#######################
# PageRank of Google 
#######################

pagerank <- function(G,method='eigen',d=.85,niter=100){
  # follows the notation of the matlab article on pagerank.
  # at http://www.mathworks.com/company/newsletters/news_notes/clevescorner/oct02_cleve.html
  # G is a connectivity matrix, with G[i,j]=1 if page i points to page j
  # method is either "power" or "eigen"
  cvec <- apply(G,2,sum)
  cvec[cvec==0] <- 1 # nodes with indegree 0 will cause problems if we divide by 0.
  gvec <- apply(G,1,sum)
  n <- nrow(G)
  delta <- (1-d)/n
  A <- matrix(delta,nrow(G),ncol(G))
  for (i in 1:n) 
    A[i,] <- A[i,] + d*G[i,]/cvec
#  print(A)
  if (method=='power'){
    x <- rep(1,n)
    for (i in 1:niter) x <- A%*%x
  } else {
    x <- Re(eigen(A)$vector[,1])
  }
  x/sum(x)
}

G <- rbind(c(0,1,1),c(0,0,1),c(1,0,0))
pagerank(G)
pagerank(G,'eigen')



# ==================
# An attempt to get at a "wikipedia" effect. Create 10 nodes, mostly 
# with random links. Then add another 20 nodes that node 1 points to and 
# which only point back to node 1. Then add random links within nodes 21...30. 

n <- 10
G <- matrix(0,n,n)
G[1,2:n] <- 1
G[2:n,1] <- 1
pagerank(G)

n <- 10
G <- matrix(0,n,n)
G[1,2:3] <- G[2:3,1] <- 1
for (i in 1:(2*n)) {
  i <- sample(1:n,1) 
  j <- sample(1:n,1)
  G[i,j] <- 1
}
G
par(mfrow=c(3,2))
image(G);
barplot(pagerank(G), ylim=c(0,1), col="orange")

G2 <- G
n2 <- 20
G2 <- cbind(G,matrix(0,n,n2))
G2 <- rbind(G2,matrix(0,n2,n+n2))
G2[n+(1:n2),1] <- G2[1,n+(1:n2)] <- 1
image(G2);
barplot(pagerank(G2),ylim=c(0,1), col="orange")

for (i in 1:(2*n2)) {
  i <- sample(1:n2,1) 
  j <- sample(1:n2,1)
  G2[i+n,j+n] <- 1
}
image(G2);
barplot(pagerank(G2),ylim=c(0,1))


#