A scale-free topology model with fault-tolerance and intrusion-tolerance in wireless sensor networks

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A scale-free topology model with fault-tolerance and intrusion-tolerance in wireless sensor networksR Haoran Liu a,b, Rongrong Yin a,b,∗, Bin Liu a,b, Yaqian Li c a

School of Information Science and Engineering, Yanshan University, Qinhuangdao, China The Key Laboratory for Special Fiber and Fiber Sensor of Hebei Province, Yanshan University, Qinhuangdao, China c Institute of Electrical Engineering, Yanshan University, Qinhuangdao, China b

a r t i c l e

i n f o

Article history: Received 22 April 2015 Revised 4 January 2016 Accepted 4 January 2016 Available online xxx Keywords: Wireless sensor network Scale-free topology Fault-tolerance Intrusion-tolerance

a b s t r a c t The scale-free topology is robust when confronted with random faults, but it is fragile when confronted with selective remove attacks. In this paper, we propose a new scalefree topology model which has both fault-tolerance against random faults and intrusiontolerance against selective remove attacks at the same time. Then the mathematical expression of the topological degree distribution is derived. Through analyzing the effect of topological degree distribution on these properties of topological fault-tolerance and topological intrusion-tolerance, the optimal scale-free topology which can keep the faulttolerance and maximize intrusion-tolerance is obtained. We performed extensive experiments on the proposed model and compared it with other existing models. The simulation results show that the new scale-free topology model can keep the character that the scale-free topology has a stronger robustness to random faults. And it also can reduce their fragility for selective remove attacks and further prolong its lifetime. © 2016 Elsevier Ltd. All rights reserved.

1. Introduction Wireless sensor networks (WSNs) have gained enormous attention for their wide range of applications. In many applications, it is impractical to replace the nodes as they work under harsh environment. However, the nodes in WSNs, which are deployed in harsh environments, are easy to break down because of energy depletion, hardware failure or invasion [1]. On node failure, the network connectivity will be reduced greatly, and the entire network even to be paralytic. After classical BA model (Barabasi–Albert Model) was put forward [2], the robustness of scale-free topology to node failure excited many scholars’ widespread interest [3]. The scale-free topology is a heterogeneous topology, that is, the degree distribution probability fulfills the power-law relation. The topological robustness is the ability of the topology to maintain its connectivity after node failures, meaning that the nodes remove [4]. It has been shown that the method of removing failure nodes (randomly or selectively) changes topology functionality [5]. In this context, the influence of random and selective node failures on the efficiency of scale-free topology is investigated. Recent researches show that the scale-free topology is robust to random faults of the node but vulnerable to selective remove attacks of the node [6]. This conclusion becomes more evident after a deal of works by Barabasi and Albert on R ∗

Reviews processed and recommended for publication to the Editor-in-Chief by Associate Editor Dr. M. H. Rehmani. Corresponding author at: School of Information Science and Engineering, Yanshan University, Qinhuangdao, China. Tel.: +86 13653369916. E-mail address: [email protected] (R. Yin).

http://dx.doi.org/10.1016/j.compeleceng.2016.01.003 0045-7906/© 2016 Elsevier Ltd. All rights reserved.

Please cite this article as: H. Liu et al., A scale-free topology model with fault-tolerance and intrusion-tolerance in wireless sensor networks, Computers and Electrical Engineering (2016), http://dx.doi.org/10.1016/j.compeleceng.2016.01.003

