# No Isolated Nodes Allowed network

### Description

Given a distance matrix, this function computes a network connecting all nodes with the minimum number of links.

### Usage

1 2 3 4 5 6 | ```
NINA.thr(dis, range = seq(0, 1, 0.01), ptPDF = TRUE, ptPDFname = "NINA_Network.pdf",
estimPDF = TRUE, estimPDFname = "NINA.ThresholdEstimation.pdf", estimOutfile = TRUE,
cex.label = 1, cex.vertex = 1, estimOutName = "NINA.ThresholdEstimation.txt",
appendOutfile = TRUE, plotALL = FALSE, bgcol = "white", label.col = "black",
label = colnames(dis), modules = FALSE, moduleCol = NA,
modFileName = "Modules_summary.txt", ncs = 4, na.rm.row.col = FALSE)
``` |

### Arguments

`dis` |
the input distance matrix |

`range` |
a numeric vector between 0 and 1, is the range of thresholds (referred to the maximum distance in the input matrix) to be screened (by default, 101 values from 0 to 1). |

`ptPDF` |
a logical, must the resulting network be saved as a pdf file? |

`ptPDFname` |
if ptPDF=TRUE, the name of the pdf file containing the network to be saved ("NINA_Network.pdf", by default) |

`estimOutfile` |
a logical, must the value of <s> for each threshold (NINA threshold estimation) be saved as a text file? |

`estimOutName` |
if estimOutfile=TRUE (default), contains the name of the text file containing the NINA threshold estimation ("PercThr Estimation.txt" by default). |

`appendOutfile` |
a logical, if estimOutfile=TRUE, it defines whether results must be appended to an existing file with the same name (TRUE) or the existing file must be replaced (FALSE). |

`estimPDF` |
a logical, must the NINA threshold estimation plot be saved as a pdf file? |

`estimPDFname` |
if estimPDF=TRUE (default), defines the name of the pdf file containing the NINA threshold estimation plot (by default). |

`cex.label` |
a numeric; the size of the node labels. |

`cex.vertex` |
a numeric; the size of the nodes. |

`plotALL` |
a logical, must all the networks calculated during the NINA threshold estimation (defined by "range" option) be saved as different pdf files? (FALSE, by default). If TRUE, for each value defined in threshold, one pdf file is generated. |

`bgcol` |
the colour of the background for each node in the network. Can be equal for all nodes (if only one colour is defined), customized (if several colours are defined), or can represent different modules (see "modules" option). |

`label.col` |
vector of strings defining the colour of labels for each node in the network. Can be equal for all nodes (if only one colour is defined) or customized (if several colours are defined). |

`label` |
vector of strings, labels for each node. By default are the column names of the distance matrix (dis). (See substr function in base package to automatically set a string subset from column names). |

`modules` |
a logical, If TRUE, nodes belonging to different modules are represented as different colours (defined by 'moduleCol'). Modules (defined as subsets of nodes that conform densely connected subgraphs) are estimated by means of random walks (see 'igraph' package for details). |

`moduleCol` |
(if modules=TRUE) a vector of strings defining the colour of nodes belonging to different modules in the network. If 'NA' (or there are less colours than haplotypes), colours are automatically selected |

`modFileName` |
(if modules=TRUE) the name of a generated file containing a summary of module results |

`ncs` |
a numeric; number of decimal places to display threshold in plot title. |

`na.rm.row.col` |
a logical; if TRUE, missing values are removed before the computation proceeds. |

### Details

In some circumstances you may get distance matrices showing off-diagonal zeros. In such cases you may consider that the existence of these off-diagonal zeros suggests that some of the groups you defined (e.g., populations) are not genetically different. Thus, you must re-define groups to get a matrix composed only by different groups using the 'mergeNodes' function and estimate a percolation network using the 'perc.thr' function. On the other hand, you may consider that, despite the off- diagonal zeros, the groups you defined are actually different. In that case you may not be able to estimate a percolation threshold, but you can represent the original distance matrix using the 'NINA.thr' or the 'zero.thr' functions.

