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

TL;DR

This paper derives an approximate analytical formula for the size of the minimum dominating set in complex networks using cavity method and Ultra-Discretization, based solely on the network's degree distribution.

Contribution

It introduces the first analytical approximation for the minimum dominating set size in complex networks, linking it to degree distribution data.

Findings

Provides an equation to estimate MDS size from degree distribution

Enables analytical insights into domination in large networks

Facilitates faster computation of dominating sets in complex systems

Abstract

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.