Critical Node Detection Problem Research Articles

Optimal design of vaccination policies: A case study for Newfoundland and Labrador

This paper proposes pandemic mitigation vaccination policies for Newfoundland and Labrador (NL) based on two compact mixed integer programming (MIP) models of the distance-based critical node detection problem (DCNDP). Our main focus is on two variants of the DCNDP that seek to minimize the number of connections with lengths of at most one (1-DCNDP) and two (2-DCNDP). A polyhedral study for the 1-DCNDP is conducted, and new aggregated inequalities are provided for the 2-DCNDP. The computational experiments show that the 2-DCNDP with aggregated inequalities outperforms the one with disaggregated inequalities for graphs with a density of at least 0.5%. We also study the strategic vaccine allocation problem as a real-world application of the DCNDP and conduct a set of computational experiments on a simulated contact network of NL. Our computational results demonstrate that the DCNDP-based strategies can have a better performance in comparison with the real-world strategies implemented during COVID-19.

A Pseudo-Deterministic Noisy Extremal Optimization algorithm for the pairwise connectivity Critical Node Detection Problem

Abstract The critical node detection problem is a central task in computational graph theory due to its large applicability, consisting in deleting $k$ nodes to minimize a certain graph measure. In this article, we propose a new Extremal Optimization-based approach, the Pseudo-Deterministic Noisy Extremal Optimization (PDNEO) algorithm, to solve the Critical Node Detection variant in which the pairwise connectivity is minimized. PDNEO uses an adaptive pseudo-deterministic parameter to switch between random nodes and articulation points during the search, as well as other features, such as noise induction to preserve diversity, greedy search to better exploit the search space and a greater search space exploration mechanism. Numerical experiments on synthetic and real-world networks show the effectiveness of the proposed algorithm compared with existing methods.

Code and Data Repository for Detecting Critical Nodes in Sparse Graphs via “Reduce-Solve-Combine” Memetic Search

This study considers a well-known critical node detection problem that aims to minimize a pairwise connectivity measure of an undirected graph via the removal of a subset of nodes (referred to as critical nodes) subject to a cardinality constraint. Potential applications include epidemic control, emergency response, vulnerability assessment, carbon emission monitoring, network security and drug design. To solve the problem, we present a "reduce-solve-combine" memetic search approach that integrates a problem reduction mechanism into the popular population-based memetic algorithm framework. At each generation, a common pattern mined from two parent solutions is first used to reduce the given problem instance, then the reduced instance is solved by a component-based hybrid neighborhood search that effectively combines an articulation point impact strategy and a node weighting strategy, and finally an offspring solution is produced by combining the mined common pattern and the solution of the reduced instance. Extensive evaluations on 42 real-world and synthetic benchmark instances show the efficacy of the proposed method, which discovers 9 new upper bounds and significantly outperforms the current state-of-the-art algorithms. Investigation of key algorithmic modules additionally discloses the importance of the proposed ideas and strategies. Finally, we demonstrate the generality of the proposed method via its adaptation to solve the node-weighted critical node problem.

Detecting Critical Nodes in Sparse Graphs via “Reduce-Solve-Combine” Memetic Search

This study considers a well-known critical node detection problem that aims to minimize a pairwise connectivity measure of an undirected graph via the removal of a subset of nodes (referred to as critical nodes) subject to a cardinality constraint. Potential applications include epidemic control, emergency response, vulnerability assessment, carbon emission monitoring, network security, and drug design. To solve the problem, we present a “reduce-solve-combine” memetic search approach that integrates a problem reduction mechanism into the popular population-based memetic algorithm framework. At each generation, a common pattern mined from two parent solutions is first used to reduce the given problem instance, then the reduced instance is solved by a component-based hybrid neighborhood search that effectively combines an articulation point impact strategy and a node weighting strategy, and finally an offspring solution is produced by combining the mined common pattern and the solution of the reduced instance. Extensive evaluations on 42 real-world and synthetic benchmark instances show the efficacy of the proposed method, which discovers nine new upper bounds and significantly outperforms the current state-of-the-art algorithms. Investigation of key algorithmic modules additionally discloses the importance of the proposed ideas and strategies. Finally, we demonstrate the generality of the proposed method via its adaptation to solve the node-weighted critical node problem. History: Accepted by Erwin Pesch, Area Editor for Heuristic Search & Approximation Algorithms. Funding: This work was supported by the National Natural Science Foundation of China [Grants 72371157, 61903144, 72031007]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2022.0130 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2022.0130 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Balancing wildlife protection and wildfire threat mitigation using a network optimization approach

