This paper discusses the resilience of networks based on graph theory and stochastic process. The electric power network where edges may fail simultaneously and the performance of the network is measured by the ratio of connected nodes is supposed for the target network. For the restoration, under the constraint that the resources are limited, the failed edges are repaired one by one, and the order of the repair for several failed edges is determined with the priority to the edge that the amount of increasing system performance is the largest after the completion of repair. Two types of resilience are discussed, one is resilience in the recovery stage according to the conventional definition of resilience and the other is steady state operational resilience considering the long-term operation in which the network state changes stochastically. The second represents a comprehensive capacity of resilience for a system and is analytically derived by Markov analysis. We assume that the large-scale disruption occurs due to the simultaneous failure of edges caused by the common cause failures in the analysis. Marshall-Olkin type shock model and α factor method are incorporated to model the common cause failures. Then two resilience measures, “operational resilience” and “operational resilience in recovery stage” are proposed. We also propose approximation methods to obtain these two operational resilience measures for complex networks.

Next generation advanced power devices show remarkable progress in wide band-gap power devices such as silicon carbide and gallium nitride devices, as well as novel silicon devices called as super junction FETs and so on. The future direction of power electronics applications is surveyed in terms of output power density as an index of future power electronics development, instead of the power conversion efficiency, taking the device progress in sight. Over the last 30 years, the output power density of power electronics apparatuses has increased by a factor of two figures. New markets, such as a power supply for future generation CPU, a compact unit inverter and a electric vehicle-driving inverter unit, are expected to grow rapidly from 2010 to 2015 with the advance in the out power density of power converter. The possibility of power electronics innovation with progress in the output power density will be discussed in conjunction with development of next generation advanced power devices and related technologies.