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.

Our society has been facing a decentralizing effect on multiscale complexity in low-carbon transitions. The purpose of this work is to evaluate the roles of transaction rate, energy and carbon market regulation, energy consumption and energy capacity in single and one-way climate policy linked system in low carbon transition for distributed energy systems through agent-based simulation. A set of agent behavioral rules for the competition of high and low carbon energies mimics the domestic and exotic local dynamics of the energy production of distributed energy systems and the energy consumption of industrial firms under the impact of energy and carbon market fluctuation. Simulation results show that a single system cannot achieve low carbon transition, while a one-way climate policy linked system can realize low carbon transition. The results also show that the larger high and low energy capacity is, the system is less likely to achieve low carbon transition in the circumstance of the same transaction rate with the constant emission policy bias. Policy implications are drawn regarding the regulation of localized carbon markets in single and climate policy linked systems.

To better understand the economic crisis impact on industry economy in low carbon transition, the simplified macroeconomic system combined with a distributed industry-energy ecosystem, namely the economy-ecosystem, was established based on an agent-based modeling approach. A set of behavioral rules for each agent involved was formalized to study the competition between high and low carbon energy under the scheme of ecological life of industrial firms, and periodical economy crisis was emerged by adjusting asymmetric adjustment speed ratio of hiring to firing and default threshold that controls the ratio between total debt and total circulating currency. The impact of the economic crisis on the ecosystem in a distributed energy system and the interaction between the economy and ecosystem in low carbon transition under different conditions of the ecosystem were investigated. Simulation results show that high carbon energy is more resistant to economic crisis than low carbon energy. Moreover, low-carbon transition with economic recession state can be reached in economy-ecosystem. Various policy implications are also given to policymakers with different technical levels and economic status.