Complexity

Volume 2018, Article ID 9826243, 15 pages

https://doi.org/10.1155/2018/9826243

## Underestimated Cost of Targeted Attacks on Complex Networks

Correspondence should be addressed to Nino Antulov-Fantulin; hc.zhte.sseg@volutna.onin

Received 25 August 2017; Revised 7 December 2017; Accepted 17 December 2017; Published 17 January 2018

Academic Editor: Ilaria Giannoccaro

Copyright © 2018 Xiao-Long Ren et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The robustness of complex networks under targeted attacks is deeply connected to the resilience of complex systems, which is defined as the ability to make appropriate response to the attack. In this paper, we study robustness of complex networks under a realistic assumption that the cost of removing a node is not constant but rather proportional to the degree of a node or equivalently to the number of removed links a removal action produces. We have investigated the state-of-the-art targeted node removing algorithms and demonstrate that they become very inefficient when the cost of the attack is taken into consideration. For the case when it is possible to attack or remove links, we propose a simple and efficient edge removal strategy named Hierarchical Power Iterative Normalized cut (HPI-Ncut). The results on real and artificial networks show that the HPI-Ncut algorithm outperforms all the node removal and link removal attack algorithms when the same definition of cost is taken into consideration. In addition, we show that, on sparse networks, the complexity of this hierarchical power iteration edge removal algorithm is only .

#### 1. Introduction

The ability of complex system to dynamically adapt to internal failures or external disturbances is called a resilience. The adaptation is connected to the robustness of the network structure [1], which is defined as the ability to maintain functionality without adaptation to internal failures or external disturbances (attacks). In this paper, we will focus on the robustness of complex networks under targeted attacks with the more realistic cost function. Robustness of connected components under random failure of nodes or links is described with the classical percolation theory [2, 3]. Percolation is the simplest process showing a continuous phase transition, scale invariance, fractal structure, and universality and it is described with just a single parameter, that is, the probability of removing a node or edge. Network science studies have demonstrated that scale-free networks [4, 5] are more robust than random networks [6, 7] under random attacks or failures but less robust under targeted attacks [8–12]. Recently, studies of network resilience have moved their focus to more realistic scenarios of interdependent networks [13], competing networks [14], different failure [15], and recovery [16, 17] mechanisms.

Although the study of network robustness has received a huge amount of attention, the majority of the targeted attack strategies are still based on the heuristic identification of influential nodes [11, 18–21] with no performance guarantees for the optimality of the solution. Finding the minimal set of nodes such that their removal maximally fragments the network is called the network dismantling problem [22, 23] and it belongs to the NP-hard class. Thus no polynomial-time algorithm has been found for it and only recently different state-of-the-art methods were proposed as approximation algorithms [22–28] for this task. Although state-of-the-art methods show promising results for network dismantling, we take one step back and analyze the implicit assumption these network dismantling algorithms have. The implicit assumption that the cost of a removing action is equivalent for all nodes regardless of their importance or centrality in network is not a realistic one. Attacking a central node, for example, a high degree node in sociotechnical systems, usually comes with the higher additional cost when compared to the same action on a low degree node. Therefore, it is more realistic to explicitly assume that the cost of an attack is heterogeneous. In this paper, we define the cost of removing a node as a function of its degree.

Recently, similar definition of the cost [29] was used to analyze fragmentation and strengthening process for a class of random network models. Under the assumption of the random network models, they found that the optimal cost for fragmentation and strengthening process consists out of the list of priorities of degrees for removed nodes which is independent of the network’s degree distribution.

In this work, we make the explicit assumption that the cost of an attack is proportional to the degree of a node or equivalently to the number of adjacent links a removed node has. We investigated different state-of-the-art node removal algorithms on real networks and results show that with respect to this concept of cost, most state-of-the-art algorithms are very inefficient and in most instances perform even worse than the random removal strategy for a fixed finite budget of cost.

