An improved vulnerability index of complex networks based on fractal dimension

An improved vulnerability index of complex networks based on fractal dimension

Li Gou, Bo Wei, Rehan Sadiq, Sankaran Mahadevan, Yong Deng111Corresponding author: Yong Deng, School of Computer and Information Science, Southwest University, Chongqing, 400715, China. Email address:; School of Computer and Information Science,
Southwest University, Chongqing 400715, China
School of Engineering, University of British Columbia Okanagan,
3333 University Way, Kelowna, BC, Canada V1V 1V7
School of Engineering, Vanderbilt University, Nashville, TN 37235, USA

With an increasing emphasis on network security, much more attention has been attracted to the vulnerability of complex networks. The multi-scale evaluation of vulnerability is widely used since it makes use of combined powers of the links’ betweenness and has an effective evaluation to vulnerability. However, how to determine the coefficient in existing multi-scale evaluation model to measure the vulnerability of different networks is still an open issue. In this paper, an improved model based on the fractal dimension of complex networks is proposed to obtain a more reasonable evaluation of vulnerability with more physical significance. Not only the structure and basic physical properties of networks is characterized, but also the covering ability of networks, which is related to the vulnerability of the network, is taken into consideration in our proposed method. The numerical examples and real applications are used to illustrate the efficiency of our proposed method.

vulnerability, fractal dimension, complex networks
journal: Safety Science

1 Introduction

Complex network are widely used to model the structure of many complex systems in nature and society kim2012analysis (); wang2009complex (); tang2011detecting (); qi2010efficiency (); zhang2013self (); holme2002attack (); MonfaredPA2014 (). An open issue is how to assess the vulnerability of complex networks holmgren2006using (); boccaletti2007multiscale (); zhang2012attack (); wang2013vulnerability (), whose main objective is to understand, predict, and even control the behavior of a networked system under vicious attacks or any types of dysfunctions boccaletti2007multiscale (); zhang2013route ().

Different approaches to characterize network vulnerability and robustness have recently been proposed, which can be grouped into two types broadly mishkovski2011vulnerability (); ouyang2014correlation (). The first type of approach is related to structural robustness mishkovski2011vulnerability (); albert2004structural (); albert2002statistical (): how topological properties of networks are affected by the removal of a finite number of vertexes or/and links, such as the degree distribution, the network connectivity level, the size of largest component, the average geodesic length and etc. The second type of method concerns dynamical robustness holme2002attack (); motter2002cascade (); crucitti2004model (); wang2009cascade (); wang2013robustness (). The removal of a vertex or link will cause the flow to redistribution with the risk that some other vertexes or links may be overloaded, which can cause a sequence of failures and even threaten the global stability. Such behavior is called cascading failures mishkovski2011vulnerability (); wang2009vulnerability (); wang2013improving (); wang2011robustness ().

One of the mostly used methods is proposed by Boccaletti boccaletti2007multiscale (). They construct a multi-scale evaluation model of vulnerability, which makes use of combined powers of the links’ betweenness. Due to the simplicity and efficiency, this method is heavily studied mishkovski2011vulnerability (). One limitation of original model is that it cannot discriminate two different networks in some situations. To solve this problem, a coefficient is introduced to improve the original model. However, a straight problem is that how to determine the coefficient . The method to determine the coefficient in Boccaletti ’ work is very complicated and lack of physical significance.

The main motivation of our work is that we believe that this coefficient should be determined by the network itself. To address this issue, we take the fractal dimension of complex network into consideration. The dimension of complex networks is one of the most fundamental quantities to characterize its structure and basic physical properties daqing2011dimension (); shanker2007graph (); wei2013new (). One has proved that the network dimension is a key concept to understand not only network topology. But also dynamical process on networks, such as diffusion and critical phenomenon including percolation, which is also used to characterize the vulnerability of network. Box covering algorithm song2005self (); song2007calculate (); wei2013box () are one of the typical ways to calculate the fractal dimension shanker2008algorithms (). In short, fractal dimension is a key parameter to represent the characters of the network. Based on this idea, we propose that the dimension of the network has a significant relation with network vulnerability in this paper.

