0    0    0    0
#> [2,]    0    0    0    0
#> [3,]    0    0    0    0
#> [4,]    0    0    0    0
#> 
#> , , 2
#> 
#>      [,1] [,2] [,3] [,4]
#> [1,]    0    0    0    0
#> [2,]    1    0    0    0
#> [3,]    1    1    0    0
#> [4,]    1    1    1    0
#> 
#> , , 3
#> 
#>      [,1] [,2] [,3] [,4]
#> [1,]    0    0    0    0
#> [2,]    1    0    0    0
#> [3,]    1    1    0    0
#> [4,]    1    1    1    0
#> 
#> , , 4
#> 
#>      [,1] [,2] [,3] [,4]
#> [1,]    0    0    0    0
#> [2,]    1    0    0    0
#> [3,]    1    1    0    0
#> [4,]    1    1    1    0
#> 
#> , , 5
#> 
#>      [,1] [,2] [,3] [,4]
#> [1,]    0    0    0    0
#> [2,]    1    0    0    0
#> [3,]    1    1    0    0
#> [4,]    1    1    1    0
#> 
#> , , 6
#> 
#>      [,1] [,2] [,3] [,4]
#> [1,]    0    0    0    0
#> [2,]    0    0    0    0
#> [3,]    0    0    0    0
#> [4,]    0    0    0    0
#> 
#> , , 7
#> 
#>      [,1] [,2] [,3] [,4]
#> [1,]    0    0    0    0
#> [2,]    1    0    0    0
#> [3,]    1    1    0    0
#> [4,]    1    1    1    0
#> 
#> , , 8
#> 
#>      [,1] [,2] [,3] [,4]
#> [1,]    0    0    0    0
#> [2,]    1    0    0    0
#> [3,]    1    1    0    0
#> [4,]    1    1    1    0
#> 
#> , , 9
#> 
#>      [,1] [,2] [,3] [,4]
#> [1,]    0    0    0    0
#> [2,]    1    0    0    0
#> [3,]    1    1    0    0
#> [4,]    1    1    1    0
#> 
#> , , 10
#> 
#>      [,1] [,2] [,3] [,4]
#> [1,]    0    0    0    0
#> [2,]    1    0    0    0
#> [3,]    1    1    0    0
#> [4,]    1    1    1    0
#> 
#> , , 11
#> 
#>      [,1] [,2] [,3] [,4]
#> [1,]    0    0    0    0
#> [2,]    0    0    0    0
#> [3,]    0    0    0    0
#> [4,]    0    0    0    0
#> 
#> , , 12
#> 
#>      [,1] [,2] [,3] [,4]
#> [1,]    0    0    0    0
#> [2,]    1    0    0    0
#> [3,]    1    1    0    0
#> [4,]    1    1    1    0
#> 
#> , , 13
#> 
#>      [,1] [,2] [,3] [,4]
#> [1,]    0    0    0    0
#> [2,]    1    0    0    0
#> [3,]    1    1    0    0
#> [4,]    1    1    1    0
#> 
#> , , 14
#> 
#>      [,1] [,2] [,3] [,4]
#> [1,]    0    0    0    0
#> [2,]    1    0    0    0
#> [3,]    1    1    0    0
#> [4,]    1    1    1    0
#> 
#> , , 15
#> 
#>      [,1] [,2] [,3] [,4]
#> [1,]    0    0    0    0
#> [2,]    1    0    0    0
#> [3,]    1    1    0    0
#> [4,]    1    1    1    0
#> 
#> , , 16
#> 
#>      [,1] [,2] [,3] [,4]
#> [1,]    0    0    0    0
#> [2,]    0    0    0    0
#> [3,]    0    0    0    0
#> [4,]    0    0    0    0
IndicFshape(Ti,ngrp,TestIndic)
#> , , 1
