3d 矩阵到 2d 邻接矩阵或边缘列表

Question

考虑一个 3 x 3 x 3 立方体，其中 27 个元素中的每一个都沿面连接到其他元素。立方体形状的元素有 6 个边，因此每个元素最多可以有 6 个连接（例如，3 x 3 x 3 立方体中最中心的元素由 6 个元素包围，并且有 6 个连接）。

然后，让m1、m2 和m3 分别为立方体的第一、第二和第三层。每个元素的名称是xyz，其中x、y、z是元素的行号、列号和层号。例如，元素213 位于立方体的第二行、第一列和第三层。该元素与其他 4 个元素相连：三个在其层中 (113, 313, 223)，一个在其上一层 (212)。

x = 3 # nrow
y = 3 # ncol
z = 3 # nlay

# print each layer as a 2D matrix
for(k in 1:z){
  m = paste0(rep(1:x, each=x), rep(1:y, times = y), k)
  print(matrix(m, nrow=x, byrow=T))
}

     [,1]  [,2]  [,3] 
[1,] "111" "121" "131"
[2,] "211" "221" "231"
[3,] "311" "321" "331"
     [,1]  [,2]  [,3] 
[1,] "112" "122" "132"
[2,] "212" "222" "232"
[3,] "312" "322" "332"
     [,1]  [,2]  [,3] 
[1,] "113" "123" "133"
[2,] "213" "223" "233"
[3,] "313" "323" "333"

igraph 中是否有开箱即用的函数或相关包，用于为这样的网络创建 邻接矩阵或边缘列表？我需要一个可以扩展到任意数量的行、列和层的解决方案。欢迎使用 Python 解决方案。

我手动创建了二维邻接矩阵，其中行和列由下面的c(m1, m2, m3) 给出：

m1 = paste0(rep(1:x, each=x), rep(1:y, times = y), 1)
m2 = paste0(rep(1:x, each=x), rep(1:y, times = y), 2)
m3 = paste0(rep(1:x, each=x), rep(1:y, times = y), 3)
c(m1, m2, m3)
 [1] "111" "121" "131" "211" "221" "231" "311" "321" "331" "112" "122" "132" "212" "222" "232" "312" "322" "332"
[19] "113" "123" "133" "213" "223" "233" "313" "323" "333"

对于这个简单的例子，邻接矩阵是稀疏的，沿对角线有 0，并且是对称的。它看起来像这样：

这里有一个dput() 给 C&P 并用于验证。

dput(temp)
structure(c(0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 
1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 
0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 
1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 
0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 
0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 
0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 
0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 
0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 
1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 
0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 
0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 
0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 
1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 
0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 
0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 
0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 
0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 
0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 
0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 
1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 
0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 
0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 
1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 
0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 
0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0), .Dim = c(27L, 
27L), .Dimnames = list(c("111", "121", "131", "211", "221", "231", 
"311", "321", "331", "112", "122", "132", "212", "222", "232", 
"312", "322", "332", "113", "123", "133", "213", "223", "233", 
"313", "323", "333"), c("111", "121", "131", "211", "221", "231", 
"311", "321", "331", "112", "122", "132", "212", "222", "232", 
"312", "322", "332", "113", "123", "133", "213", "223", "233", 
"313", "323", "333")))

Answer 1

当节点之间的曼哈顿距离为1时有一条边，因此可以在R中使用dist()创建邻接矩阵：

cube_mat = expand.grid(
    x = 1:3,
    y = 1:3,
    z = 1:3
)

m_dist = as.matrix(dist(cube_mat[, 1:3], method = "manhattan", diag = TRUE))
# Zero out any distances != 1
m_dist[m_dist != 1] = 0
rownames(m_dist) = paste0(cube_mat$x, cube_mat$y, cube_mat$z)
colnames(m_dist) = paste0(cube_mat$x, cube_mat$y, cube_mat$z)
# Plot of the adjacency matrix (looks reversed because 111 is in the bottom left):
image(m_dist)

Answer 2

如果你只想使用来自igraph 的包函数：

#adj <- my.adjacency.matrix
as_edgelist(graph.adjacency(adj))

一般而言，您可以使用 igraph 包中的函数在边列表、邻接矩阵之间切换，还可以使用 plot.igraph 生成图形。这是默认立方体：

plot.igraph(graph.adjacency(adj))