Some Combinatorial Problems in Power-Law Graphs

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.

Automated summary

In this paper, several combinatorial problems for two power-law graphs with the same characteristics are studied, and it is found that power-law behavior itself cannot fully characterize the combinatorial properties of heterogeneous graphs. This is important for the application of combinatorial problems in different fields.