#> 
#>      [,1] [,2] [,3] [,4]
#> [1,] "0"  "0"  "0"  "0" 
#> [2,] "0"  "0"  "0"  "0" 
#> [3,] "0"  "0"  "0"  "0" 
#> [4,] "0"  "0"  "0"  "0" 
#> 
#> , , 2
#> 
#>      [,1] [,2] [,3] [,4]
#> [1,] "0"  "0"  "0"  "0" 
#> [2,] "a1" "0"  "0"  "0" 
#> [3,] "a2" "a1" "0"  "0" 
#> [4,] "a3" "a2" "a1" "0" 
#> 
#> , , 3
#> 
#>      [,1] [,2] [,3] [,4]
#> [1,] "0"  "0"  "0"  "0" 
#> [2,] "a1" "0"  "0"  "0" 
#> [3,] "a2" "a1" "0"  "0" 
#> [4,] "a3" "a2" "a1" "0" 
#> 
#> , , 4
#> 
#>      [,1] [,2] [,3] [,4]
#> [1,] "0"  "0"  "0"  "0" 
#> [2,] "a1" "0"  "0"  "0" 
#> [3,] "a2" "a1" "0"  "0" 
#> [4,] "a3" "a2" "a1" "0" 
#> 
#> , , 5
#> 
#>      [,1] [,2] [,3] [,4]
#> [1,] "0"  "0"  "0"  "0" 
#> [2,] "a1" "0"  "0"  "0" 
#> [3,] "a2" "a1" "0"  "0" 
#> [4,] "a3" "a2" "a1" "0" 
#> 
#> , , 6
#> 
#>      [,1] [,2] [,3] [,4]
#> [1,] "0"  "0"  "0"  "0" 
#> [2,] "0"  "0"  "0"  "0" 
#> [3,] "0"  "0"  "0"  "0" 
#> [4,] "0"  "0"  "0"  "0" 
#> 
#> , , 7
#> 
#>      [,1] [,2] [,3] [,4]
#> [1,] "0"  "0"  "0"  "0" 
#> [2,] "a1" "0"  "0"  "0" 
#> [3,] "a2" "a1" "0"  "0" 
#> [4,] "a3" "a2" "a1" "0" 
#> 
#> , , 8
#> 
#>      [,1] [,2] [,3] [,4]
#> [1,] "0"  "0"  "0"  "0" 
#> [2,] "a1" "0"  "0"  "0" 
#> [3,] "a2" "a1" "0"  "0" 
#> [4,] "a3" "a2" "a1" "0" 
#> 
#> , , 9
#> 
#>      [,1] [,2] [,3] [,4]
#> [1,] "0"  "0"  "0"  "0" 
#> [2,] "a1" "0"  "0"  "0" 
#> [3,] "a2" "a1" "0"  "0" 
#> [4,] "a3" "a2" "a1" "0" 
#> 
#> , , 10
#> 
#>      [,1] [,2] [,3] [,4]
#> [1,] "0"  "0"  "0"  "0" 
#> [2,] "a1" "0"  "0"  "0" 
#> [3,] "a2" "a1" "0"  "0" 
#> [4,] "a3" "a2" "a1" "0" 
#> 
#> , , 11
#> 
#>      [,1] [,2] [,3] [,4]
#> [1,] "0"  "0"  "0"  "0" 
#> [2,] "0"  "0"  "0"  "0" 
#> [3,] "0"  "0"  "0"  "0" 
#> [4,] "0"  "0"  "0"  "0" 
#> 
#> , , 12
#> 
#>      [,1] [,2] [,3] [,4]
#> [1,] "0"  "0"  "0"  "0" 
#> [2,] "a1" "0"  "0"  "0" 
#> [3,] "a2" "a1" "0"  "0" 
#> [4,] "a3" "a2" "a1" "0" 
#> 
#> , , 13
#> 
#>      [,1] [,2] [,3] [,4]
#> [1,] "0"  "0"  "0"  "0" 
#> [2,] "a1" "0"  "0"  "0" 
#> [3,] "a2" "a1" "0"  "0" 
#> [4,] "a3" "a2" "a1" "0" 
#> 
#> , , 14
#> 
#>      [,1] [,2] [,3] [,4]
#> [1,] "0"  "0"  "0"  "0" 
#> [2,] "a1" "0"  "0"  "0" 
#> [3,] "a2" "a1" "0"  "0" 
#> [4,] "a3" "a2" "a1" "0" 
#> 
#> , , 15
#> 
#>      [,1] [,2] [,3] [,4]
#> [1,] "0"  "0"  "0"  "0" 
#> [2,] "a1" "0"  "0"  "0" 
#> [3,] "a2" "a1" "0"  "0" 
#> [4,] "a3" "a2" "a1" "0" 
#> 
#> , , 16
#> 
#>      [,1] [,2] [,3] [,4]
#> [1,] "0"  "0"  "0"  "0" 
#> [2,] "0"  "0"  "0"  "0" 
#> [3,] "0"  "0"  "0"  "0" 
#> [4,] "0"  "0"  "0"  "0"

