Some Combinatorial Problems in Power-law Graphs

Abstract

The power-law behavior is ubiquitous in a majority of real-world networks, and it was shown to have a strong effect on various combinatorial, structural, and dynamical properties of graphs. For example, it has been shown that in real-life power-law networks, both the matching number and the domination number are relatively smaller, compared with homogeneous graphs. In this paper, we study analytically several combinatorial problems for two power-law graphs with the same number of vertices, edges, and the same power exponent. For both graphs, we determine exactly or recursively their matching number, independence number, domination number, the number of maximum matchings, the number of maximum independent sets, and the number of minimum dominating sets. We show that power-law behavior itself cannot characterize the combinatorial properties of a heterogenous graph. Since the combinatorial properties studied here have found wide applications in different fields, such as structural controllability of complex networks, our work offers insight in the applications of these combinatorial problems in power-law graphs.

References in corpus (9)

• Emergence of scaling in random networks
• The structure and function of complex networks
• Error and attack tolerance of complex networks
• Self-similarity of complex networks
• Control Principles of Complex Networks
• Controlling edge dynamics in complex networks
• Inverted Berezinskii-Kosterlitz-Thouless Singularity and High-Temperature Algebraic Order in an Ising Model on a Scale-Free Hierarchical-Lattice Small-World Network
• Domination number and minimum dominating sets in pseudofractal scale-free web and Sierpiński graph
• Maximum matchings in scale-free networks with identical degree distribution