In boreal forests of North America, land managers often carry out preventive treatments of forest fuel for the protection of human infrastructure from wildfires. However, these treatments may negatively affect other ecosystem services, such as the capacity to sustain wildlife populations. Here, we examine the efficacy of a strategy aimed at preserving a critical movement corridor for boreal woodland caribou (Rangifer tarandus caribou) in northern Québec, Canada, by raising high-voltage power line conductors above the forest canopy. To assess the interplay between the caribou protection objectives and a reduction in power line's exposure to wildfires, we developed an optimization model that combines the objectives of protecting the power line from wildfires via fuel treatments and maintaining a suitable movement corridor for caribou. The model combines a critical node detection (CND) problem with a habitat connectivity problem that allocates a minimum-resistance fixed-width habitat corridor between isolated wildlife refuges. Our results identify the best locations to perform fire fuel treatments to lessen the threat of fire damage to human infrastructure while maintaining a connectivity corridor for caribou in present and future climate scenarios. The selected fuel treatment locations aimed to mitigate wildfire exposure to a power line. In small-budget solutions, the exposure of power line infrastructure to wildfires was reduced by 36–39% in current climate conditions and by 20–31% in future climate, compared with no-treatment scenarios. Despite the detrimental effects of wildfire on both the industrial asset and caribou habitat, the approach provides strategies that help achieve a compromise between these two values. Such knowledge is timely to help mitigate the negative impacts of climate change on human livelihoods and natural ecosystems.

The critical node detection problem in hypergraphs using weighted node degree centrality.

Network analysis is an indispensable part of today's academic field. Among the different types of networks, the more complex hypergraphs can provide an excellent challenge and new angles for analysis. This study proposes a variant of the critical node detection problem for hypergraphs using weighted node degree centrality as a form of importance metric. An analysis is done on both generated synthetic networks and real-world derived data on the topic of United States House and Senate committees, using a newly designed algorithm. The numerical results show that the combination of the critical node detection on hypergraphs with the weighted node degree centrality provides promising results and the topic is worth exploring further.

An Interactive Co-Evolutionary Framework for Multi-Objective Critical Node Detection on Large-Scale Complex Networks

Multi-objective evolutionary algorithms (MOEAs) have been proved to be competitive for solving the critical node detection problem (CNDP) on complex networks in both non-cascading scenario and cascading scenario. With the continuous expansion of network scale, the search space of the existing MOEAs will increase exponentially. To this end, we propose an interactive co-evolutionary framework (ICoEF) to solve the multi-objective critical node detection on large-scale complex networks, where a set of local populations as well as a global population are interactively evolved to detect the critical nodes with high qualities. Specifically, we first use a community detection algorithm to divide the original network into multiple local communities, where each local community generates a local population for quickly finding local solutions independently. Then, the latent good solutions can be easily obtained by combining the returned local solutions. After that, one global population generated by the original network can refine these latent good solutions by using the proposed two-stage replacement search strategy. Through the above local and global evolutionary interaction, a set of high-quality solutions can be finally obtained. Experiments on synthetic and real-world networks demonstrate the superiority of the proposed framework over several state-of-the-art baselines in both non-cascading and cascading scenarios.