Those \(F\) matrices are lower diagonal ones to enforce that an observed value at a given time can only be predicted by a value that was observed in the past only (i.e. neither at the same moment or in the future).

The plotF is convenient to display F matrices. Here are the the displays of the three \(F\) matrices we have just introduced.

plotF(Fshape,choice="Fshape")

plotF(CascadeFshape(4,4),choice="Fshape")
plotF(IndicFshape(Ti,ngrp,TestIndic),choice="Fshape")

We now fit the model with an \(F\) matrix that is designed for cascade networks.

Specific Fshape

Net_inf_P_S <- Patterns::inference(M, Finit=CascadeFinit(4,4), Fshape=CascadeFshape(4,4))
#> We are at step :  1
#> Computing Group (out of 4) : 
#>  1
#>  2.........................
#>  3.........................
#>  4.........................
#> The convergence of the network is (L1 norm) : 0.0074
#> We are at step :  2
#> Computing Group (out of 4) : 
#>  1
#>  2.........................
#>  3.........................
#>  4.........................
#> The convergence of the network is (L1 norm) : 0.00314
#> We are at step :  3
#> Computing Group (out of 4) : 
#>  1
#>  2.........................
#>  3.........................
#>  4.........................
#> The convergence of the network is (L1 norm) : 0.0019
#> We are at step :  4
#> Computing Group (out of 4) : 
#>  1
#>  2.........................
#>  3.........................
#>  4.........................
#> The convergence of the network is (L1 norm) : 0.00131
#> We are at step :  5
#> Computing Group (out of 4) : 
#>  1
#>  2.........................
#>  3.........................
#>  4.........................
#> The convergence of the network is (L1 norm) : 0.00101
#> We are at step :  6
#> Computing Group (out of 4) : 
#>  1
#>  2.........................
#>  3.........................
#>  4.........................
#> The convergence of the network is (L1 norm) : 0.00081

Plot of the inferred F matrix

plot(Net_inf_P_S, choice="F")

Heatmap of the coefficients of the Omega matrix of the network. They reflect the use of a special \(F\) matrix. It is an example of an F matrix specifically designed to deal with cascade networks.

stats::heatmap(Net_inf_P_S@network, Rowv = NA, Colv = NA, scale="none", revC=TRUE)

There are many fitting functions provided with the Patterns package in order to search for specific features for the inferred network such as sparsity, robust links, high confidence links or stable through resampling links. :

  • LASSO, from the lars package
  • LASSO2, from the glmnet package. An unweighted and a weighted version of the algorithm are available
  • SPLS, from the spls package
  • ELASTICNET, from the elasticnet package
  • stability.c060, from the c060 package implementation of stability selection
  • stability.c060.weighted, a new weighted version of the c060 package implementation of stability selection
  • robust, lasso from the lars package with light random Gaussian noise added to the explanatory variables
  • selectboost.weighted, a new weighted version of the selectboost package implementation of the selectboost algorithm to look for the more stable links against resampling that takes into account the correlated structure of the predictors. If no weights are provided, equal weigths are for all the variables (=non weighted case).
Net_inf_P_Lasso2 <- Patterns::inference(M, Finit=CascadeFinit(4,4), Fshape=CascadeFshape(4,4), fitfun="LASSO2")
#> We are at step :  1
#> Computing Group (out of 4) : 
#>  1
#>  2.........................
#>  3.........................
#>  4.........................
#> The convergence of the network is (L1 norm) : 0.0069
#> We are at step :  2
#> Computing Group (out of 4) : 
#>  1
#>  2.........................
#>  3.........................
#>  4.........................
#> The convergence of the network is (L1 norm) : 0.00229
#> We are at step :  3
#> Computing Group (out of 4) : 
#>  1
#>  2.........................
#>  3.........................
#>  4.........................
#> The convergence of the network is (L1 norm) : 0.00153
#> We are at step :  4
#> Computing Group (out of 4) : 
#>  1
#>  2.........................
#>  3.........................
#>  4.........................
#> The convergence of the network is (L1 norm) : 0.00114
#> We are at step :  5
#> Computing Group (out of 4) : 
#>  1
#>  2.........................
#>  3.........................
#>  4.........................
#> The convergence of the network is (L1 norm) : 0.00086