'mergeNodes' select all rows (and columns) showing a distance equal to zero and generates a new row (and column). The distance between the new merged and the remaining rows (or columns) in the matrix is estimated as the arithmetic mean of the selected elements. The biological interpretation of the new matrix could be hard if the original matrix shows a large number of off-diagonal zeros.

'perc.thr' estimates a threshold to represent a distance matrix as a network. To estimate this threshold, the algorithm represents as a link all distances lower than a range of thresholds (by default, select 101 values from 0 to 1), defined as the percentage of the maximum distance in the input matrix. For each threshold a network is built and the number of clusters (that is, the number of isolated groups of nodes) in the network is also estimated. Finally, the algorithm selects the lower threshold connecting a higher number of nodes. Note that the resulting network may show isolated nodes if it is necessary to represent a large number of links to connect a low number of nodes.

**'NINA.thr'** is identical to 'perc.thr', but, in the last step, the algorithm selects the lower threshold connecting all nodes in a single cluster. The information provided by this function may be limited if the original distance matrix shows high variation.

'zero.thr' represents as a link only distances equal to zero. The information provided by this function may be limited if the original matrix shows few off-diagonal zeros.

### Author(s)

A. J. Muñoz-Pajares

### See Also

`mergeNodes`

, `zero.thr`

, `perc.thr`

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 | ```
#EXAMPLE 1: FEW OFF-DIAGONAL ZEROS
#Generating a distance matrix:
Dis1<-matrix(c(0.00,0.77,0.28,0.94,0.17,0.14,0.08,0.49,0.64,0.01,0.77,0.00,0.12,0.78,0.97,
0.02,0.58,0.09,0.36,0.33,0.28,0.12,0.00,0.70,0.73,0.06,0.50,0.79,0.80,0.94,
0.94,0.78,0.70,0.00,0.00,0.78,0.04,0.42,0.25,0.85,0.17,0.97,0.73,0.00,0.00,
0.30,0.55,0.12,0.68,0.99,0.14,0.02,0.06,0.78,0.30,0.00,0.71,1.00,0.64,0.88,
0.08,0.58,0.50,0.04,0.55,0.71,0.00,0.35,0.84,0.76,0.49,0.09,0.79,0.42,0.12,
1.00,0.35,0.00,0.56,0.81,0.64,0.36,0.80,0.25,0.68,0.64,0.84,0.56,0.00,0.62,
0.01,0.33,0.94,0.85,0.99,0.88,0.76,0.81,0.62,0.00),ncol=10)
colnames(Dis1)<-c(paste("Pop",c(1:10),sep=""))
row.names(Dis1)<-colnames(Dis1)
# No percolation threshold can be found.
#perc.thr(Dis1)
#Check Dis1 and merge populations showing distances equal to zero:
Dis1
Dis1_Merged<-mergeNodes(dis=Dis1)
#Check the merged matrix. A new "population" has been defined merging populations 4 and 5.
#Distances between the merged and the remaining populations are estimated as the arithmetic mean.
Dis1_Merged
# It is now possible to estimate a percolation threshold
perc.thr(dis=Dis1_Merged,ptPDF=FALSE, estimPDF=FALSE, estimOutfile=FALSE)
# EXAMPLE 2: TOO MANY OFF-DIAGONAL ZEROS
#Generating a distance matrix:
Dis2<-matrix(c(0.00,0.77,0.28,0.00,0.17,0.14,0.00,0.49,0.64,0.01,0.77,0.00,0.12,0.00,0.97
,0.02,0.00,0.09,0.36,0.33,0.28,0.12,0.00,0.70,0.73,0.06,0.50,0.79,0.00,0.94
,0.00,0.00,0.70,0.00,0.00,0.78,0.04,0.00,0.00,0.00,0.17,0.97,0.73,0.00,0.00
,0.30,0.55,0.12,0.00,0.00,0.14,0.02,0.06,0.78,0.30,0.00,0.71,1.00,0.64,0.00
,0.00,0.00,0.50,0.04,0.55,0.71,0.00,0.35,0.84,0.00,0.49,0.09,0.79,0.00,0.12
,1.00,0.35,0.00,0.56,0.81,0.64,0.36,0.00,0.00,0.00,0.64,0.84,0.56,0.00,0.62
,0.01,0.33,0.94,0.00,0.00,0.00,0.00,0.81,0.62,0.00),ncol=10)
colnames(Dis2)<-c(paste("Pop",c(1:10),sep=""))
row.names(Dis2)<-colnames(Dis2)
# No percolation threshold can be found
#perc.thr(Dis2)
#Check Dis2 and merge populations showing distances equal to zero:
Dis2
Dis2_Merged<-mergeNodes(dis=Dis2)
#Check the merged matrix. Many new "populations" have been defined and both the new matrix
#and the resulting network are difficult to interpret:
Dis2_Merged
perc.thr(dis=Dis2_Merged,ptPDF=FALSE, estimPDF=FALSE, estimOutfile=FALSE)
#Instead of percolation network, representing zeros as the lowest values may be informative:
zero.thr(dis=Dis2,ptPDF=FALSE)
# Adjusting sizes and showing modules:
zero.thr(dis=Dis2,ptPDF=FALSE,cex.label=0.8,cex.vertex=1.2,modules=TRUE)
#In the previous example, the 'zero.thr' method is unuseful:
zero.thr(dis=Dis1,ptPDF=FALSE)
#In both cases, the 'No Isolation Nodes Allowed' method yields an informative matrix:
NINA.thr(dis=Dis1)
NINA.thr(dis=Dis2)
``` |

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