This paper is organized as follows. Section 2 introduces the preliminaries. In Section 3 we calculate the vulnerability of some networks using the proposed method. In Section 4 we compare the proposed method with the existing methods in other papers by calculating network vulnerability. Finally, we summarize our results in Section 5.

2 Preliminaries

In this section, we introduce Boccaletti ’s modelboccaletti2007multiscale () and three other methods holme2002attack (); holmgren2006using (); mishkovski2011vulnerability (). In general, the complex networks can be represented by an undirected and unweighted graph , where is the set of vertices and is the set of edges. Each edges connects exactly one pari of vertices, and a vertex-pair can be connected by maximally one edge, i.e. loop is not allowed.

In Boccaletti ’s work boccaletti2007multiscale (), the original method to evaluate the vulnerability is represented by the average edge betweenness, which is defined as:


where is the number of the edges, and is the edge betweenness of the edge , define as:


where is the number of geodesics(shortest path) from to that contain the link , and is the total number of geodesics from to .

However, this evaluation of gives no relevant new information about the vulnerability of the network. For example, two networks referred in boccaletti2007multiscale () shown in Fig. 1 can’t be distinguished using this method. By evaluating the vulnerability according to Eq. 1, one gets . It’s absolute that the “bat” graph is more vulnerable than the “umbrella” graph , but Eq. 1 gives the same evaluation result.

Figure 1: The “bat” graph and the “umbrella” graph boccaletti2007multiscale ().

In order to overcome the original method’s limitation of failing to distinguish some networks, the coefficient was introduced to evaluate vulnerability of complex network, which is called multi-scale evaluation of vulnerability boccaletti2007multiscale () and shown as below:


for each value of . If we want to compare two networks and , first computes . If , then is more robust than . On the other hand, if then one takes and computes until .

To get the coefficient , Boccaletti define a relative function of like:


The coefficient is obtained when the function has a maximal value. For the more detailed information to determine the coefficient , refer boccaletti2007multiscale (). It’s clear that, the coefficient ’s definition is complicated and lack of physical significance. The coefficient should reflect the complex network itself.

For the sake of comparison, three other methods to calculate vulnerability are described as follows. The first method is the average inverse geodesic length holme2002attack ():


where is the length of the geodesic between and (). The larger is, more robust the network is.

The second method is the largest component size () holmgren2006using (), which quantifies the number of nodes in the largest connected subgraph and defined as follows:


where is the size of the largest connected subgraph.

And the third method is the normalized average edge betweenness mishkovski2011vulnerability (), which is on the base of the Eq. 3 while and is defined as:


where is a complete graph and is a path graph.

3 Proposed vulnerability model

In this section, the proposed method is detailed. As mentioned in introduction section, We think that the coefficient should be determined by the network itself. In addition, this coefficient should also has the direct relation to the vulnerability of this network. In our opinion, the fractal dimension of the network is a promising alternative. For a given network and box size , a box is a set of nodes where all distances between any two nodes and in the box are smaller than . The minimum number of boxes required to cover the entire network is denoted by . The detailed illustration referred in song2007calculate () of the calculation of the fractal dimension is given in Fig. 2. The fractal dimension or box dimension calculated with the box covering algorithm is given as follows song2005self (); song2007calculate ():

Figure 2: Illustration of the box-covering algorithms. Starting from (upper left panel), a dual network (upper right panel) was constructed for a given box size (here ), where two nodes are connected if they are at a distance . A greedy algorithm was used for vertex colouring in , which is then used to determine the box covering in , as shown in the plotsong2007calculate ().

It is very known that the fractal dimension can characterize the network structure and basic physical properties which reflects the covering ability. For a given network, the higher the fractal dimension, the higher the covering ability, which means that there are more edges between the nodes in this network. We also know that given certain nodes in the network, the more edges, the more robust of this network. As a result, the fractal dimension not only reflects the characters of the network structure, but also partially reflects the vulnerability of the network. According to this idea, we use the fractal dimension to redefine . So the proposed method to calculate network vulnerability is given as follows:


where is the fractal dimension of the complex networks.

