Urban critical infrastructure systems serve as lifelines for modern society, yet their vulnerability to deliberate attacks underscores the urgency of enhancing resilience. Although extensive research has focused on pre-allocating resources against external threats, a key gap persists: the lack of a unified framework that simultaneously addresses external attacks and internal cascading failures under resource constraints. To address this, we propose a novel tri-level dynamic game-theoretic framework integrating defense resource deployment, attack path prediction with cascading effects, and recovery strategy optimization. A nested heuristic algorithm is developed to validate the model and algorithm’s effectiveness.

Operation optimization of Water Distribution Systems (WDSs) in large metropolitan areas is complex and timeconsuming. Effective and reliable scheduling of WDSs can enhance the efficiency of urban water supply and ensure WDS resilience. This paper developed a historical discretization tensor-based optimization (HDTO) method from a new perspective, integrating experience of scheduling staff to sift the search solution space and guarantee the reliability and accessibility of scheduling. Results showed that HDTO method can achieve a 90.0 % reduction in running time relative to stochastic heuristic optimization and pruning method, while enhancing energy and hydraulic performance. Furthermore, the newly developed HDTO method demonstrated remarkable efficacy in optimizing the operation of 24-hour extended period operations, potentially contributing to real-time WDS control. Finally, through the application of emergency and managed scenario analyses, the HDTO method demonstrated significant improvements in both the social and economic sustainability of WDS.

Poisson equation is a partial differential equation with broad utility in theoretical physics. Dirichlet problem of Poisson Equations can be solved by Green’s function. It is a very attractive problem to look for analogous results of the above problem in the fractal context. This paper studies the network set Higher-Dimensional Sierpinski Gaskets, solves Dirichlet problem of Poisson Equations on them by expressing Green’s function explicitly.

With increasing interdependence of various lifeline networks, natural disasters, especially seismic disasters damage will further degrade overall network performance. Hence, based on the concept of risk, reliability and vulnerability, some scholars assessed the resilience of multi-layer lifeline networks. However few studies considered bidirectional interdependence of lifelines, which would immensely exacerbates the consequences of seismic disaster. And the existing research of resilience assessment generally assumed that lifeline networks can recover to their initial performance rather than limited recovery which is more practical. Therefore, this study presents a framework and method for assessing the resilience of lifeline networks. And a performance response function for interdependent lifelines is proposed based on its functional characteristics. Then, we use an IEEE 30-bus and Belgium 20-node gas interdependent network as examples to verify the effectiveness of the framework. The results showed that, for each network, the recovery resource and budget had various effects on the resilience. And our framework can provide a quick reference for optimal decision-making in various scenarios under seismic disasters.

Urban water treatment plants (WTPs) threatened by terrorist attacks is critical to residents’ security and economic development. They are treating and transit hubs that connect the upstream water sources and downstream urban distribution network. Due to the limitation of defensive resources, how to appropriately allocate defensive resources to protect WTPs against terrorist attack is still a challenging issue for urban municipal department. As of now, very few studies have investigated this problem, which often involve strategic game between the attacker and defender. This study presents a game-theoretic method and operational framework to optimize the allocation of defensive resource to protect urban WTPs against terrorist physical attacks. This method fully considers the direct economic loss, systematic loss and human loss generated by the attack on WTPs, and then model the strategic interactions between defenders and terrorists by integrating game theory with risk assessment. In the case study of typical urban water distribution network including five WTPs in central China, we figure out the optimal allocation strategy of defensive resources in Nash equilibrium using proposed game-theoretic method, and then thoroughly compare it with three conventional allocation strategies solely based on risk assessment. The results show that the resources allocation strategy and the attack strategy in Nash equilibrium are optimal strategies for both defender and attacker respectively. Furthermore, the feasibility and availability of the proposed method and operational framework in identifying the optimal allocation strategy of defensive resources is also tested and confirmed in the case study.

Many real-world complex systems, such as power grids, the Internet, transportation networks, and complex information systems networks, which can be modeled abstractly with complex networks, may exhibit coupling or interdependence when subjected to random failure or deliberate attacks, i.e., cascading failures. Therefore, we want to understand the robustness of complex networks against cascading failures. The cascading failure has a failure propagation process, and it is necessary to establish an appropriate propagation model as the research basis. Hence, we adopted a new and more universal cascading failure propagation model based on discrete dynamic system. Based on this propagation model and combined with risk assessment, we proposed several cascading robustness index for complex networks. Then we conducted numerical experiments on several types of existing complex networks and real-world network. These experiments confirm the feasibility and rationality of our proposed indicators to reflect the cascading robustness, which can provide a reference and value for simulation and protection of real-world interdependent networks.

A new method is applied to calculating fractal dimensions of fractional calculus of some fractal functions, by this method, we obtain Box dimension of Weyl-Marchaud fractional derivative of Weierstrass function.