Plot of the inferred F matrix

plot(Net_inf_P_Lasso2, choice="F")

Heatmap of the coefficients of the Omega matrix of the network

stats::heatmap(Net_inf_P_Lasso2@network, Rowv = NA, Colv = NA, scale="none", revC=TRUE)

We create a weighting vector to perform weighted lasso inference.

Weights_Net=slot(Net,"network")
Weights_Net[Net@network!=0]=.1        
Weights_Net[Net@network==0]=1000
Net_inf_P_Lasso2_Weighted <- Patterns::inference(M, Finit=CascadeFinit(4,4), Fshape=CascadeFshape(4,4), fitfun="LASSO2", priors=Weights_Net)
#> We are at step :  1
#> Computing Group (out of 4) : 
#>  1
#>  2.........................
#>  3.........................
#>  4.........................
#> The convergence of the network is (L1 norm) : 0.0075
#> We are at step :  2
#> Computing Group (out of 4) : 
#>  1
#>  2.........................
#>  3.........................
#>  4.........................
#> The convergence of the network is (L1 norm) : 4e-04

Plot of the inferred F matrix

plot(Net_inf_P_Lasso2_Weighted, choice="F")

Heatmap of the coefficients of the Omega matrix of the network

stats::heatmap(Net_inf_P_Lasso2_Weighted@network, Rowv = NA, Colv = NA, scale="none", revC=TRUE)
Net_inf_P_SPLS <- Patterns::inference(M, Finit=CascadeFinit(4,4), Fshape=CascadeFshape(4,4), fitfun="SPLS")
#> We are at step :  1
#> Computing Group (out of 4) : 
#>  1
#>  2.........................
#>  3.........................
#>  4.........................
#> The convergence of the network is (L1 norm) : 0.0075
#> We are at step :  2
#> Computing Group (out of 4) : 
#>  1
#>  2.........................
#>  3.........................
#>  4.........................
#> The convergence of the network is (L1 norm) : 0.00229
#> We are at step :  3
#> Computing Group (out of 4) : 
#>  1
#>  2.........................
#>  3.........................
#>  4.........................
#> The convergence of the network is (L1 norm) : 0.00164
#> We are at step :  4
#> Computing Group (out of 4) : 
#>  1
#>  2.........................
#>  3.........................
#>  4.........................
#> The convergence of the network is (L1 norm) : 0.00123
#> We are at step :  5
#> Computing Group (out of 4) : 
#>  1
#>  2.........................
#>  3.........................
#>  4.........................
#> The convergence of the network is (L1 norm) : 0.00104
#> We are at step :  6
#> Computing Group (out of 4) : 
#>  1
#>  2.........................
#>  3.........................
#>  4.........................
#> The convergence of the network is (L1 norm) : 0.00088

Plot of the inferred F matrix

plot(Net_inf_P_SPLS, choice="F")

