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Graph Partitioning Induced Phase Transitions

Gerald Paul·Reuven Cohen·Sameet Sreenivasan·Shlomo Havlin·H. Eugene Stanley
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Abstract

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 non-optimal) that partitions the graph into 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. Additionally, we find that $S$ undergoes multiple non-percolation transitions for $f<f_c$.

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Gerald Paul, Reuven Cohen, Sameet Sreenivasan, Shlomo Havlin, H. Eugene Stanley. 2007-02-17. Graph Partitioning Induced Phase Transitions. https://doi.org/10.1103/physrevlett.99.115701

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