Because the infrastructure network itself has the characteristics of cascading failures, deliberate attackers will often take advantage of these internal dependencies to perform long-term multi-target strikes, which the paper calls consecutive multi-target attacks. This kind of continuous external destruction on the time scale will increase the chain effect of cascading failures within the system. Among these infrastructure networks, electric power systems are particularly critical, because most of the other infrastructures need electricity for their operation and management, and the cascading failure of the power network is more serious. Thus, this paper comprehensively considers external consecutive multi-target attacks and internal cascading failure propagation, establishes a defense-attack-recovery tri-level game optimization model, and study how to allocate defensive and restorative resources from a macro perspective to improve the strategic resilience of the power supply system. In order to solve the optimization model, the paper proposes a nested heuristic algorithm, and proposes a performance response function that combines the power flow operating characteristics and external failure events, and then uses the IEEE 39-bus network as a case study to verify the effectiveness of the model and algorithm.

Interdependent networks can practically model several complex systems. In such networks, particularly critical infrastructure networks, cascading failure is the most common phenomenon observed, and hence, has gradually become a prevalent research topic in recent years. We combine statistical methods to build a cascading failure propagation model based on discrete dynamical systems. Based on a discrete-time model, a failure pattern is proposed wherein the units in a system continuously fail due to cascading events or malevolent attacks. Then, we analyze the model in combination with practical scenarios.

The aim of this paper is to advance the field of network interdiction analysis by introducing an application to the urban water distribution networks (WDNs), deploying protective resources against intentional attacks. The resource allocation problem for urban water supply systems is considered as a three-player (i.e., defender–attacker–operator) game, in which the attacker aims to maximize disruption impacts via interdicting water plants in the network, the defender aims to minimize the worst-case disruption impacts achieved by the attacker while the system operators fulfill the water demand in the residual urban water supply network. Considering the operating characteristics of the water supply network, we adopted the method of hydraulic analysis in the third level to obtain its reliability, and use this as the game equilibrium index of the first two levels. An effective modified variable neighborhood search method is devised to obtain the solution to the game. Finally, a case study was conducted based on the data of water supply network of a certain city in China to evaluate the effectiveness of protection resources against intentional attacks.

As an important part of urban lifeline networks, the urban power network is threatened by multiple types of intentional attackers, such as criminals, terrorists, illegal employees, etc. Each type of attackers differs remarkably in purposes, strategies and valuations of targets. In order to improve the defense capability of urban power network under the threat of different types of attackers, this study proposed an optimal allocation method of defensive resources based on Bayesian game model to reduce the expected loss of the defender. In the case study of IEEE 30-bus network, the reliability and applicability of the resource optimization allocation method are verified through the comparison of the expected loss under different strategy combinations. It is proved that the Bayesian Nash equilibrium defense strategy is the best choice for the defender when there are different types of attackers.

Harmonic functions possess the mean value property, that is, the value of the function at any point is equal to the average value of the function in a domain that contain this point. It is a very attractive problem to look for analogous results in the fractal context. In this paper, we establish a similar results of the mean value property for the harmonic functions on the higher-dimensional Sierpinski gasket.

This paper studied the level-3 Sierpinski gasket. We solved Dirichlet problem of Poisson equations and proved variational principle on the level-3 Sierpinski gasket by expressing Green’s function explicitly.

In accordance with the requirements of Technical guidelines for environmental impact assessment―groundwater environment (HJ 610-2016), the process of groundwater environmental impact assessment and prediction was analyzed. Groundwater Model System (GMS) was used to simulate Groundwater flow field and solute transport, the following main conclusions were draw. As long as the project conforms to the specifications and design and the management is in place during operation, the production area and tailings area in normal working conditions will not cause obvious adverse effects on the quality of groundwater. Under different assumptions, groundwater pollution of the production area and tailings area in abnormal conditions will not directly affect the drinking water, farmland irrigation water and surface water of surrounding residents.

In this paper, we study the mean value property for both the harmonic functions and the functions in the domain of the Laplacian on the tetrahedral Sierpinski gasket. This paper is a continuation of the work of Strichartz and the first author (Qiu and Strichartz, J Fourier Anal Appl 19:943–966, 2013)where the same property on p.c.f. self-similar sets with Dihedral-3 symmetry was considered.

In recent years, because complex networks can be used to model real-world complex systems, such as the Internet, urban infrastructure networks, and gene interaction networks, such research has been widely applied in engineering, social sciences, and life sciences and has caused widespread concern. Fractal dimension, as a concept concerning the filling ability and complexity of an object space, has great significance for the study of the robustness of complex networks. This paper studies the relationship between fractal dimension and the robustness of different types of complex networks from the perspective of network structure and network scale. We find that fractal dimension is strongly correlated with robustness under certain conditions and can be used as an important index to evaluate the robustness of complex networks.

This paper investigates the Hadamard fractional calculus of a fractal function. It is proved that there exists some linear relationship between the order of Hadamard fractional calculus and the fractal dimension of the Weierstrass function including Box dimension, [Formula: see text]-dimension and Packing dimension. Furthermore, numerical result is given to show the linear relationship.

Pipe bursts are a universal problem in water supply systems,which can severely disrupt daily life and urban development.This study correlates flow measurements collected from inlets and outlets of a district metering area (DMA) to quickly identify bursts.The analysis uses a clustering algorithm to detect abnormal flow data i.e.,outliers.Then,some outliers are identified as bursts according to the inlet and outlet flow fluctuation characteristics.The results indicate that the number and locations of the inlets and outlets affect the detection performance.The system can accurately detect bursts and insure a low false positive rate when there are a relatively small number of inlets and outlets and the locations are well placed.