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 stateoftheart 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 (HPINcut). The results on real and artificial networks show that the HPINcut 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
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 [
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 [
Recently, similar definition of the cost [
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 stateoftheart node removal algorithms on real networks and results show that with respect to this concept of cost, most stateoftheart 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
The structure of this paper is organized as follows. First, in Section
In this section, we describe data sets and some existing stateoftheart targeted attack algorithms. Among them, the node removalbased 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 costfragmentation effectiveness (DCFE) as an index to measure the performance of different attacking methods. At last, we introduce the degree costfragmentation effectiveness measure and present the HPINcut method.
To evaluate the performances of the network dismantling (fragmentation) algorithms, we used both real networks and synthetic networks in this paper: (a)
Basic statistical features of the GCCs of the eight real and synthetic networks.
Network  Nodes  Links  Avg. degree  Sparsity 

Political Blogs (PB)  1222  16714  27.36 

Petsterhamster (PH)  2000  16098  16.10 

Power Grid (PG)  4941  6594  2.67 

Autonomous Systems (AS)  6474  12572  3.88 

ER  2500  12500  10.00 

SF ( 
10000  23423  4.68 

SF ( 
10000  11761  2.35 

SBM  4232  5503  2.60 

In this subsection, we will briefly introduce stateoftheart node removal attack algorithms and some edge evaluation methods which are used in this paper. We also employ several baselines methods for edge based attacks, which are based on the random edge removal and sequential removal of edges with high betweenness and bridgeness measures.
Percolation method: in the study of the network attacks, percolation [
High degree (HD) method [
Equal graph partitioning (EGP) algorithm: EGP algorithm [
Collective Influence (CI) algorithm: CI algorithm [
MinSum algorithm: the threestage MinSum algorithm [
CoreHD algorithm: inspired by MinSum algorithm, CoreHD algorithm [
Belief propagationguided decimation (BPD) [
Edge betweenness [
Bridgeness [
The robustness of the network structure can be measured in different means, but a common way is to characterize the function of the size of the largest connected component (GCC) with respect to the ratio of the removed nodes or edges, that is, cost. Characterization of this function was done in two distinct ways: (i) by the value of critical point when the largest component completely collapses [
We make explicit assumption that the cost of removing a node is proportional to the degree or to the number of the adjacent edges that have to be removed. Let us define the function
Here we define the
In this section, we introduce and describe the Hierarchical Power Iterative Normalized cut for edge removal strategy (HPINcut). Thus, if the edge removal actions on networks are applicable, we compare them with the same definition of the cost to the nodebased strategies. The link fragmentation problem can be narrated as follows: if we have a budget of
We applied the spectral strategy for edge attack problem, which fall in the class of wellknown spectral clustering and partitioning algorithms [
Now, we describe the hierarchical iterative algorithm for edge removing. This algorithm hierarchically applies the spectral bisection algorithm, which has the same objective function as the normalized cut algorithm [
Input: adjacency matrix
Output: a separated set of edges that partition the network into two disconnected clusters
compute the eigenvector
put all the nodes with
The clusters that we obtained by this method had usually very balanced sizes. If, however, it is very important to get clusters of exactly the same size, one could put those
Input: adjacency matrix of a network
Output: partition of the network into small groups:
partition the GCC of the network into two disconnected clusters
if the budget for link removal has not been overrun and if the GCC is not yet small enough, partition
The reason why we cluster hierarchically is because this allows us to refine the fragmentation gradually. For example, if, after partitioning the network into
Input: adjacency matrix
Output: the eigenvector
draw
set
for
The main reason we used this objective function is that it minimizes the number of links that are removed and the total sum of node degree centralities in both partitions
In this section, we compare existing node targeting attack strategies with respect to the new definition of cost. We make explicit assumption that the cost of removing a node is proportional to the number of the adjacent edges that have to be removed. This suggests that the nodes with higher degree have higher associated removal cost.
By taking into the account the degreebased cost in targeted attacks, the results can be highly counterintuitive. The performances of the stateoftheart node removalbased methods are in some cases even worse than the naive process of random removal of nodes (site percolation), when we take into account the attack cost, as shown in Figures
The size of the GCC of the networks versus the link removing proportion, comparing with classical node removalbased methods on real networks. The results of the site percolation are obtained after 100 independent runs.
Power Grid
Political Blogs
Petsterhamster
Autonomous Systems
The size of the GCC of the networks versus link removing proportion, comparing with classical node removalbased methods on artificial networks. The results of the site percolation are obtained after 100 independent runs.
ER
SF
SF
SBM with ten clusters
Table
DCFE, that is, the area under the curve of the size of the GCC after attacking by different algorithms.
DCFE 

HD  HDA  EGP  CI  MinSum  CoreHD  BPD 

Betw  Bridg  HPINcut 

PB  0.638  0.920  0.861  0.619  0.657  0.726  0.815  0.722  0.843  0.597  0.910 

PH  0.627  0.677  0.696  0.747  0.687  0.675  0.736  0.661  0.817  0.536  0.689 

PG  0.371  0.260  0.293  0.263  0.219  0.130  0.256  0.113  0.305  0.145  0.420 

AS  0.567  0.576  0.604  0.592  0.567  0.567  0.576  0.561  0.605  0.527  0.618 

ER  0.601  0.547  0.647  0.502  0.441  0.268  0.647  0.247  0.753  0.387  0.542 

SF ( 
0.619  0.700  0.706  0.671  0.650  0.636  0.660  0.632  0.683  0.672  0.694 

SF ( 
0.406  0.231  0.228  0.343  0.214  0.202  0.227  0.192  0.298  0.312  0.352 

SBM  0.487  0.419  0.378  0.397  0.348  0.284  0.374  0.274  0.384  0.348  0.512 

The improvement of the DCFE of each algorithm, comparing with the baseline, that is, site percolation method. The best performing algorithm in each column is emphasized in bold.
Improvement 

