An improved vulnerability index of complex networks based on fractal dimension
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.
keywords:vulnerability, fractal dimension, complex networks
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 et.al 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 et.al ’ 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.
In this section, we introduce Boccaletti et.al ’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 et.al ’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.
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 et.al 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.
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 ():
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:
(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.
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.
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 et.al are applied to these networks. A comparison of the proposed method and Boccaletti et.al ’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 et.al ’s method gives a order about the robustness.
It’s absolute that, the coefficient obtained by Boccaletti et.al ’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 et.al ’s|
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).
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