Random Graph Partitioning

Question

I'm trying to test some models of graph partitioning (these come from the real world, where a graph slowly self-partitions). To do this, I need to be able to uniformly randomly partition this graph into contiguous components (we are given the graph is initially connected, as well). Were the contiguity criterion not required I believe this would be the problem of randomly partitioning a set, which can be combinatorially analyzed. Does anyone know of any way to randomly partition graphs into subgraphs (i.e. randomly sample one partition), or, if no such method is known, to randomly sample a set of elements? The method of randomizing the number of partitions and then randomizing membership won't work because there are different numbers of possible partitions for each partition size.

Answer 1

You have to differentiate edge-cut partitioning and vertex-cut partitioning, where you divide the graph along the edges or vertices. This significantly impacts your problem as the number of different vertex-cuts is much larger than the number of edge-cuts. The reason is that you exclusively assign edges to partitions in vertex-cut - as opposed to edge-cut where you assign vertices to partitions - and there are much more edges than vertices (e.g. O(n^2) edges for n vertices). Hence, the combinatorially larger vertex-cut leads to a larger number of subgraphs that have to be checked for connectivity. A naive method for randomization is to enumerate all partitionings, iteratively select one partitioning, and check connectivity of all subgraphs in the selected partitioning. Then you just take the first one. In this case, all solutions have equal probability (uniformly random).

Answer 2

I have come across the same problem in work I am doing. I have two solutions to randomly partition a graph into m contiguous components:

1. Spanning Tree Approach. Randomly choose a spanning tree of your graph (e.g. Using Wilson's algorithm which chooses uniformly amongst all spanning trees). Then randomly select m-1 edges (without replacements) and remove them from the spanning tree. This will give m components which are each connected in the original graph.

2. Edge contraction approach. Randomly choose an edge and contract it, renaming the (new) vertex as the union of the two previous vertices. Repeat until you have only m vertices left. Identify each vertex with the subset of (original) vertices that were contracted into it.