Seeking critical nodes in digraphs

The Critical Node Detection Problem (CNDP) consists in finding the set of nodes, defined critical, whose removal maximally degrades the graph. In this work we focus on finding the set of critical nodes whose removal minimizes the pairwise connectivity of a direct graph (digraph). Such problem has been proved to be NP-hard, thus we need efficient heuristics to detect critical nodes in real-world applications. We aim at understanding which is the best heuristic we can apply to identify critical nodes in practice, i.e., taking into account time constrains and real-world networks. We present an in-depth analysis of several heuristics we ran on both real-world and on synthetic graphs. We define and evaluate two different strategies for each heuristic: standard and iterative. Our main findings show that an algorithm recently proposed to solve the CNDP and that can be used as heuristic for the general case provides the best results in real-world graphs, and it is also the fastest. However, there are few exceptions that are thoroughly analyzed and discussed. We show that among the heuristics we analyzed, few of them cannot be applied to very large graphs, when the iterative strategy is used, due to their time complexity. Finally, we suggest possible directions to further improve the heuristic providing the best results.

Comparative analysis of centrality measures for identifying critical nodes in complex networks

One of the fundamental tasks in complex networks is detecting critical nodes whose removal significantly disrupts network connectivity. Identifying critical nodes can help analyze the topological characteristics of the network, such as vulnerability and robustness. This work considers a well-known critical node detection problem variant, Maximize the Number of Connected Components Problem, which aims to find a set of nodes whose removal maximizes the number of connected components and compares the centrality measures for detecting these nodes. While the existing literature focused only on small datasets, this work analyzes the widely used topology-based centrality measures on several synthetic and real-world networks. Our findings show that degree-like centralities are more relevant measures than path-like centralities for disconnecting networks into several connected components. However, our results also indicate that the traditional centrality measures cannot detect the most vital critical nodes. To overcome this drawback, a new centrality measure, namely Isolating Centrality, that aims to identify the nodes that significantly impact network connectedness is presented. The comprehensive computational study demonstrates that the proposed measure outperforms traditional measures in identifying critical nodes.

Approaching the bi-objective critical node detection problem with a smart initialization-based evolutionary algorithm.

Determining the critical nodes in a complex network is an essential computation problem. Several variants of this problem have emerged due to its wide applicability in network analysis. In this article we study the bi-objective critical node detection problem (BOCNDP), which is a new variant of the well-known critical node detection problem, optimizing two objectives at the same time: maximizing the number of connected components and minimizing the variance of their cardinalities. Evolutionary multi-objective algorithms (EMOA) are a straightforward choice to solve this type of problem. We propose three different smart initialization strategies which can be incorporated into any EMOA. These initialization strategies take into account the basic properties of the networks. They are based on the highest degree, random walk (RW) and depth-first search. Numerical experiments were conducted on synthetic and real-world network data. The three different initialization types significantly improve the performance of the EMOA.

The minimum cost network upgrade problem with maximum robustness to multiple node failures

The design of networks which are robust to multiple failures is gaining increasing attention in areas such as telecommunications. In this paper, we consider the problem of upgrading an existent network in order to enhance its robustness to events involving multiple node failures. This problem is modeled as a bi-objective mixed linear integer formulation considering both the minimization of the cost of the added edges and the maximization of the robustness of the resulting upgraded network. As the robustness metric of the network, we consider the value of the Critical Node Detection (CND) problem variant which provides the minimum pairwise connectivity between all node pairs when a set of c critical nodes are removed from the network. We present a general iterative framework to obtain the complete Pareto frontier that alternates between the minimum cost edge selection problem and the CND problem. Two different approaches based on a cover model are introduced for the edge selection problem. Computational results conducted on different network topologies show that the proposed methodology based on the cover model is effective in computing Pareto solutions for graphs with up to 100 nodes, which includes four commonly used telecommunication networks.

A heuristic approach for the distance-based critical node detection problem in complex networks

The distance-based critical node problem involves identifying a subset of nodes in a network whose removal minimises a pre-defined distance-based connectivity measure. Having the classical critical node problem as a special case, the distance-based critical node problem is computationally challenging. In this article, we study the distance-based critical node problem from a heuristic algorithm perspective. We consider the distance-based connectivity objective whose goal is to minimise the number of node pairs connected by a path of length at most k, subject to budgetary constraints. We propose a centrality based heuristic which combines a backbone-based crossover procedure to generate good offspring solutions and a centrality-based neighbourhood search to improve the solution. Extensive computational experiments on real-world and synthetic graphs show the effectiveness of the developed heuristic in generating good solutions when compared to exact solution. Our empirical results also provide useful insights for future algorithm development.