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scale-free topology [7]. So how to design and optimize the scale-free topology and how to make the scale-free topology have the strong capability of fault-tolerance to random faults and intrusion-tolerance to selective remove attacks simultaneously is particularly important and urgent. In order to improve the robustness of the scale-free topology against selective remove attacks on the basis of strong fault-tolerance, this paper proposes a new scale-free topology model BA-E (BA-Evolution) and obtains an optimal scalefree topology which can assure the topological fault-tolerance against random faults and maximize topological intrusiontolerance against selective remove attacks. Our main contributions are summarized as follows. 1. The proposed scale-free topology model BA-E has the adjustable scaling exponent for the degree distribution. 2. Two criterions of measuring the topological properties (i.e. one for fault-tolerance against random faults, the other for intrusion-tolerance against selective remove attacks) and the effect of topological degree distribution on these properties are discussed mathematically. Then the optimal parameter of the BA-E model is derived. 3. The proposed BA-E model with the obtained optimal parameter is implemented through simulation for fault-tolerance against random faults and intrusion-tolerance against selective remove attacks. 4. Comparison of simulation experimental results to demonstrate superiority of the proposed BA-E model over the existing models. The rest of the paper is organized as follows. In the next section, related work is described. Section 3 puts forward the BA-E model and the degree distribution of the model is deduced. Section 4 constructs the optimized mathematical model about these properties of both fault-tolerance and intrusion-tolerance, then the effect of degree distribution characteristics on topological fault-tolerance and intrusion-tolerance is discussed in detail. Section 5 makes the simulation verification. Finally, Section 6 concludes the paper. 2. Related works Scale-free topology is the most important technique used for robustness optimization of WSNs. Considering the major constraints of WSNs is the limited energy sources of the nodes, EAEM (Energy-Aware Evolution Model) investigated how to build the energy efficient scale-free topology [8]. It introduced the node residual energy to the preferential attachment mechanism, and improved the energy efficiency of the scale-free topology. In [9], Qi et al. proposed the scale-free topology model CHMM (scale-free topology based on Clustering Hierarchy Modularity Measure). It considered the residual energy and the node fitness during the topology evolution, which made the scale-free topology has a good robustness against energy exhaustion and random faults. In [10], Chen et al. constructed the scale-free topology among cluster heads by the random walkers. In [11], Saffre et al. constructed the scale-free topology using the local geographic information of nodes. They both considered the limited transmission of sensors. Similarly, in [12], Liu et al. proposed the scale-free topology model with small-world feather. It combined more characteristics of sensors, including residual energy, degree saturation and maximum communication radius. In [13], Holme et al. proposed the scale-free topology model with a tunable clustering coefficient. It considered the triad formation process of links, and showed that the robustness of scale-free topology can be steadily enhanced at a slightly increased clustering coefficient. All these methods manifest a good performance in improving the scale-free topology robustness in harsh environment. Until now these methods can still be improved. There are more constraints being added on the scale-free topology to make it accord with the characteristics of WSNs in varied applications [14,15], which are needed to optimize the topology robustness, but the scale-free topologies don’t optimize the capability of intrusion-tolerance against selective remove attacks. Now most of the researches on the intrusion-tolerance of scale-free topology use a method that redundancy of the key node and the key link. In [16], Xiao et al. show that incomplete global information has different impacts on the selective remove attacks in different circumstances, while local information-based attacks can be actually highly efficient. The proposed method includes hiding or partially hiding key nodes. In redundancy, the determining method about key node and key link, and the backups scheduling are required. So this method also increases the costs of the scale-free topology significantly. In [17], Du et al. found that the scale-free topology is robust to random faults and easily destroyed when faces selective remove attacks. However, the random topology is just the opposite. Based on this, In [18], Liu et al. built a new model in which the topological parameter is adjustable. The classical BA scale-free topology model and the random topology model can be obtained through adjusting the parameter. Because this model did not keep the power-law property of scale-free topology, it didn’t ensure the strong fault-tolerance of topology unchanged. In [19], Beygelzimer et al. present empirical results that how robustness is affected by several different strategies that alter the topology by rewiring a fraction of the links or by adding new links. They enhance the topological robustness by link insertion between low-degree nodes without changing its degree distribution. Another work similar to this work is introduced by Xiao et al. [20]. They study the effects of connections patterns on the robustness of scale-free topology and propose a simple local repairing strategy by restoring some of the links that have been cut when a node is fail by rewiring each of them to another node. In [21], Ghamry et al. show that highly-heterogeneous topologies have less robustness compared with lightly-heterogeneous topologies. In [22], an indicator of measuring the topological heterogeneity is obtained by utilizing the degree distribution entropy and the topology structure entropy respectively. But they did not take into account the scaling exponent’s influence on the topological robustness when the scale-free topology was faced with random faults and selective remove attacks simultaneously. Please cite this article as: H. Liu et al., A scale-free topology model with fault-tolerance and intrusion-tolerance in wireless sensor networks, Computers and Electrical Engineering (2016), http://dx.doi.org/10.1016/j.compeleceng.2016.01.003

