An elitism-based multi-objective evolutionary algorithm for min-cost network disintegration

Highlights

• A multi-objective disintegration algorithm considering actual cost factor is proposed.

• By combining cost and node importance, a new unit cost importance measure is defined.

• The combination information of nodes is added by ingenious elitism and update strategy.

• Our algorithm performs better than five other state-of-the-art disintegration methods.

Abstract

Network disintegration or strengthening is a significant problem, which is widely used in infrastructure construction, social networks, infectious disease prevention and so on. But most studies assume that the cost of attacking anyone node is equal. In this paper, we investigate the robustness of complex networks under a more realistic assumption that costs are functions of degrees of nodes. A multi-objective, elitism-based, evolutionary algorithm (MOEEA) is proposed for the network disintegration problem with heterogeneous costs. By defining a new unit cost influence measure of the target attack node and combining with an elitism strategy, some combination nodes’ information can be retained. Through an ingenious update mechanism, this information is passed on to the next generation to guide the population to move to more promising regions, which can improve the rate of convergence of the proposed algorithm. A series of experiments have been carried out on four benchmark networks and some model networks, the results show that our method performs better than five other state-of-the-art attack strategies. MOEEA can usually find min-cost network disintegration solutions. Simultaneously, through testing different cost functions, we find that the stronger the cost heterogeneity, the better performance of our algorithm.

Introduction

Modern societies have many key networks such as transportation networks, power transmission networks, interconnected social networks, etc., which have huge impacts on the quality of life; thus protecting their robustness and integrity becomes crucially important [1], [2]. On the other hand, infectious disease networks, criminal networks, and terrorist organization networks also exist. Such undesirable networks should be controlled and disintegrated so as to minimize their detrimental effects on society. Therefore, studying the robustness and the weakness of networks is of practical importance [3], [4].

Network analysis suggests that the network integrity appears to be heavily related to a small number of skeleton nodes (key nodes), which seems to maintain the framework and the performance of network [5], [6]. In essence, the problem of network disintegration is equivalent to finding the optimal (minimum) set of these key nodes that may strongly influence the structural integrity of the network [7], [8]. Consequently, we can enhance the robustness of the network by protecting these key nodes to ensure the smooth running of the transportation and logistics networks. On the other hand, the removal of some key nodes can maximally fragment some undesirable networks [9], which potentially provides key insight into the ways of controlling diseases and isolating certain network nodes [10].

Network robustness has received much attention in the past. The network disintegration is a very challenging problem; in fact, it is a non-deterministic polynomial-time (NP) hard. Thus, it is unlikely to have any efficient methods, most network attack strategies are still based on heuristic ranking to identify influencing nodes [11]. Subsequently, various methods were proposed for tackling this problem, which were usually approximation algorithms based on different theories or assumptions, including the optimal percolation theory [12], module-based attacks [13], the collective influence (CI) algorithm [14], or the Min-Sum algorithm [7].

These methods based on the centrality of nodes treat the optimal dismantling set as a collection of “well-performing” nodes. However, This problem is essentially a collective problem [7]. Moreover, these methods are not global optimization methods and cannot guarantee the optimality of the solution [15]. In addition, studies have shown that many weakly connected nodes that may be critical to the network structure were previously ignored [16]. Subsequently, metaheuristic algorithms were used to solve the network dismantling problem [17], [18]. These black-box types of global optimization algorithms were considered to be more suitable for this problem.

However, an important issue is that most of the existing studies on network robustness have an implicit assumption: the removal cost of any node in the network is equal, regardless of its centrality or importance. This assumption is not valid for many real-world networks. For example, the removal of a hub node can be more costly than the other nodes. Also, it is more difficult and costly to arrest the leader of a terrorist organization than its ordinary members. In addition, when an infectious disease breaks out, it is more effective to isolate super-spreaders without symptoms than the simple isolation of ordinary patients. In the case of the limited resources and environment, it is more realistic to take account of the heterogeneous costs in the strategy or model. Recent attempts in this respect have been carried out, the cost of protecting or attacking a node was defined as a function of its degree [15], [19], but there are still some shortcomings, which we will explain in detail later.

A crucial issue is that existing methods do not really incorporate the cost factor into the optimization process. When the cost function of the nodes changes, the solutions obtained by these methods will not change accordingly. So it is difficult to find a low-cost and efficient set of key nodes. It can be expected that attacking a higher-degree node in a social system can usually incur a higher cost than the same operation on a lower-degree node. A reasonable attack strategy should find the optimal node or nodes with lower costs, and the removal may cause more damage to the network. However, the crux of the matter is that the lower costs and higher damage seem to be conflicting. Existing methods struggle to deal with this type of problem.

In fact, multi-objective metaheuristic algorithms are very powerful for solving these challenging problems [20] and good at dealing with several conflicting objectives [21], [22]. In contrast with single objective optimization, there are multiple optimal solutions in multi-objective optimization, which form the so-called Pareto Front. When the decision-maker needs to consider some practical constraints, such as the execution budget and conditions, Pareto optimal solutions can provide different choices or options for the decision-maker.

Motivated by the above challenges, we now take a multi-objective metaheuristic approach to deal with such a problem. This paper proposes an elitism-based multi-objective evolutionary algorithm for network disintegration problem with heterogeneous costs. In essence, the problem is treated as a bi-objective problem: one objective is the cost of protecting or removing/attacking nodes; the other is the extent to which the network is disintegrated after removing the nodes. We assume that the removal cost of a node can be either an exponential function or a power function of the node’s degree. Our proposed approach intends to find a set of nodes with the lowest removal costs for network collapse and simultaneously try to provide more suitable choices for decision-makers with limited resources. Thus, the main contributions of this work can be summarized as follows:

• The network disintegration problem with heterogeneous costs is formulated for the first time as a bi-objective problem by incorporating the cost as one of the objective functions. A multi-objective, elitism-based, evolutionary algorithm (MOEEA) is proposed to solve this problem, which can find a set of better solutions efficiently.

• A new unit cost importance measure is defined, which combines attack cost and node importance to provide a measure for the comparison and selection of nodes in multi-objective problems.

• The combination influence of nodes has been considered to the proposed algorithm. A reservation mechanism combining a unit cost influence measure is proposed for elite individuals. The reserved key node combination information will participate in each offspring’s individual generation process and guide the population to move to more promising regions.

• The parameter $N_s$ is used to convert between local search and global search to achieve a balance between exploitation and exploration.

Therefore, this paper is organized as follows. Section 2 discusses the recent developments, whereas Section 3 introduces some background concerning the estimation of measures and the optimization formulation for complex network disintegration with heterogeneous costs. Section 4 focuses on the proposed elitism-based multi-objective evolutionary algorithm, including the initialization, elitism strategy, update mechanism, and the realization of non-dominant solutions and selection mechanisms. Then, Section 5 presents a series of experiments on real networks and widely used network models, followed by a summary of the comparison with five other methods. Algorithm analysis, including parametric studies, convergence property and complexity analysis are carried out in Section 6. Then, some discussions about preprocessing are conducted in Section 7. Finally, Section 8 draws some conclusions and discusses briefly the relevant topics for further research.