The robustness of complex networks was one of the first phenomena studied after the inception of network science. However, many contemporary presentations of this theory do not go beyond the original papers. Here we revisit this topic with the aim of providing a deep but didactic introduction. We pay attention to some complications in the computation of giant component sizes that are commonly ignored. Following an intuitive procedure, we derive simple formulas that capture the effect of common attack scenarios on arbitrary (configuration model) networks. We hope that this easy introduction will help new researchers discover this beautiful area of network science.

Let G be a connected graph with vertex set V(G). The resistance distance between any two vertices u, v ∈ V(G) is the net effective resistance between them in the electric network constructed from G by replacing each edge with a unit resistor. Let S ⊂ V(G) be a set of vertices such that all the vertices in S have the same neighborhood in G − S, and let G[S] be the subgraph induced by S. In this note, by the {1}-inverse of the Laplacian matrix of G, formula for resistance distances between vertices in S is obtained. It turns out that resistance distances between vertices in S could be given in terms of elements in the inverse matrix of an auxiliary matrix of the Laplacian matrix of G[S], which derives the reduction principle obtained in [J. Phys. A: Math. Theor. 41 (2008) 445203] by algebraic method.

Based on the community discovery method in complex network theory, a power grid partition method considering generator nodes and network weightings is proposed. Firstly, the weighted network model of a power system is established, an improved Fast-Newman hierarchical algorithm and a weighted modular Q function index are introduced, and the partitioning algorithm process is practically improved combined with the characteristics of the actual power grid. Then, the partition results of several IEEE test systems with the improved algorithm and with the Fast-Newman algorithm are compared to demonstrate its effectiveness and correctness. Subsequently, on the basis of subnet partition, two kinds of network attack strategies are proposed. One is attacking the maximum degree node of each subnet, and the other is attacking the maximum betweenness node of each subnet. Meanwhile, considering the two traditional intentional attack strategies, that is, attacking the maximum degree nodes or attacking the maximum betweenness nodes of the whole network, the cascading fault survivability of different types of networks under four attack strategies is simulated and analyzed. It was found that the proposed two attack strategies based on subnet partition are better than the two traditional intentional attack strategies.

In real-world scenarios, networks do not exist in isolation but coupled together in different ways, including dependent, multi-support, and inter-connected patterns. And, when a coupled network suffers from structural instability or dynamic perturbations, the system with different coupling patterns shows rich phase transition behaviors. In this review, we present coupled network models with different coupling patterns developed from real scenarios in recent years for studying the system robustness. For the coupled networks with different coupling patterns, based on the network percolation theory, this paper mainly describes the influence of coupling patterns on network robustness. Moreover, for different coupling patterns, we here show readers the research background, research context, and the latest research results and applications. Furthermore, different approaches to improve system robustness with various coupling patterns and future possible research directions for coupled networks are explained and considered.