Furthermore, when edge removal attacks are possible, we compare them to the node removal strategies with respect to the same definition of cost, that is, the number of removed links needed to fragment the network. Note that removing a node is equivalent to removing all the edges of that node, and therefore all node removal actions can be reproduced with the edge removal strategy but vice versa does not hold. Therefore, we also make highlight that the comparisons between node and edge based strategies are only interpretable in cases when edge based attacks are possible. In that case, we propose and use an edge removal strategy, named the* Hierarchical Power Iterative Normalized cut* (HPI-Ncut) as one of the possible solutions to overcome the large fragmentation cost. Although edge based strategies have higher degree of freedom as they can remove only a fraction of edges adjacent to the node, still we find cases where node-based strategies can outperform the edge based strategies. However, our proposed method (HPI-Ncut) always outperforms all the state-of-the-art targeted node-based attack algorithms and edge removal strategies [18, 27, 30].

The structure of this paper is organized as follows. First, in Section 2 (“Materials and Methods”), we introduce the empirical and artificial networks that are used in this paper (Section 2.1), present and describe current targeted attack strategies (Section 2.2), define a degree cost-fragmentation measure (Section 2.3), and describe the proposed HPI-Ncut method (Section 2.4). Then, in Section 3 (“Results and Discussions”), we quantify the cost of the state-of-the-art node removal strategies and show that in most cases the cost of such attacks is inefficient with respect to the degree-based definition of cost (Section 3.1). These results have important impact for real world scenarios of network fragmentations where cost budget is limited. Finally, when it is possible to remove single edges (e.g., shielding communication links, removing power lines, cutting off trading relationships, or others), we use the proposed HPI-Ncut method and compare its performances with other strategies (Section 3.2). The effect of edge removal HPI-Ncut method as an immunization measure for the epidemic spreading process on networks is presented (Section 3.3).

#### 2. Materials and Methods

In this section, we describe data sets and some existing state-of-the-art targeted attack algorithms. Among them, the node removal-based attack algorithms are designed to dismantle the network into pieces with no thought for the cost of the attacking. In other words, these algorithms consider all the nodes have uniform cost. We also introduce the edge betweenness and bridgeness, which originally proposed evaluateing the importance of nodes, as two comparable link attacking methods. Then, we define the degree cost-fragmentation effectiveness (DCFE) as an index to measure the performance of different attacking methods. At last, we introduce the degree cost-fragmentation effectiveness measure and present the HPI-Ncut method.

##### 2.1. Data Sets

To evaluate the performances of the network dismantling (fragmentation) algorithms, we used both real networks and synthetic networks in this paper: (a)* Political Blogs* [31] which is an undirected social network that was collected around the time of the US presidential election in 2004. This network is a relatively dense network whose average degree is 27.36; (b)* Petster-hamster* which is an undirected social network which contains friendships and family links between users of the website http://hamsterster.com. This network data set can be downloaded from KONECT (http://konect.uni-koblenz.de/networks/petster-hamster); (c)* Power Grid* [32] which is an undirected Power Grid network in which a node is either a generator, a transformer, or a substation, while a link represents a transmission line. This network data set can also be downloaded from KONECT (http://konect.uni-koblenz.de/networks/opsahl-powergrid); (d)* Autonomous Systems* is an undirected network from the University of Oregon Route Views Project [33]. This network data set can be downloaded from SNAP (https://snap.stanford.edu/data/as.html); (e) Erdös and Rényi (ER) network [34] is constructed with 2500 nodes. Its average degree is 20 and the connection probability is 0.01; (f) Scale-free (SF) network with size 10,000, exponent 2.5, and average degree 4.68; (g) Scale-free (SF) network with size 10,000, exponent 3.5, and average degree 2.35; (h) stochastic block model (SBM) with ten clusters is an undirected network with 4232 nodes and average degree 2.60. The basic properties of these networks are listed in Table 1.