我正在尝试使用拉普拉斯矩阵解决网络上的流量问题。我开始在这里测试这个问题:https ://rosettacode.org/wiki/Resistor_mesh#Python
当所有权重为 1 时,R = 1.6089 无论如何,它完美地得出了解决方案。我希望能够解决不等于 1 的电阻!然后能够获得每个电阻器上的电流,所以我尝试为电导(权重)生成一组随机值并检查它是否正常工作,看看来自注入电流的节点的电流总和是否相等注入应有的电流。不幸的是,事实并非如此。我检查了拉普拉斯算子,它看起来与我的预期相符,但除此之外,我完全迷失了,有人能说明我是否在这里找到了正确的拉普拉斯算子吗?或者如果我遗漏了一些非常明显的东西,比如一些不正确的索引?
代码如下(请原谅我生成网格的非常业余的方式,任何提示和技巧也很感激!):
library(igraph)
library(SparseM)
# setup edgelist + grid to look at
sample_grid = function(N_x, N_y, xlim = c(-1,1), ylim = c(-1,1)) {
# bounding box
min_x <- xlim[1]
max_x <- xlim[2]
min_y <- ylim[1]
max_y <- ylim[2]
x_locs <- seq(0, N_x)
y_locs <- seq(0, N_y)
N_grid <- N_x * N_y
x <- rep(0, length(N_grid))
y <- rep(0, length(N_grid))
for(i in 1:N_x){
for(j in 1:N_y) {
x[N_y * (i-1) + j] <- x_locs[i]
y[N_y * (i-1) + j] <- y_locs[j]
}
}
locations <- cbind(x,y)
return(locations)
}
N_x <- 10
N_y <- 10
# node locations
Vert <- sample_grid(N_x,N_y)
# edges...
# horizontally...
fromH <- c()
for(i in 1:(N_x-1)){
fromH <- c(fromH, seq(0,(N_y*N_x-N_x), length.out = (N_y)) + i)
}
toH <- fromH + 1
# vertically...
fromV <- 1:(N_x*N_y-N_x)
toV <- fromV + N_x
# --------------------------------------------------------------------------------------------
# crux
Edges <- data.frame(from = c(fromH, fromV), to = c(toH, toV)) # , weights = rep(1,(2*N_x*N_y - (N_x+N_y))))
# change the weights up
set.seed(1)
weight <- rlnorm(n = nrow(Edges), meanlog = 0, sdlog = 1)
Edges$weight <- weight
the_graph <- graph_from_data_frame(Edges, directed = FALSE)
lo <- layout.norm(as.matrix(Vert))
plot(the_graph, layout = lo, directed = FALSE, edge.arrow.size=0)
# solving
L <- laplacian_matrix(the_graph, weights = weight)
# boundary conditions on 68, 12:
# draw 1 amp @ 12
# inject 1 amp @ 68
q <- matrix(rep(0, nrow(Vert)), ncol = 1)
q[68,] <- +1
q[12,] <- -1
# solve
p <- solve(L,q)
R <- p[68,] - p[12,]
# investigating why the weights aren't working! (wrong first attempt)
# neighbours <- c(78,58,67,69) # neighbours of node 68
# neighbours_weights <- weight[neighbours]
# neighbours_potential_diffs <- p[68,] - p[neighbours,]
# neighbours_currents <- neighbours_potential_diffs * neighbours_weights
neighbours <- which(Edges$from == 68 | Edges$to == 68)
potential_diffs <- p[Edges$from,] - p[Edges$to,]
currents <- potential_diffs * Edges$weight
what <- cbind(Edges,p[Edges$from,], p[Edges$to,], currents)
what[neighbours,]
# exact solution for effective resistance between 68 and 12 with 10x10 and all 1ohm
exact <- 455859137025721/283319837425200
问题是计算出的电流没有像预期的那样总和为 1:
> neighbours_currents
[1] 0.05035087 0.09044874 0.03309549 0.21782845