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According to the above analysis, our proposed model has the advantage over the existing ones. The proposed BA-E model addresses the problem of designing scale-free topology to tolerate both random faults and selective remove attacks. It is different from any other published solutions (i.e. redundancy, damage in power-law property), because the topological degree distribution derived by BA-E model obeys the adjustable power-law distribution, which keeps the robustness of faulttolerance, together with strong fault-tolerance, the strong intrusion-tolerance can also be obtained by adjusting the powerlaw distribution. It means that BA-E model improves the intrusion-tolerance of scale-free topology against selective remove attacks without breaking the strong fault-tolerance against random faults. 3. Scale-free topology model (BA-E) In order to evolve the scale-free topology with the good performance of both fault-tolerance and intrusion-tolerance in WSNs, we can use the improved growth and preferential attachment to build the scale-free topology model with the local-area idea [23]. Then the evolutional degree distribution of the model is discussed. 3.1. BA-E evolution model In [2], BA model uses the method of growth and preferential attachment. In the growth process, the number of nodes in the network grows continuously. In the preferential attachment process, the probability of a new node to be linked to an existing node i depends on the degree ki of node i, and obeys the following rules



k

(ki ) =  i

(1)

kj

j

The growth rule and preferential attachment rule above lead to a skewed degree distribution for topology, it constructs the BA scale-free topology whose degree distribution follows p(k ) ∝ k−3 . This BA scale-free topology has a good faulttolerance against random faults and a poor intrusion-tolerance against selective remove attacks. In order to improve the robustness of the scale-free topology against both random faults and selective remove attacks, the variable parameter p is introduced. When p varies from 1 to 0, the topology changes from random topology to BA scale-free topology. The BA-E model which is similar with the BA model can be divided into two steps: Growth: starting with a small number of nodes m0 , at every time, a new node with m(m ≤ m0 ) links that will be connected to the nodes already exist in the network is added. Preferential attachment: when a new node comes into the network, it will choose some nodes in its local-area to connect.  (ki ) of the new node to be connected to node i depends on its degree ki and the preferential And the probability local

probability factor p, which is defined as



(1−p)ki + p (ki ) =  (1 − p)k j + p local

(0 ≤ p ≤ 1 )

(2)

j∈local

where the node i and j are within the local-area of the new node. The model evolves to BA scale-free topology when p = 0, and it becomes an approximately random topology when p = 1. For 0 < p < 1, we call this evolving model as BA-E. It is worthwhile to note that the adjustable degree distribution of BA-E model follows the power-law statistics of the scalefree topology, which can improve the intrusion-tolerance of topology against selective remove attacks without breaking the strong fault-tolerance against random faults. Theorem 1. BA-E model runs in O((N − m0 )(1 + m )) time. Let G = (V, E ) be the topology evolved by BA-E model with N nodes. The topology is created starting from m0 nodes and by then growing the topology by adding new nodes and links. At each time in the creation of the BA-E model, one node and m outgoing links from the new node are added to the topology. Since the last new node and its links can be added in O((N − m0 − 1 )(1 + m )) by using the BA-E model, the theorem follows. 3.2. BA-E degree distribution characteristics The evolution of BA-E model is shown in Fig. 1. The initial network consists of m0 nodes and e0 links. It is supposed that, node n joins the network at time t, Rn is the communication radius of node n, R0 is the initial network radius, R0 +t is the network radius at time t. Considering the degree distribution of BA-E model, the probability that the node n builds communication link in its local-area is decided by Eq. (2) and m new links are formed during each time. According to the mean-field theory, we get

 ∂ ki (ki ) =m ∂t local

(3)

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Fig. 1. The BA-E evolution model.