Heatmap of the coefficients of the Omega matrix of the network

stats::heatmap(Net_inf_P_SPLS@network, Rowv = NA, Colv = NA, scale="none", revC=TRUE)
Net_inf_P_ELASTICNET <- Patterns::inference(M, Finit=CascadeFinit(4,4), Fshape=CascadeFshape(4,4), fitfun="ELASTICNET")
#> We are at step :  1
#> Computing Group (out of 4) : 
#>  1
#>  2.........................
#>  3.........................
#>  4.........................
#> The convergence of the network is (L1 norm) : 0.0074
#> We are at step :  2
#> Computing Group (out of 4) : 
#>  1
#>  2.........................
#>  3.........................
#>  4.........................
#> The convergence of the network is (L1 norm) : 0.00402
#> We are at step :  3
#> Computing Group (out of 4) : 
#>  1
#>  2.........................
#>  3.........................
#>  4.........................
#> The convergence of the network is (L1 norm) : 0.00289
#> We are at step :  4
#> Computing Group (out of 4) : 
#>  1
#>  2.........................
#>  3.........................
#>  4.........................
#> The convergence of the network is (L1 norm) : 0.00223
#> We are at step :  5
#> Computing Group (out of 4) : 
#>  1
#>  2.........................
#>  3.........................
#>  4.........................
#> The convergence of the network is (L1 norm) : 0.00178
#> We are at step :  6
#> Computing Group (out of 4) : 
#>  1
#>  2.........................
#>  3.........................
#>  4.........................
#> The convergence of the network is (L1 norm) : 0.00142
#> We are at step :  7
#> Computing Group (out of 4) : 
#>  1
#>  2.........................
#>  3.........................
#>  4.........................
#> The convergence of the network is (L1 norm) : 0.00121
#> We are at step :  8
#> Computing Group (out of 4) : 
#>  1
#>  2.........................
#>  3.........................
#>  4.........................
#> The convergence of the network is (L1 norm) : 0.00107
#> We are at step :  9
#> Computing Group (out of 4) : 
#>  1
#>  2.........................
#>  3.........................
#>  4.........................
#> The convergence of the network is (L1 norm) : 0.00095

Plot of the inferred F matrix

plot(Net_inf_P_ELASTICNET, choice="F")

Heatmap of the coefficients of the Omega matrix of the network

stats::heatmap(Net_inf_P_ELASTICNET@network, Rowv = NA, Colv = NA, scale="none", revC=TRUE)
Net_inf_P_stability <- Patterns::inference(M, Finit=CascadeFinit(4,4), Fshape=CascadeFshape(4,4), fitfun="stability.c060")
#> We are at step :  1
#> Computing Group (out of 4) : 
#>  1mc.cores=2
#>  2mc.cores=2.........................
#>  3mc.cores=2.........................
#>  4mc.cores=2.........................
#> The convergence of the network is (L1 norm) : 0.0042
#> We are at step :  2
#> Computing Group (out of 4) : 
#>  1mc.cores=2
#>  2mc.cores=2.........................
#>  3mc.cores=2.........................
#>  4mc.cores=2.........................
#> The convergence of the network is (L1 norm) : 0.00095

Plot of the inferred F matrix

plot(Net_inf_P_stability, choice="F")

Heatmap of the coefficients of the Omega matrix of the network

stats::heatmap(Net_inf_P_stability@network, Rowv = NA, Colv = NA, scale="none", revC=TRUE)
Net_inf_P_StabWeight <- Patterns::inference(M, Finit=CascadeFinit(4,4), Fshape=CascadeFshape(4,4), fitfun="stability.c060.weighted", priors=Weights_Net)
#> We are at step :  1
#> Computing Group (out of 4) : 
#>  1mc.cores=2 
#>  2mc.cores=2 .........................
#>  3mc.cores=2 .........................
#>  4mc.cores=2 .........................
#> The convergence of the network is (L1 norm) : 0.0075
#> We are at step :  2
#> Computing Group (out of 4) : 
#>  1mc.cores=2 
#>  2mc.cores=2 .........................
#>  3mc.cores=2 .........................
#>  4mc.cores=2 .........................
#> The convergence of the network is (L1 norm) : 0.00034

Plot of the inferred F matrix

plot(Net_inf_P_StabWeight, choice="F")

