Minimum Dominating Sets in Scale-Free Network Ensembles

We study the scaling behavior of the size of minimum dominating set (MDS) in scale-free networks, with respect to network size $N$ and power-law exponent $\gamma$, while keeping the average degree fixed. We study ensembles generated by three different network construction methods, and we use a greedy algorithm to approximate the MDS. With a structural cutoff imposed on the maximal degree ($k_{\max}=\sqrt{N}$) we find linear scaling of the MDS size with respect to $N$ in all three network classes. Without any cutoff ($k_{\max}=N-1$) two of the network classes display a transition at $\gamma \approx 1.9$, with linear scaling above, and vanishingly weak dependence below, but in the third network class we find linear scaling irrespective of $\gamma$. We find that the partial MDS, which dominates a given $z<1$ fraction of nodes, displays essentially the same scaling behavior as the MDS.