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```
#' calculate transitivity measurements for a matrix
#'
#' \code{transitivity} calculate transitivity measurements for a matrix
#'
#' @param conf an N-by-N conflict matrix whose \code{(i,j)}th element is the number of times \code{i} defeated \code{j}
#'
#' @param strict a logical vector of length 1 (TRUE or FALSE). It is used in transitivity definition for alpha estimation.
#' It should be set to TRUE when a transitive triangle is defined as all pathways in the triangle go to the same direction;
#' it should be set to FALSE when a transitive triangle is defined as PRIMARY pathways in the triangle go to the same direction.
#' Strict = FALSE by default.
#'
#' @return A list of four elements.
#'
#' \item{transitive}{The number of transitive triangles.}
#'
#' \item{intransitive}{The number of intransitive triangles.}
#'
#' \item{transitivity}{transitivity, the proportion of transitive triangles.}
#'
#' \item{alpha}{The value of alpha corresponding to this value of transitivity.}
#'
#'
#' @details \code{transitivity} is calculated as the proportion transitive triangles in the total of transitive and intransitive triangles.
#' transitivity is used to estimate alpha, which is used in turn in imputing information from indirect pathways as to what degree we can trust information from indirect pathways.
#' Greater transitivity is associated with assigning higher weight to information from indirect pathways.
#'
#' @seealso \code{\link{countPaths}}, \code{\link{findIDpaths}}, \code{\link{conductance}}
#' @examples
#' # convert an edgelist to conflict matrix
#' confmatrix <- as.conflictmat(sampleEdgelist)
#' # transitivity calculation
#' conftrans <- transitivity(confmatrix, strict = FALSE)
#' conftrans$transitive
#' conftrans$intransitive
#' conftrans$transitivity
#' conftrans$alpha
#' @export
transitivity = function(conf, strict = FALSE){
# making sure conf is of conf.mat
if (!("conf.mat" %in% class(conf))){
conf = as.conflictmat(conf)
}
N = nrow(conf)
### These lines set up the transitivity calculation.
### We are making a matrix of all the possible sets of three subjects.
### We won't really need to refer to this matrix.
### The important part of the calculation is at the end.
numrows = 0
ctr = 0
for(i in 1:(N-2)){
ctr = ctr + i
numrows = numrows + ctr
}
firstrow = numeric(0)
for(i in 1:(N-2)){
temp = rep(i, (N-1-i)*(N-i)/2)
firstrow = c(firstrow, temp) # first ID
}
secondrow = numeric(0)
for(i in 1:(N-2)){ # first row number
for(j in (i+1):(N-1)){
temp = rep(j, N-j)
secondrow = c(secondrow, temp) # Second ID
}
}
thirdrow = numeric(0)
for(i in 3:N){
for(j in i:N){
thirdrow = c(thirdrow, seq(j, N, 1)) # third ID
}
}
triples = matrix(0, numrows, 3)
triples[,1] = firstrow
triples[,2] = secondrow
triples[,3] = thirdrow
## Here's where the actual transitivity calculation begins.
transitive = 0
intransitive = 0
strictTransitive = 0
tList = matrix(0, 0, 4)
iList = matrix(0, 0, 4)
stList = matrix(0, 0, 4)
for(i in 1:nrow(triples)){
tA = triples[i,1] # first ID
tB = triples[i,2] # second ID
tC = triples[i,3] # third ID
AB = conf[tA, tB] - conf[tB, tA]
AC = conf[tA, tC] - conf[tC, tA]
BC = conf[tB, tC] - conf[tC, tB]
### See if the triangle is transitive...
if((AC > 0 & BC > 0 & AB != 0) | # BAC, ABC
(AB < 0 & AC < 0 & BC != 0) | # CBA, BCA
(AB > 0 & BC < 0 & AC != 0)){ # CAB, ACB
transitive = transitive + 1
tList = rbind(tList, c(triples[i,], i))
}
### See if the triangle is intransitive...
if((AB > 0 & BC > 0 & AC < 0) |
(AB < 0 & AC > 0 & BC < 0)){
intransitive = intransitive + 1
iList = rbind(iList, c(triples[i,], i))
}
ABstrict = conf[tA, tB]
ACstrict = conf[tA, tC]
BCstrict = conf[tB, tC]
BAstrict = conf[tB, tA]
CAstrict = conf[tC, tA]
CBstrict = conf[tC, tB]
### See if the triangle is strict transitive
if(
((BAstrict > 0 | ABstrict > 0) & ACstrict > 0 & BCstrict > 0 &
CAstrict == 0 & CBstrict == 0) |
(BAstrict > 0 & CAstrict > 0 & (BCstrict > 0 | CBstrict > 0) &
ABstrict == 0 & ACstrict == 0)|
(ABstrict > 0 & CBstrict > 0 & (ACstrict > 0 | CAstrict > 0) &
BAstrict == 0 & BCstrict == 0)) {
strictTransitive = strictTransitive + 1
stList = rbind(stList, c(triples[i,], i))
}
}
if (strict) {
T1 = strictTransitive / (transitive + intransitive)
strictIntransitive = (transitive + intransitive) - strictTransitive
alpha = (2 * sqrt(T1) - 1) / (1 - sqrt(T1))
return(list(transitive = strictTransitive,
intransitive = strictIntransitive,
transitivity = T1,
alpha = alpha))
} else {
### T1 is the order-1 transitivity.
T1 = transitive / (transitive + intransitive)
### From the paper, we estimate alpha as follows.
alpha = (2 * sqrt(T1) - 1) / (1 - sqrt(T1))
return(list(transitive = transitive, intransitive = intransitive, transitivity = T1, alpha = alpha))
}
}
```

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