Heatmap of the coefficients of the Omega matrix of the network

stats::heatmap(Net_inf_P_StabWeight@network, Rowv = NA, Colv = NA, scale="none", revC=TRUE)
Net_inf_P_Robust <- Patterns::inference(M, Finit=CascadeFinit(4,4), Fshape=CascadeFshape(4,4), fitfun="robust")
#> We are at step :  1
#> Computing Group (out of 4) : 
#>  1
#>  2.........................
#>  3.........................
#>  4.........................
#> The convergence of the network is (L1 norm) : 0.0069
#> We are at step :  2
#> Computing Group (out of 4) : 
#>  1
#>  2.........................
#>  3.........................
#>  4.........................
#> The convergence of the network is (L1 norm) : 0.00366
#> We are at step :  3
#> Computing Group (out of 4) : 
#>  1
#>  2.........................
#>  3.........................
#>  4.........................
#> The convergence of the network is (L1 norm) : 0.00203
#> We are at step :  4
#> Computing Group (out of 4) : 
#>  1
#>  2.........................
#>  3.........................
#>  4.........................
#> The convergence of the network is (L1 norm) : 0.00142
#> We are at step :  5
#> Computing Group (out of 4) : 
#>  1
#>  2.........................
#>  3.........................
#>  4.........................
#> The convergence of the network is (L1 norm) : 0.00117
#> We are at step :  6
#> Computing Group (out of 4) : 
#>  1
#>  2.........................
#>  3.........................
#>  4.........................
#> The convergence of the network is (L1 norm) : 0.00086

Plot of the inferred F matrix

plot(Net_inf_P_Robust, choice="F")

Heatmap of the coefficients of the Omega matrix of the network

stats::heatmap(Net_inf_P_Robust@network, Rowv = NA, Colv = NA, scale="none", revC=TRUE)
Weights_Net_1 <- Weights_Net
Weights_Net_1[,] <- 1
Net_inf_P_SelectBoost <- Patterns::inference(M, Finit=CascadeFinit(4,4), Fshape=CascadeFshape(4,4), fitfun="selectboost.weighted",priors=Weights_Net_1)
#> We are at step :  1
#> Computing Group (out of 4) : 
#>  1
#> Loading required namespace: SelectBoost
#> 
#>  2.........................
#>  3.........................
#>  4.........................
#> The convergence of the network is (L1 norm) : 0.0057
#> We are at step :  2
#> Computing Group (out of 4) : 
#>  1
#>  2.........................
#>  3.........................
#>  4.........................
#> The convergence of the network is (L1 norm) : 0.00174
#> We are at step :  3
#> Computing Group (out of 4) : 
#>  1
#>  2.........................
#>  3.........................
#>  4.........................
#> The convergence of the network is (L1 norm) : 0.00125
#> We are at step :  4
#> Computing Group (out of 4) : 
#>  1
#>  2.........................
#>  3.........................
#>  4.........................
#> The convergence of the network is (L1 norm) : 0.00094
detach("package:Patterns", unload=TRUE)
library(Patterns)
#> 
#> Attaching package: 'Patterns'
#> The following object is masked from 'package:igraph':
#> 
#>     compare

Plot of the inferred F matrix

plot(Net_inf_P_SelectBoost, choice="F")

Heatmap of the coefficients of the Omega matrix of the network

stats::heatmap(Net_inf_P_SelectBoost@network, Rowv = NA, Colv = NA, scale="none", revC=TRUE)
Net_inf_P_SelectBoostWeighted <- Patterns::inference(M, Finit=CascadeFinit(4,4), Fshape=CascadeFshape(4,4), fitfun="selectboost.weighted",priors=Weights_Net)
#> We are at step :  1
#> Computing Group (out of 4) : 
#>  1
#>  2.........................
#>  3.........................
#>  4.........................
#> The convergence of the network is (L1 norm) : 0.007
#> We are at step :  2
#> Computing Group (out of 4) : 
#>  1
#>  2.........................
#>  3.........................
#>  4.........................
#> The convergence of the network is (L1 norm) : 0.00047
detach("package:Patterns", unload=TRUE)
library(Patterns)
#> 
#> Attaching package: 'Patterns'
#> The following object is masked from 'package:igraph':
#> 
#>     compare

Plot of the inferred F matrix

plot(Net_inf_P_SelectBoostWeighted, choice="F")

Heatmap of the coefficients of the Omega matrix of the network

stats::heatmap(Net_inf_P_SelectBoostWeighted@network, Rowv = NA, Colv = NA, scale="none", revC=TRUE)

###Post inference network analysis Such an analysis is only required if the model was not fitted using the stability selection or the selectboost algorithm.