We apply our method to six networks to calculate the vulnerability index. Two are synthetic networks, Erdős-Rényi(ER) random networks erdos1960evolution () and Barabási-Albert(BA) model of scale-free networks barabasi1999emergence (). Four are real networks: US airport networks colizza2007reaction (), network of e-mail interchanges Guimera2003email (), protein-protein interaction network ppi_data () and German highway system kaiser2004spatial ().

The vulnerability of these networks are calculated according to the follow steps:

(1) calculate the fractal dimension of these networks above using box-covering algorithm song2005self (); song2007calculate (), i.e. Eq. 8. The results are illustrated in Fig. 3.

Figure 3: The versus of some complex networks obtained in a log-log scale: (a) the ER network with the size , the average degree . (b) the BA network with , the average degree . (c) US airport network. (d) network of e-mail interchanges. (e) protein-protein interaction network. (f) German highway system. The vertical ordinate of every subplot is the mean value of for 100 times, and the horizonal ordinate represents the box size . The absolute value of the slope is the fractal dimension.

(2) Calculate the average edge betweenness according to Eq. 2, and normalized by .

(3) Calculate the vulnerability in accordance with Eq. 9.

Table 1 shows the result. The larger the , the more vulnerable the network. So the results illustrate the order of vulnerability , and the robustness of networks correspond to the inverse order.

  ER 1500 6 3.711 0.0011
  BA 1500 4.8 2.05 0.0014
  AP 500 11.9 4.048 0.0079
  EI 1134 9.6 4.235 0.0013
  PPI 2375 9.8 4.486 0.0037
  GH 1168 2.1 1.34 0.0184
Table 1: General characteristics of several complex networks. For each network we list the number of nodes , the average degree , the fractal dimension , and the vulnerability obtained by the proposed method. ER, BA, AP, EI, PPI and GH denote the ER network, the BA network, US airport network, network of e-mail interchanges, protein-protein interaction network and the German highway system.

4 Comparison and Discussion

In this section, to testify the correctness of the results obtained by the proposed method, three other methods presented in section 2 are applied to these networks to calculate the vulnerability, that is, the average inverse geodesic length , the largest component size and the normalized average edge betweenness . All three methods can reflect static topological properties of networks, in order to get the vulnerability reflecting the dynamical overall characteristics of networks, we apply the RB attack strategy holme2002attack () to networks when calculating them. RB attack strategy means that one should remove the node which has the highest betweenness value and recalculate the betweenness at every vertices-removing step. In this paper, , , are computed after 1% of vertices are removed. Table 2 shows the results.

  ER 0.0011 0.9788 0.9886 0.0666
  BA 0.0014 0.8152 0.9613 0.4874
  AP 0.0079 0.6259 0.746 -0.3563
  EI 0.0013 0.9466 0.9841 0.1490
  PPI 0.0037 0.7681 0.9175 0.0912
  GH 0.0184 0.5119 0.9144 0.7644
Table 2: The normalized average inverse geodesic length , normalized largest component size and the average edge betweenness is computed after 1% of vertices are removed. All of them are normalized by the values of the initial networks.

All the methods can give a rank about the vulnerability of these networks. The gives a order about the robustness, and a robustness order judging from , whereas ranks in point of vulnerability. One can see that, The German highway system has the largest vulnerability and for all the methods. The proposed method and the shows a completely identical order. So the proposed method is an effect way to quantify the network vulnerability.

In a addition, the multi-scale model to calculate vulnerability proposed by Boccaletti are applied to these networks. A comparison of the proposed method and Boccaletti ’s are illustrated in Table 3 and Fig. 4. As mentioned in section 2, we should firstly compute to judge if can distinguish these networks. Through computing, we found that , which mean that the coefficient should be recalculated according to the relative function (Eq. 4). When the relative function has a maximal value, is obtained. We get for BA and AP networks, . Boccaletti ’s method gives a order about the robustness.

