As the number of cores on a chip increases, power consumed by the communication structures takes a significant portion of the overall power budget. Many application-specific systems on chips (SoCs) involve heterogeneous cores with varied functionality and communication requirements, such as those in mobile-phone systems. If a regular network-on-chip is designed to fit the requirements of few high-communicative components, it will be largely over-designed with respect to the needs of the remaining components. Consequently, irregular network architectures might be necessary for realising application-specific SoCs. The authors propose a power-aware topology construction method, which can construct application-specific low-power interconnection topologies according to the traffic characteristics of SoCs. They take several multimedia applications as case studies and experimental results show the power savings of power-aware topology approximate to 49% of the interconnection architecture. They also implement a simulator to experiment more general large scale systems, and the results show that customised irregular networks are clearly superior to traditional regular architectures in terms of performance and energy.