Fortification Against Cascade Propagation Under Uncertainty

Network cascades represent a number of real-life applications: social influence, electrical grid failures, viral spread, and so on. The commonality between these phenomena is that they begin from a set of seed nodes and spread to other regions of the network. We consider a variant of a critical node detection problem dubbed the robust critical node fortification problem, wherein the decision maker wishes to fortify nodes (within a budget) to limit the spread of cascading behavior under uncertain conditions. In particular, the arc weights—how much influence one node has on another in the cascade process—are uncertain but are known to lie in some range bounded by a worst-case budget uncertainty. This problem is shown to be [Formula: see text]-hard even in the deterministic case. We formulate a mixed-integer program (MIP) to solve the deterministic problem and improve its continuous relaxation via nonlinear constraints and convexification. The robust problem is computationally more difficult, and we present an MIP-based expand-and-cut exact solution algorithm, in which the expansion is enhanced by cutting planes, which are themselves tied to the expansion process. Insights from these exact solutions motivate two novel (interrelated) centrality measures, and a centrality-based heuristic that obtains high-quality solutions within a few seconds. Finally, extensive computational results are given to validate our theoretical developments as well as provide insights into structural properties of the robust problem and its solution.

Efficient methods for the distance-based critical node detection problem in complex networks

An important problem in network survivability assessment is the identification of critical nodes. The distance-based critical node detection problem addresses the issues of internal cohesiveness and actual distance connectivity overlooked by the traditional critical node detection problem. In this study, we consider the distance-based critical node detection problem which seeks to optimise some distance-based connectivity metric subject to budgetary constraints on the critical node set. We exploit the structure of the problem to derive new path-based integer linear programming formulations that are scalable when compared to an existing compact model. We develop an efficient algorithm for the separation problem that is based on breadth first search tree generation. We also study some valid inequalities to strengthen the formulations and a heuristic to improve primal bounds. We have applied our models and algorithm to two different classes of the problems determined by the distance based connectivity functions. Extensive computational experiments on both real-world and randomly generated network instances, show that the proposed approach is computationally more efficient than the existing compact model especially for larger instances where connections between nodes consist of a small number of hops. Our computational experiments on both classes of distance-based critical node detection problem provide good numerical evidence to support the importance of defining appropriate metrics for specific network applications.

MEA-CNDP: A Membrane Evolutionary Algorithm for Solving Biobjective Critical Node Detection Problem.

The critical node detection problem (CNDP) refers to the identification of one or more nodes that have a significant impact on the entire complex network according to the importance of each node in a complex network. Most methods consider the CNDP as a single-objective optimization problem, which requires more prior knowledge to a certain extent. This paper proposes a membrane evolution algorithm MEA-CNDP to solve biobjective CNDP. MEA-CNDP includes a population initialization strategy based on the evaluation of decision variables, a strategy to transform the main objective, a strategy to update the membrane inherited pool, and four membrane evolutionary operators. The numerical experiments on 16 benchmark problems with random and logarithmic weights show that MEA-CNDP outperforms other algorithms in most cases. In particular, MEA-CNDP has unique advantages in dealing with large-scale sparse bi-CNDP.

Efficient methods for the distance-based critical node detection problem in complex networks

Efficient methods for the distance-based critical node detection problem in complex networks

EIA-CNDP: An exact iterative algorithm for critical node detection problem

In designing reliable and impermeable networks, the robustness of the network is evaluated against the removal and failure of the node or edge where the network robustness (network connectivity) is measured using various metrics (objective functions) such as the number of connected components, size of the largest connected component, and pairwise connectivity. Critical node detection problem (CNDP) is one of the main issues in this literature, which aims to find a set of vertices whose removal maximizes or minimizes some objective function. In this paper, the focus is on solving CNDP, considering the size of the largest connected component as its objective function. In this regard, we introduce a new problem called K-Group-Division-Problem and present a mixed integer linear programming model to solve it. We prove that under certain circumstances, any optimal solution of the new problem is also an optimal solution of CNDP. Analyzing the performance of the proposed model on solving CNDP, indicates that this model is highly competitive against the base model in the literature. Furthermore, a novel exact algorithm is introduced which improves the proposed mixed integer linear programming model to address CNDP more efficiently. The results show that the proposed algorithm is much more efficient, and, compared with the base model, it can solve the problem on networks with a higher number of nodes.

