  We study ensemble-based graph-theoretical methods aiming to approximate the
size of the minimum dominating set (MDS) in scale-free networks. We analyze
both analytical upper bounds of dominating sets and numerical realizations for
applications. We propose two novel probabilistic dominating set selection
strategies that are applicable to heterogeneous networks. One of them obtains
the smallest probabilistic dominating set and also outperforms the
deterministic degree-ranked method. We show that a degree-dependent
probabilistic selection method becomes optimal in its deterministic limit. In
addition, we also find the precise limit where selecting high-degree nodes
exclusively becomes inefficient for network domination. We validate our results
on several real-world networks, and provide highly accurate analytical
estimates for our methods.