HD  HDA  EGP  CI  MinSum  CoreHD  BPD  Betw  Bridg  HPINcut 

PB  −32%  −44%  −35%  3%  −3%  −14%  −28%  −13%  6%  −43% 

PH  −30%  −8%  −11%  −19%  −10%  −8%  −17%  −5%  15%  −10% 

PG  18%  30%  21%  29%  41%  65%  31%  70%  61%  −13% 

AS  −7%  −2%  −7%  −4%  0  0  −2%  1%  7%  −9% 

ER  −25%  9%  −8%  17%  27%  55%  −8%  59%  36%  10% 


−10%  −13%  −14%  −8%  −5%  −3%  −7%  −2%  −9%  −12% 


27%  43%  44%  15%  47%  50%  44%  53%  23%  13% 

SBM  20%  13%  22%  18%  28%  41%  23%  44%  28%  −6% 



Average  −5%  3%  2%  6%  16%  23%  5%  26%  21%  −9% 

In this section, we will compare the proposed edge removalbased attack strategy, HPINcut algorithm, with random uniform attack, edge betweenness, bridgeness, and some classical node removing strategies (see the details in Section
In general case, each attack strategy algorithm could generate a ranking list of all (or partial) nodes or links of the network. After removing the nodes or links one after another, the size of the GCC of the residual network characterizes the effectiveness of each algorithm. The removal process will cease when the size of the GCC is smaller than a given threshold (here we use 0.01). In this paper, to test the effectiveness of this spectral edge removal algorithm, HPINcut, we plot the size of the GCC versus the removal fraction of links, for both real networks (Figures
The size of the GCC of the networks versus link removing proportion, comparing with existing link removalbased methods on real networks. The results of the bond percolation are obtained after 100 independent runs.
Power Grid
Political Blogs
Petsterhamster
Autonomous Systems
The size of the GCC of the networks versus link removing proportion, comparing with existed link removalbased methods on artificial networks. The results of the bond percolation are obtained after 100 independent runs.
ER
SF
SF
SBM with ten clusters
In Figures
To conclude the results of Figures
In Figures
To more intuitively display the ability of the HPINcut to make immunization of links, we studied the susceptibleinfectiousrecovery (SIR) [
The spreadability of the networks before and after the removing of 10% edges by HPINcut algorithm. The
Political Blogs
Power Grid
Petsterhamster
Autonomous Systems
To summarize, we investigated some stateoftheart node target attack algorithms and found that they are very inefficient when the degreebased cost of the attack is taken into consideration. The cost of removing a node is defined as the number of links that have to be removed in the attack process.
We found some highly counterintuitive results; that is, the performances of the stateoftheart node removalbased methods are even worse than the naive site percolation method with respect to the limited cost. This demonstrates that the current stateoftheart node targeted attack strategies underestimate the heterogeneity of the cost associated with the nodes in complex networks.
Furthermore, in cases when the link removal strategies are possible, we compared the performances of the nodecentric (HD, HDA, EGP, CI, CoreHD, BPD, and MinSum) and edge removal strategies (edge betweenness and bridgeness strategy) based on the cost of their attacks, which are measured in the same units, that is, the ratio of the removed links. We propose a hierarchical power iterative algorithm (HPINcut) to fragment a network, which has the same objective function with the Ncut [
The underestimated cost of current stateoftheart algorithms with respect to the degreebased cost has high influence on the development and design of better robustness and resilience mechanisms in complex systems. Furthermore, more accurate estimation of robustness under realistic conditions will allow better allocation of response resources.
Let
Finding such a set
Find
where we have imposed the condition
Set
and define
The idea behind this method is that
One can show that a solution to (
If
The complexity of the spectral bisection algorithm is the same as the complexity of the power iteration method. The complexity of the power iteration method equals the number of iterations
Assuming that the spectral bisection algorithm always produces clusters of equal size, the complexity of the hierarchical spectral clustering algorithm is then given by the sum of
the complexity of applying spectral bisection once on the whole network
The complexity of applying it on each of the two clusters that we obtained from the first application of spectral bisection and which will have size
The complexity of applying it on each of the 4 clusters that we obtained from the previous step and which will have size
The complexity of applying it on each of the
That is, in total at most
The choice of the function
Another condition that might slow down the computation of
Due to this fast convergence, one can expect asymptotically good partitions when
Previous sections give us a clear picture about the performances of different attack algorithms. Some algorithms work quite well, such as HPINcut algorithm, MinSum algorithm, and edge betweenness algorithm, while others are not. What causes such a difference? Figure
The schematic diagram of the removed links in a SBM network with two clusters. (a) is the original network with all the links. (b)–(f) are the top 10% links (i.e., 373 links) removed by different algorithms.
Original network
HDA
EGP
CI
Betweenness
HPINcut
In the previous sections, the default target number of the disconnected clusters in HPINcut algorithm is set to 2. Figure
The size of the GCC of the networks versus link removing proportion, comparing of different quantities of target disconnected clusters in HPINcut algorithm.
SBM with two clusters
SBM with ten clusters
The authors declare that they have no conflicts of interest.
The work of Nino AntulovFantulin has been funded by the EU Horizon 2020 SoBigData project under Grant Agreement no. 654024. The work of Dijana Tolić is funded by the Croatian Science Foundation IP2013119623 “Machine Learning Algorithms for Insightful Analysis of Complex Data Structures.” XiaoLong Ren acknowledges the support from China Scholarship Council (CSC).