R, igraph walktrap.community splits all nodes in the case of a fully connected graph

Question

Using the walktrap.community approach for defining communities within my graph works great - of all the algorithms I tested it performs the best. The caveat is that in the case of a fully connected graph with no self linkages (every node connects to each other node, but not itself) each node is assigned its own community.

I am not experienced in network analysis but this seems like an interesting case and its certainly not desired behavior. How can I avoid this splitting in my actual data?

library(igraph)
match.mat = matrix(T, nrow=8, ncol=8)

diag(match.mat)[1:8] = T
topology = which(match.mat, arr.ind=T)
g = graph.data.frame(topology, directed=F)
cm = walktrap.community(g)
membership(cm)

# 2 3 4 5 6 7 8 1 
# 1 1 1 1 1 1 1 1 

plot(cm, g)

diag(match.mat)[1:8] = F
topology = which(match.mat, arr.ind=T)
g = graph.data.frame(topology, directed=F)
cm = walktrap.community(g)
membership(cm)

#2 3 4 5 6 7 8 1 
#1 2 3 4 5 6 7 8 

plot(cm, g)

Conceptually I'm not sure how the lack of self linkages would lead to every node being split - maybe possible communities are all tied and therefore split? But the case of all self linkages would seem equivalent in that regard.

Thanks!

http://www-rp.lip6.fr/~latapy/Publis/communities.pdf

Answer 1

If you have read the paper carefully, you will note that the Walktrap builds a node distance measure based on the random walk transition matrix. However, this transition matrix needs to be ergodic, therefore its underlying adjacency matrix needs to be connected and non-bipartite. Non-bipartiteness is achieved by adding self loops to the nodes. Therefore, you need to add self loops to each node in your graph. Maybe it will be a good idea for the future to include this correction in the igraph package, but as far as I know they are using the C implementation of Latapy and Pons and for this one the graph needs to have self loops. Hope this answers your question!