Figure 4: for Barabási-Albert(BA) model of scale-free networks (dot line) and US airport networks (dash line) as functions of . (solid line) as a relative function of has a unique maximum at

It’s absolute that, the coefficient obtained by Boccaletti ’s method is lack of physical meaning. Comparing rank orders obtained by these method, it’s easy to found that the proposed method gives a more reasonable order and a more effective evaluation.

network () of the of Boccaletti ’s
proposed method method
ER 3.711 0.0011 1
BA 2.05 0.0014 0.001 12 0.0035
AP 4.048 0.0079 0.001 12 0.0234
EI 4.235 0.0013 1
PPI 4.486 0.0037 1
GH 1.34 0.0184 0.0156 1 0.0156
Table 3: Some hypothetical scenarios to demonstrate the comparison of two methods.

5 Conclusions

The coefficient used in the multi-scale model plays an important role in the vulnerability evaluation. How to determine the coefficient is still an open issue. The existing method is complex and lack of physical significance. To address this issue, an improved vulnerability index is proposed based on the fractal dimension of complex networks. The fractal dimension is one of the fundamental properties of complex networks, which can not only characterize the physical properties of networks, but also reflect the covering ability of networks. As a result, the new model has more meaning in physical aspect compared with existing methods. The numerical examples and real applications are used to illustrate the efficiency of our proposed method.


The work is partially supported by National Natural Science Foundation of China (Grant No. 61174022), R D Program of China (2012BAH07B01), National High Technology Research and Development Program of China (863 Program) (Grant No. 2013AA013801).


