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.
UAV swarm are often subjected to random interference or malicious attacks during the execution of their tasks, resulting in UAV failure or communication interruption. When the UAV swarm is out of interference or the repair command is executed, the performance of the UAV swarm will be restored to a certain extent. However, how to measure the changes of UAV swarm’s performance during this process will be very important, and it is also crucial to determine whether the UAVs can continue to perform its mission. Based on this motivation, we propose a resilience assessment framework for UAV swarm considering load balancing after UAV swarm suffer from disturbances. We analyze the effects of different topologies and different parameters on the resilience of UAV swarm. The study found that attack intensity is the most important factor affecting UAV swarm performance. As the attack intensity increases, the performance of the UAV swarm decreases rapidly. At the same time, topology also has a very important impact on UAV swarm resilience.
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.
Network robustness is the ability of a network to maintain a certain level of structural integrity and its original functions after being attacked, and it is the key to whether the damaged network can continue to operate normally. We define two types of robustness evaluation indicators based on network maximum flow: flow capacity robustness, which assesses the ability of the network to resist attack, and flow recovery robustness, which assesses the ability to rebuild the network after an attack on the network. To verify the effectiveness of the robustness indicators proposed in this study, we simulate four typical networks and analyze their robustness, and the results show that a high-density random network is stronger than a low-density network in terms of connectivity and resilience; the growth rate parameter of scale-free network does not have a significant impact on robustness changes in most cases; the greater the average degree of a regular network, the greater the robustness; the robustness of small-world network increases with the increase in the average degree. In addition, there is a critical damage rate (when the node damage rate is less than this critical value, the damaged nodes and edges can almost be completely recovered) when examining flow recovery robustness, and the critical damage rate is around 20%. Flow capacity robustness and flow recovery robustness enrich the network structure indicator system and more comprehensively describe the structural stability of real networks.
Water consumption has been one of the most important topics in the field of environment and economy. Even though the driving factors of water consumption have been well studied, it is still a daunting task to reveal the influence of the status of provinces in the entire supply chain. By combining the multi-regional input-output (MRIO) model and complex network theory, an inter-provincial virtual water transfer (V WT) network was constructed to analyze the overall structural characteristics of the network model and identify the structural roles of each province. The constructed inter-provincial V WT network exhibited the characteristics of a small-world network, that is, virtual water can be easily transferred from one province to another. Moreover, network analysis revealed that provinces with different positions in the V WT network played discrepant structural roles. Panel regression analysis was further used to quantify the impact of provincial structural roles on their water consumption. The results showed that water consumption in China largely depended on some structural role characteristics in the V WT network. Out-degree and out-strength characterizing the ability of direct exporting virtual water exerted significant positive influences, while in-closeness featuring the indirect virtual water importing rate had a significant negative effect on water usage. This indicated that adjusting the uneven provincial consumption structure, the direct production demand of downstream provinces and the indirect production activities in the supply chain would help reduce water consumption. Therefore, to come true the goal of water conservation in China, it would be necessary to improve the trade structure between direct and indirect exporters and importers in the entire supply chain.
Frontiers in Physics
Exploring Human Interactions through Sociophysics: Dynamics of Opinion Formation