Create an animation of the network with increasing cutoffs with an animated .gif format or a html webpage. See the the webpage of the Patterns package for the animation results at the .gif and .html formats.

data(network)
sequence<-seq(0,0.2,length.out=20)
evolution(network,sequence,type.ani = "gif")
evolution(network,sequence,type.ani = "html")

Evolution of some properties of a reverse-engineered network with increasing cut-off values.

data(Net)
data(Net_inf_PL)

#Comparing true and inferred networks
Crit_values=NULL

#Here are the cutoff level tested
test.seq<-seq(0,max(abs(Net_inf_PL@network*0.9)),length.out=200)
for(u in test.seq){
    Crit_values<-rbind(Crit_values,Patterns::compare(Net,Net_inf_PL,u))
}
matplot(test.seq,Crit_values,type="l",ylab="Criterion value",xlab="Cutoff level",lwd=2)
legend(x="topleft", legend=colnames(Crit_values), lty=1:5,col=1:5,ncol=2,cex=.9)

We switch to data that were derived from the inferrence of a real biological network and try to detect the optimal cutoff value: the best cutoff value for a network to fit a scale free network. The cutoff was validated only single group cascade networks (number of actors groups = number of timepoints) and for genes dataset. Instead of the cutoff function, manual curation or the stability selection or the selectboost algorithm should be used.

data("networkCascade")
set.seed(1)
cutoff(networkCascade)
#> [1] "This computation may be long"
#> [1] "1/10"
#> [1] "2/10"
#> [1] "3/10"
#> [1] "4/10"
#> [1] "5/10"
#> [1] "6/10"
#> [1] "7/10"
#> [1] "8/10"
#> [1] "9/10"
#> [1] "10/10"
#>  [1] 0.000 0.007 0.423 0.388 0.236 0.631 0.952 0.976 0.825 0.362
#> $p.value
#>  [1] 0.000 0.007 0.423 0.388 0.236 0.631 0.952 0.976 0.825 0.362
#> 
#> $p.value.inter
#>  [1] -0.04643835  0.14848083  0.28125541  0.36247646  0.33538710  0.60021707
#>  [7]  0.92531630  0.98522398  0.79881816  0.37310221
#> 
#> $sequence
#>  [1] 0.00000000 0.04444444 0.08888889 0.13333333 0.17777778 0.22222222
#>  [7] 0.26666667 0.31111111 0.35555556 0.40000000

Analyze the network with a cutoff set to the previouly found 0.133 optimal value.