  • (1) H.-J. Kim, Analysis of a complex network of physics concepts, Modern Physics Letters B 26 (28).
  • (2) J. Wang, H. Yang, Complex network-based analysis of air temperature data in china, Modern Physics Letters B 23 (14) (2009) 1781–1789.
  • (3) Q. Tang, J. Zhao, T. Hu, Detecting chaos time series via complex network feature, Modern Physics Letters B 25 (23) (2011) 1889–1896.
  • (4) X. Qi, Z.-G. Shao, J. Qi, L. Yang, Efficiency dynamics on scale-free networks with communities, Modern Physics Letters B 24 (14) (2010) 1549–1557.
  • (5) H. Zhang, X. Lan, D. Wei, S. Mahadevan, Y. Deng, Self-similarity in complex networks: from the view of the hub repulsion, Modern Physics Letters B 27 (28).
  • (6) P. Holme, B. J. Kim, C. N. Yoon, S. K. Han, Attack vulnerability of complex networks, Physical Review E 65 (5) (2002) 056109.
  • (7) M. A. S. Monfared, M. Jalili, Z. Alipour, Topology and vulnerability of the iranian power grid, Physica A - (-) (2014) –. doi:10.1016/j.physa.2014.03.031.
  • (8) Å. J. Holmgren, Using graph models to analyze the vulnerability of electric power networks, Risk analysis 26 (4) (2006) 955–969.
  • (9) S. Boccaletti, J. Buldú, R. Criado, J. Flores, V. Latora, J. Pello, M. Romance, Multiscale vulnerability of complex networks, Chaos: An Interdisciplinary Journal of Nonlinear Science 17 (4) (2007) 043110–043110.
  • (10) J. Zhang, X. Xu, L. Hong, S. Wang, Q. Fei, Attack vulnerability of self-organizing networks, Safety science 50 (3) (2012) 443–447.
  • (11) S. Wang, L. Hong, M. Ouyang, J. Zhang, X. Chen, Vulnerability analysis of interdependent infrastructure systems under edge attack strategies, Safety science 51 (1) (2013) 328–337.
  • (12) X. Zhang, Z. Zhang, Y. Zhang, D. Wei, Y. Deng, Route selection for emergency logistics management: A bio-inspired algorithm, Safety Science 54 (2013) 87–91.
  • (13) I. Mishkovski, M. Biey, L. Kocarev, Vulnerability of complex networks, Communications in Nonlinear Science and Numerical Simulation 16 (1) (2011) 341–349.
  • (14) M. Ouyang, Z. Pan, L. Hong, L. Zhao, Correlation analysis of different vulnerability metrics on power grids, Physica A: Statistical Mechanics and its Applications 396 (2014) 204–211.
  • (15) R. Albert, I. Albert, G. L. Nakarado, Structural vulnerability of the north american power grid, Physical review E 69 (2) (2004) 025103.
  • (16) R. Albert, A.-L. Barabási, Statistical mechanics of complex networks, Reviews of modern physics 74 (1) (2002) 47.
  • (17) A. E. Motter, Y.-C. Lai, Cascade-based attacks on complex networks, Physical Review E 66 (6) (2002) 065102.
  • (18) P. Crucitti, V. Latora, M. Marchiori, Model for cascading failures in complex networks, Physical Review E 69 (4) (2004) 045104.
  • (19) J.-W. Wang, L.-L. Rong, Cascade-based attack vulnerability on the us power grid, Safety Science 47 (10) (2009) 1332–1336.
  • (20) J. Wang, Robustness of complex networks with the local protection strategy against cascading failures, Safety Science 53 (2013) 219–225.
  • (21) J.-W. Wang, L.-L. Rong, Vulnerability of effective attack on edges in scale-free networks due to cascading failures, International Journal of Modern Physics C 20 (08) (2009) 1291–1298.
  • (22) J. Wang, C. Jiang, J. Qian, Improving robustness of coupled networks against cascading failures, International Journal of Modern Physics C 24 (11).
  • (23) J.-W. Wang, L.-L. Rong, Robustness of the western united states power grid under edge attack strategies due to cascading failures, Safety science 49 (6) (2011) 807–812.
  • (24) L. Daqing, K. Kosmidis, A. Bunde, S. Havlin, Dimension of spatially embedded networks, Nature Physics 7 (6) (2011) 481–484.
  • (25) O. Shanker, Graph zeta function and dimension of complex network, Modern Physics Letters B 21 (11) (2007) 639–644.
  • (26) D. Wei, B. Wei, Y. Hu, H. Zhang, Y. Deng, A new information dimension of complex networks, Physics Letters A (2013) doi:10.1016/j.physleta.2014.02.010.
  • (27) C. Song, S. Havlin, H. A. Makse, Self-similarity of complex networks, Nature 433 (7024) (2005) 392–395.
  • (28) C. Song, L. K. Gallos, S. Havlin, H. A. Makse, How to calculate the fractal dimension of a complex network: the box covering algorithm, Journal of Statistical Mechanics: Theory and Experiment 2007 (03) (2007) P03006.
  • (29) D. Wei, Q. Liu, H. Zhang, Y. Hu, Y. Deng, S. Mahadevan, Box-covering algorithm for fractal dimension of weighted networks, Scientific reports 3 (2013) doi:10.1038/srep03049.
  • (30) O. Shanker, Algorithms for fractal dimension calculation, Modern Physics Letters B 22 (07) (2008) 459–466.
  • (31) P. Erdos, A. Rényi, On the evolution of random graphs, Publ. Math. Inst. Hung. Acad. Sci 5 (1960) 17–61.
  • (32) A.-L. Barabási, R. Albert, Emergence of scaling in random networks, science 286 (5439) (1999) 509–512.
  • (33) V. Colizza, R. Pastor-Satorras, A. Vespignani, Reaction–diffusion processes and metapopulation models in heterogeneous networks, Nature Physics 3 (4) (2007) 276–282.
  • (34) R. Guimera, L. Danon, A. Diaz-Guilera, F. Giralt, A. Arenas, Physical Review E 68 (3) (2003) 065103.
  • (35) http : //
  • (36) M. Kaiser, C. C. Hilgetag, Spatial growth of real-world networks, Physical Review E 69 (3) (2004) 036103.
Comments 0
Request Comment
You are adding the first comment!
How to quickly get a good reply:
  • Give credit where it’s due by listing out the positive aspects of a paper before getting into which changes should be made.
  • Be specific in your critique, and provide supporting evidence with appropriate references to substantiate general statements.
  • Your comment should inspire ideas to flow and help the author improves the paper.

The better we are at sharing our knowledge with each other, the faster we move forward.
The feedback must be of minimum 40 characters and the title a minimum of 5 characters
Add comment
Loading ...
This is a comment super asjknd jkasnjk adsnkj
The feedback must be of minumum 40 characters
The feedback must be of minumum 40 characters

You are asking your first question!
How to quickly get a good answer:
  • Keep your question short and to the point
  • Check for grammar or spelling errors.
  • Phrase it like a question
Test description