Graph Partitioning Induced Phase Transitions

We study the percolation properties of graph partitioning on random regular graphs with $N$ vertices of degree $k$. Optimal graph partitioning is directly related to optimal attack and immunization of complex networks. We find that for any partitioning process (even if nonoptimal) that partitions the graph into essentially equal sized connected components (clusters), the system undergoes a percolation phase transition at $f=f_c=1-2/k$ where $f$ is the fraction of edges removed to partition the graph. For optimal partitioning, at the percolation threshold, we find $S\sim N^{0.4}$ where $S$ is the size of the clusters and $\ell \sim N^{0.25}$ where $\ell$ is their diameter. Also, we find that $S$ undergoes multiple nonpercolation transitions for $f<f_c$.