analyze_network(networkCascade,nv=0.133)
#>     node betweenness degree    output  closeness
#> 1      1           0      3 0.6442821  2.5337730
#> 2      2           8      4 1.3784717  5.4211261
#> 3      3           0      6 1.3308218  6.1428682
#> 4      4           0      9 1.7531528 19.4060065
#> 5      5           0      2 0.6400128  2.5169831
#> 6      6           0      1 0.2725100  3.4520368
#> 7      7           0      0 0.0000000  0.0000000
#> 8      8           0      1 0.1483077 10.4595738
#> 9      9           4      4 0.7969945  3.4810740
#> 10    10          20      7 1.3485173  6.0949803
#> 11    11           0      4 0.8054122 16.2236448
#> 12    12           0     11 2.4624877 10.7417814
#> 13    13           0      0 0.0000000  0.0000000
#> 14    14           0      1 0.1882265  3.8504312
#> 15    15           0      5 1.5808956 13.7135886
#> 16    16          81     24 6.7716649 29.1131750
#> 17    17           0      9 2.0059124 11.2249512
#> 18    18           0      9 1.6083138  6.6506106
#> 19    19           0      0 0.0000000  0.0000000
#> 20    20           0      0 0.0000000  0.0000000
#> 21    21           0      0 0.0000000  0.0000000
#> 22    22           0      0 0.0000000  0.0000000
#> 23    23           0      0 0.0000000  0.0000000
#> 24    24           0      1 0.1607163  0.6320503
#> 25    25           0      0 0.0000000  0.0000000
#> 26    26           0      0 0.0000000  0.0000000
#> 27    27           0      8 3.2710799 13.9045595
#> 28    28           0      2 0.3957891  2.1118782
#> 29    29           0      0 0.0000000  0.0000000
#> 30    30           0      1 0.2255093  0.8868623
#> 31    31           0      0 0.0000000  0.0000000
#> 32    32           0      0 0.0000000  0.0000000
#> 33    33           0      0 0.0000000  0.0000000
#> 34    34           0      0 0.0000000  0.0000000
#> 35    35           0      6 1.4397935  5.6622868
#> 36    36           0      0 0.0000000  0.0000000
#> 37    37           0      0 0.0000000  0.0000000
#> 38    38           0      0 0.0000000  0.0000000
#> 39    39           0      0 0.0000000  0.0000000
#> 40    40           0      0 0.0000000  0.0000000
#> 41    41           0      0 0.0000000  0.0000000
#> 42    42           2      1 0.5205955  2.0473501
#> 43    43           0      0 0.0000000  0.0000000
#> 44    44           0      0 0.0000000  0.0000000
#> 45    45           0      0 0.0000000  0.0000000
#> 46    46           0      0 0.0000000  0.0000000
#> 47    47           0      0 0.0000000  0.0000000
#> 48    48           0      1 0.5096457  2.0042877
#> 49    49           1      1 0.1814386  0.7135450
#> 50    50           0      0 0.0000000  0.0000000
#> 51    51           0      0 0.0000000  0.0000000
#> 52    52           0      0 0.0000000  0.0000000
#> 53    53           0      0 0.0000000  0.0000000
#> 54    54           0      0 0.0000000  0.0000000
#> 55    55           0      0 0.0000000  0.0000000
#> 56    56           0      0 0.0000000  0.0000000
#> 57    57           5      1 0.7142387  2.8088919
#> 58    58           0      0 0.0000000  0.0000000
#> 59    59           0      0 0.0000000  0.0000000
#> 60    60           0      0 0.0000000  0.0000000
#> 61    61           0      0 0.0000000  0.0000000
#> 62    62           0      0 0.0000000  0.0000000
#> 63    63           0      1 0.1668072  0.6560038
#> 64    64           0      0 0.0000000  0.0000000
#> 65    65           0      0 0.0000000  0.0000000
#> 66    66           0      0 0.0000000  0.0000000
#> 67    67           0      0 0.0000000  0.0000000
#> 68    68           0      0 0.0000000  0.0000000
#> 69    69           0      0 0.0000000  0.0000000
#> 70    70           0      0 0.0000000  0.0000000
#> 71    71           0      0 0.0000000  0.0000000
#> 72    72           0      0 0.0000000  0.0000000
#> 73    73           4      1 0.1937648  0.7620205
#> 74    74           0      0 0.0000000  0.0000000
#> 75    75           0      0 0.0000000  0.0000000
#> 76    76           0      0 0.0000000  0.0000000
#> 77    77           0      0 0.0000000  0.0000000
#> 78    78           0      0 0.0000000  0.0000000
#> 79    79           0      0 0.0000000  0.0000000
#> 80    80           0      0 0.0000000  0.0000000
#> 81    81           0      0 0.0000000  0.0000000
#> 82    82           4      2 0.4875471  1.9173803
#> 83    83           4      1 0.2849991  1.1208180
#> 84    84           5      1 0.4477398  1.7608296
#> 85    85           4      1 0.2126269  0.8361995
#> 86    86           0      0 0.0000000  0.0000000
#> 87    87           0      0 0.0000000  0.0000000
#> 88    88           0      0 0.0000000  0.0000000
#> 89    89           0      0 0.0000000  0.0000000
#> 90    90           5      1 0.1522237  0.5986513
#> 91    91           3      1 0.2857613  1.1238157
#> 92    92           0      0 0.0000000  0.0000000
#> 93    93           0      1 0.1784006  0.7015976
#> 94    94           0      0 0.0000000  0.0000000
#> 95    95           0      0 0.0000000  0.0000000
#> 96    96           0      0 0.0000000  0.0000000
#> 97    97           0      0 0.0000000  0.0000000
#> 98    98           0      0 0.0000000  0.0000000
#> 99    99           0      0 0.0000000  0.0000000
#> 100  100           0      0 0.0000000  0.0000000
#> 101  101           0      0 0.0000000  0.0000000
#> 102  102           0      0 0.0000000  0.0000000
data(Selection)
plot(networkCascade,nv=0.133, gr=Selection@group)