查找节点 u 和 v 之间通过节点 g 的最短路径总数

Question

I have the following matrix that generates an undirected network diagram:

  a b c d e f g h i j
a 0 1 1 0 0 0 0 0 0 0
b 1 0 1 0 0 0 0 0 0 0
c 1 1 0 1 1 0 1 0 0 0
d 0 0 1 0 1 0 0 0 0 0
e 0 0 1 1 0 1 0 0 0 0
f 0 0 0 0 1 0 1 0 0 0
g 0 0 1 0 0 1 0 1 0 0
h 0 0 0 0 0 0 1 0 1 1
i 0 0 0 0 0 0 0 1 0 0
j 0 0 0 0 0 0 0 1 0 0

m <- structure(c(0L, 1L, 1L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 1L, 0L, 1L, 
0L, 0L, 0L, 0L, 0L, 0L, 0L, 1L, 1L, 0L, 1L, 1L, 0L, 1L, 0L, 0L, 
0L, 0L, 0L, 1L, 0L, 1L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 1L, 1L, 0L, 
1L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 1L, 0L, 1L, 0L, 0L, 0L, 0L, 
0L, 1L, 0L, 0L, 1L, 0L, 1L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 1L, 
0L, 1L, 1L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 1L, 0L, 0L, 0L, 0L, 0L, 
0L, 0L, 0L, 0L, 1L, 0L, 0L), .Dim = c(10L, 10L), .Dimnames = list(
    c("a", "b", "c", "d", "e", "f", "g", "h", "i", "j"), c("a", 
    "b", "c", "d", "e", "f", "g", "h", "i", "j")))

library(igraph)
g3n <- graph.adjacency(m)

I'm interested in manually calculating the betweeness of node 'g', which requires the finding of shortest paths among all possible nodes as the denominator and the numerator as the number of shortest paths contain node 'g'.

I used the following code to generate the lengths of shortest paths among all nodes:

shortest.paths(g3n, v=V(g3n), to=V(g3n))

Shortest path matrix:

  a b c d e f g h i j
a 0 1 1 2 2 3 2 3 4 4
b 1 0 1 2 2 3 2 3 4 4
c 1 1 0 1 1 2 1 2 3 3
d 2 2 1 0 1 2 2 3 4 4
e 2 2 1 1 0 1 2 3 4 4
f 3 3 2 2 1 0 1 2 3 3
g 2 2 1 2 2 1 0 1 2 2
h 3 3 2 3 3 2 1 0 1 1
i 4 4 3 4 4 3 2 1 0 2
j 4 4 3 4 4 3 2 1 2 0

Is there a way to count the number of times shortest path between 2 nodes contains node 'g' as a matrix or just in any other ways in R?

Answer 1

So, I'm not sure how elegant this solution is, but the following should work:

#initialize a list to populate with all the shortest paths in the graphy
allpaths <- list()


#Assuming this is an undirected graph, we don't want to calculate both a %--% b and b %--% a  
for(x in V(g3n)$name){
  for(y in V(g3n)$name){
    if(x < y){
      shortest_path_options <- all_shortest_paths(g3n, x, y)$res

      #sometimes there are multiple shortest paths, we will include them all
      for(z in shortest_path_options){
        allpaths[[length(allpaths)+1]] <- z$name
      }
    }
}

#create a boolean of whether a shortest path contains 'g' or not
allpaths_bool <- sapply(allpaths, function(x){
  ('g' %in% x) & (head(x, 1) != 'g') & (tail(x, 1) != 'g')
  })

#Show all the paths that contain 'g'
allpaths[allpaths_bool]

You can calculate this for every vertex by wrapping everything into a sapply function.

sapply(V(g3n)$name, function(x){
  temp_bool <- sapply(allpaths, function(y){
    (x %in% y) & (head(y, 1) != x) & (tail(y, 1) != x)
  })
  length(allpaths[temp_bool])
})

There has to be an easier way, but I'm not really sure of it. There might be a way to infer this information by using the betweeness function which provides the betweeness centrality measurements, but I'm not that well read up on graph theory.