Analytical Solution for the Size of the Minimum Dominating Set in Complex Networks

Domination is the fastest-growing field within graph theory with a profound diversity and impact in real-world applications, such as the recent breakthrough approach that identifies optimized subsets of proteins enriched with cancer-related genes. Despite its conceptual simplicity, domination is a classical NP-complete decision problem which makes analytical solutions elusive and poses difficulties to design optimization algorithms for finding a dominating set of minimum cardinality in a large network. Here, we derive for the first time an approximate analytical solution for the density of the minimum dominating set (MDS) by using a combination of cavity method and ultra-discretization (UD) procedure. The derived equation allows us to compute the size of MDS by only using as an input the information of the degree distribution of a given network.