如何找到矩阵中节点的度中心性？

Question

How does one find the degree centrality of nodes in table like,

article   users
         u1  u2  u3  u4 u5 u6 u7
 1        1   1   1   0  0  0  0
 2        0   1   0   1  1  0  0
 3        1   0   0   1  0  1  1

This is just an example of my data I have a very large file consisting of 1533 articles and about 52000 users.

I want to find the centrality of articles and centrality of users in the matrix.

Answer 1

Degree centrality simply counts the number of other nodes that each node is "connected" to. So to do this for users, for example, we have to define what it means to be connected to another user. The simplest approach asserts a connection if a user has at least one article in common with another user. A slightly more complex (and probably better) approach weights connectivity by the number of articles in common. So if user 1 has 10 articles in common with user 2 and 3 articles in common with user 3, we say that user 1 is "more connected" to user 2 than to user 3. In what follows, I'll use the latter approach.

This code creates a sample matrix with 15 articles and 30 users, sparsely connected. It then calculates a 30 X 30 adjacency matrix for users where the [i,j] element is the number of articles user i has in common with user j. Then we create a weighted igraph object from this matrix, and let igraph calculate the degree centrality.

Since degree centrality does not take the weights into account, we also calculate eigenvector centrality (which does take the weights into account). In this very simple example, the differences are subtle but instructive.

# this just set up the sample - you have the matrix M already
n.articles <- 15
n.users    <- 30
set.seed(1)    # for reproducibility
M <- matrix(sample(0L:1L,n.articles*n.users,p=c(0.8,0.2),replace=T),nc=n.users)

# you start here...
m.adj <- matrix(0L,nc=n.users,nr=n.users)
for (i in 1:(n.users-1)) {
  for (j in (i+1):n.users) {
    m.adj[i,j] <- sum(M[,i]*M[,j])
  }
}
library(igraph)
g <- graph.adjacency(m.adj,weighted=T, mode="undirected")
palette <- c("purple","blue","green","yellow","orange","red")
par(mfrow=c(1,2))
# degree centrality
c.d   <- degree(g)
col <- as.integer(5*(c.d-min(c.d))/diff(range(c.d))+1)
set.seed(1)
plot(g,vertex.color=palette[col],main="Degree Centrality",
     layout=layout.fruchterman.reingold)

# eigenvalue centrality
c.e   <- evcent(g)$vector
col <- as.integer(5*(c.e-min(c.e))/diff(range(c.e))+1)
set.seed(1)
plot(g,vertex.color=palette[col],main="Eigenvalue Centrality",
     layout=layout.fruchterman.reingold)

So in both cases node 15 has the highest centrality. However, node 28 has a higher degree centrality and a lower eigenvalue centrality than node 27. This is because node 28 is connected to more nodes, but the strength of the connections is lower.

The same approach can of course be used to calculate article centrality; just use the transpose of M.

This approach will not work with 52,000 users - the adjacency matrix will contain > 2.5 billion elements. I'm not aware of a workaround for this - perhaps someone else is, I'd like to hear it. So if you need to tablulate a centrality score for each of the 52,000 users, I can't help you. On the other hand if you want to see patterns, it might be possible to carry out the analysis on a random sample of users (say, 10%).