Critical node detection problem for complex network in undirected weighted networks

Detection of critical nodes in complex networks has recently received extensive attention. Currently, studies of the critical nodes problem (CNP) mainly focus on two problem types: “critical nodes problem/positive” (CNP-Pos) and “critical nodes problem/negative” (CNP-Neg). However, to the best of our knowledge, few studies have been conducted on CNP-Neg for weighed networks. In this paper, we investigate CNP-Neg in undirected weighted networks. We first propose a novel metric DFW to evaluate network fragmentation. Then, we formulate a new nonconvex mixed-integer quadratic programming model, named MIQPM, that aims to simultaneously minimize pairwise connectivity and maximize the weights between the nodes. After that, a general greedy algorithm is employed to solve the corresponding optimization problem. Finally, comparison experiments are carried out for several synthetic networks and four real-world networks to demonstrate the effectiveness of the proposed approaches.

Efficient Benders decomposition for distance-based critical node detection problem

This paper addresses the critical node detection problem which seeks a subset of nodes for removal in order to maximize the disconnectivity of the residual graph with respect to a specific distance-based measure, namely the Wiener index. Such a measure is defined based on the all-pair shortest path distances in the residual graph so that the longer the total length of shortest paths, the greater the value of the disconnectivity measure. In the literature, a mixed integer linear programming model and an exact iterative-based method have been presented for this problem; however, both approaches become very time-consuming on graphs having large diameter and non-unit edge lengths. To overcome this shortcoming, in this paper, we present a new formulation for the problem and solve it by Benders decomposition algorithm. We improve the performance of Benders algorithm by several techniques (including analytical calculation of dual variables, generation of good-quality initial optimality cuts, considering master's optimality cuts as lazy constraints, etc.) to reduce the total running time. The extensive computational experiments on instances, taken from the literature or generated randomly, confirm the effectiveness of the new approaches.

The bi-objective critical node detection problem with minimum pairwise connectivity and cost: theory and algorithms

An effective way to analyze and apprehend structural properties of networks is to find their most critical nodes. This paper studies a bi-objective critical node detection problem, denoted as Bi-CNDP. In this variant, we do not make any assumptions on the psychology of decision makers and seek to find a set of solutions which minimize the pairwise connectivity of the induced graph and the cost of removing these critical nodes at the same time. After explicitly stating the formulation of Bi-CNDP, we first prove the NP-hardness of this problem for general graphs and the existence of a polynomial algorithm for constructing the $$\varepsilon $$ -approximated Pareto front for Bi-CNDPs on trees. Then different approaches of determining the mating pool and the replacement pool are proposed for the decomposition-based multi-objective evolutionary algorithms. Based on this, two types of decomposition-based multi-objective evolutionary algorithms (MOEA/D and DMOEA- $$\varepsilon \hbox {C}$$ ) are modified and applied to solve the proposed Bi-CNDP. Numerical experiments on sixteen famous benchmark problems with random and logarithmic weights are firstly conducted to assess different types of the mating pool and the replacement pool. Besides, computational results between two improved algorithms, i.e., I-MOEA/D and I-DMOEA- $$\varepsilon \hbox {C}$$ , demonstrate that they behave differently on these instances and I-DMOEA- $$\varepsilon \hbox {C}$$ shows better performance on the majority of test instances. Finally, a decision-making process from the perspective of minimizing the pairwise connectivity of the induced graph given a constraint on the cost of removing nodes is presented for helping decision makers to identify the most critical nodes for further protection or attack.