## 1 Introduction  
Modern power systems are expanding in size, complexity and new technologies, which has brought significant improvement to the way we generate, transmit and consume electricity [1; 2]. Nonetheless, these ever-growing power systems, at the same time, have created new and more pressing challenges for society, such as instability caused by massive scale-up [3] and uncertainties and risks due to the wide range of features and functions that modern grids boost [4]. Cascading failure is one of severest power outages in power systems, which can cause an astronomical amount of financial loss and affect countless people [5]. Cascading failures are normally caused by initial incidents, such as an electric fault at a particular transmission line, and through fast propagation, they can lead to large-scale blackouts in modern complex power systems [6; 7; 8]. On 14 August 2003, for instance, an initial disturbance caused by the summer heat in Ohio triggered the largest blackout in the history of the USA [9], which finally affected an estimated 10 million people in Ontario and 45 million people in eight states in the USA. Moreover, it was reported that there were about 30 billion dollars of economic loss in this incident. The largest blackout in history, branded \"the biggest power failure\", happened on 31 July 2012 in India, which left 620 million people, half the Indian population, living without power for two days. This blackout was also a result of cascading failure caused by upgrading a backup power line in Bina-Gwalior-Agra in India [10]. The blackouts that happened all over the world have drawn considerable attention to study the cascading failure mechanism and investigate the robustness and vulnerability of power grids [11]. Investigations into these major blackouts have shown that almost all of them were triggered by an initial disturbance [12], which caused a series of cascading failures with rapid propagation. In order to understand the mechanism of cascading failures in power systems, it is necessary to comprehend how large-scale grids behave when different kinds of failures occur. Electric grids, in principle, can be abstracted and considered as a type of complex network, with unique assumptions that describe the power system features. Numerous researchers have carried out relevant studies on complex networks; for example, investigations of a communication network via complex networks to increase reliability [13], the study of an epidemic complex network to efficiently destruct epidemic system [14], complex networks for musical compositions [15] and complex network approaches to optimizing the smart grid structure [16], just to list a few. There are three typical forms of complex networks, namely scale-free networks, smart-world networks and Erdos-Renyi (ER) random networks. A scale-free network is a network whose degree of distribution, at least, asymptotically follows a power law [15]; a small-world network is a type of mathematical graph in which most nodes are not neighbours of one another, but the neighbours of any given node are likely to be neighbours of each other [16]; and an ER random network is either of the two closely-related models for generating random graphs with asymptotic stochastic graphs [17]. Real-world networks are usually combinations of these three typical networks, and thus, investigating these typical networks can provide innovative and impactful solutions to optimization problems and reliability and risk assessments. For instance, it has been possible to study and enhance the stability of communication networks through such analysis [18], and the issue that causes such instability can be effectively eliminated through vigorous research of complex networks [19]. The knowledge and research experience of complex network have enabled us to investigate the mechanism of cascading failures in power systems.