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) = root 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 approximate to 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.