如何绘制在网络中链接的代理的唯一对？

Question

I have a number of agents who are linked through a network and now I am trying to draw random pairs of linked agents. The final outcome should be a binary matrix (ncol = nrow = number of agents). Each agent may show up only once in a linked pair.

This is what I have done so far. First I draw random pairs

library(sna,dummies)
pairs  = matrix(sample(1:10,10,replace=F), ncol=2)

Then I convert it to a matrix

meet = cbind(merge(c(1:10), pairs,  by.x = 1, by.y = 1, all.x = TRUE),
merge(c(1:10), pairs,  by.x = 1, by.y = 2, all.x = TRUE))
meet[,2] = ifelse(is.na(meet[,2]), meet[,4], meet[,2])
meet = meet[,-c(3,4)]
meet = data.matrix(dummy.data.frame(meet, names = "V2"))
meet = meet[,-1]
colnames(meet) = c(1:10)

Let's say agents are linked through the following network

nw = rgraph(10, mode = "graph", tprob = 0.5)

If I multiply meet and nw I get a matrix of randomly drawn pairs by eliminating pairs of agents who are not linked. However, in this case I end up with a very small number of pairs...

Is there a way of drawing pairs conditioned on a link between agents?

Answer 1

What is the purpose of this? is it important to find the largest possible number of pairs? Basically, what you describe is related to the concept of matching in graphs ( http://en.wikipedia.org/wiki/Matching_%28graph_theory%29 ). There is a distinction between "maximal matching" and "maximum matching".

The first is easy to find (and is probably what you try to do. The "trick" for doing it is choosing edges rather than nodes (agents) in your graph. start with a random edge (which adds your first pair of agents), then delete all the edges that are connected to those agents, and choose (randomly) an edge from the remaining edges. repeat the process until no edges are left. This is "maximal" in that you can't add to it any more pairs. However, you may still end up with a limited number of pairs.

To get the maximum number of pairs possible for the given graph, you need to work harder (see the wikipedia link for some resources on that), but is it what you really want? (note: the maximum matching is never much bigger compared to any of the easy-to-find maximal matching matching.

You might want to look at the igraph package which has a couple of functions that deal with matchings (especially of bipartite graphs). But it might be, that for your goals, a much simpler algorithm is required (the research literature is full of algorithms for matching in special cases).

I hope this helps.