The mixing matrix of a graph gives the density of edges between vertices with different characteristics. The mixing matrix for a given igraph object can be calculated using the following function:
| # calculate the mixing matrix of in igraph graph object ‘mygraph’, by some vertex attribute ‘attrib’ | |
| # can change the default use.density=FALSE to return a matrix with raw number of edges rather than density | |
| mixmat <- function(mygraph, attrib, use.density=TRUE) { | |
| require(igraph) | |
| # get unique list of characteristics of the attribute | |
| attlist <- sort(unique(get.vertex.attribute(mygraph,attrib))) | |
| numatts <- length(attlist) | |
| # build an empty mixing matrix by attribute | |
| mm <- matrix(nrow=numatts, | |
| ncol=numatts, | |
| dimnames=list(attlist,attlist)) | |
| # calculate edge density for each matrix entry by pairing type | |
| # lends itself to parallel if available | |
| el <- get.edgelist(mygraph,names=FALSE) | |
| for (i in 1:numatts) { | |
| for (j in 1:numatts) { | |
| mm[i,j] <- length(which(apply(el,1,function(x) { | |
| get.vertex.attribute(mygraph, attrib, x[1] ) == attlist[i] && | |
| get.vertex.attribute(mygraph, attrib, x[2] ) == attlist[j] } ))) | |
| } | |
| } | |
| # convert to proportional mixing matrix if desired (ie by edge density) | |
| if (use.density) mm/ecount(mygraph) else mm | |
| } |
The assortativity coefficient, based on Newman’s paper, can be calculated from the mixing matrix by the following:
| # calculate the assortativity coefficient for a mixing matrix of a graph | |
| # ref: MEJ Newman, ‘Mixing patterns in networks’, Phys Rev E 67, 026126 (2003) | |
| # | |
| # define assortativity coefficient as | |
| # trace (m) – sum (m^2) | |
| # ac = ————————- | |
| # 1 – sum (m^2) | |
| # | |
| # where m is the mixing matrix of a graph | |
| assortcoeff <- function(m) { | |
| tr <- sum(diag(m)) | |
| sumsq <- sum (rowSums(m)*colSums(m)) | |
| (tr – sumsq) / (1 – sumsq) | |
| } |
Here is an example (be sure to load the functions mentioned above):
| # sample calculating mixing matrix and assortativity coefficient with igraph | |
| require(igraph) | |
| set.seed(12) | |
| # create a random graph | |
| g <- random.graph.game(1000,0.15) | |
| # assign some characteristics | |
| V(g)$color <- c(‘red‘,‘white‘,‘blue‘,‘orange‘,‘green‘) | |
| # calculate the mixing matrix | |
| m <- mixmat(g,‘color‘) | |
| # now calculate the assortativity coefficient | |
| assortcoeff(m) |
Cool.
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