Assuming that the nodes were distributed uniformly, we considered that when the new node n comes into the network, it will choose some nodes in its local-area to connect. The local-area of node n consisted of nodes within node n’s communication radius. So the probability that new node n connects with the node i in its local-area is associated with the proportion of area value, 12 π (Rn )2 to π (R0 + t )2 . This means the Eq. (2) can be approximated as Eq. (4)



1 (Rn ) (1 − p)ki + p (1 − p)ki + p (1 − p)ki + p = = 2 (ki ) =  · (1 − p)k j + p (R0 + t )2 N 12 (Rn )2 [(1 − p)k + p] Nt [(1 − p)kt + p] local t t j∈local (R +t )2 2

(4)

0

where, Nt is the number of nodes in the network at time t, kt is the average degree of nodes, and we get Nt = m0 + t and 2mt m +t . According to Eqs. (3) and (4), we have

kt =

0

m[(1 − p)ki + p] ∂ ki (1 − p)ki + p  = =m 2mt ∂t 2m(1 − p)t + pt (m0 + t ) (1 − p) m0 +t + p

(5)

Because the initial condition ki (ti ) = m, by solving the differential equation, Eq. (5) is described as follows:



ki (t ) = m +

p 1− p

1−p)  t 2m+m((1−2m )p

ti

−

p 1− p

(6)

Thus the probability when ki (t) is smaller than k can be described by





k + p/(1 − p) p[ki (t ) < k] = p ti > t m + p/(1 − p) Set b =

p , m(1−p)

 p

−[2+ m(1−p )]

(7)

then Eq. (7) can be simplified as



p[ki (t ) < k] = p ti > t



k/m + b 1+b

−(2+b)  (8)

Assuming that we add a node to the network at equal time intervals, the probability density at the time ti obeys the uniform distribution as following:

p(ti ) =

1 m0 + t

(9)

Combining Eq. (8) with Eq. (9), we have

  

k/m+b −(2+b)  −(3+b) 1 ∂ p[ki (t ) < k] ∂ 1 − p ti ≤ t ( 1+b ) k/m + b = = p( k ) = ∂k ∂k m (1 + b ) 1+b

(10)

According to Eq. (10), the topological degree distribution derived by BA-E model is always obeying the power-law distrip bution with scaling exponent λ = 3 + b, where b = m(1−p ) . When m is a constant, the scaling exponent λ can be changed Please cite this article as: H. Liu et al., A scale-free topology model with fault-tolerance and intrusion-tolerance in wireless sensor networks, Computers and Electrical Engineering (2016), http://dx.doi.org/10.1016/j.compeleceng.2016.01.003

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within [3, +∞ ) by adjusting the parameter p. When p → 0, we can get p(k ) ∼ k−3 , then the degree distribution of BA-E model is closing in on the BA scale-free topology which has the good capability of fault-tolerance against random faults; when p → 1, we can get p(k ) ∝ e−k/m , then the degree distribution of BA-E model is closing in on the random topology which has the good capability of intrusion-tolerance against selective remove attacks. So it can be concluded that the BA-E model has the power-law distribution that keeps its robustness of fault-tolerance, together with strong fault-tolerance, a strong intrusion-tolerance can also be obtained by adjusting the parameter p. 4. Mathematical optimization model of BA-E Based on the analysis of the degree distribution of BA-E model in Section 3, the topology derived by BA-E model exists the optimal value p (which can withstand node failures, that is, the random faults and the selective remove attacks). In this section, we analyze the effect of degree distribution characteristics on topological fault-tolerance and topological intrusiontolerance, then find the optimal p. Based on p, an optimal BA-E scale-free topology is derived which keeps the topological fault-tolerance and maximizes the topological intrusion-tolerance. 4.1. BA-E fault-tolerance index Based on the percolation theory, Cohen et al. have studied the properties of the percolation phase transition, and found that there is a critical point removal ratio hr [24], and hr can be used as the fault-tolerance strength criterion of the scalefree topology. When the removal ratio of random nodes is more than hr , the topology will collapse into many smaller disconnected parts. And Ref. [24] applied a general criterion for the existence of a spanning part, and this criterion can be written as

 2 k

k

=2

(11)

When an h(0 < h < 1) ratio of nodes is randomly removed, for a node with initial degree k0 chosen from an initial distribution p(k0 ), the connectivity distribution of the node is changed from the original distribution p(k0 ) to a new distribution p˜ (k ). The new distribution p˜ (k ) is given by

p˜ (k ) =

K 

  k p(k0 ) 0 (1 − h )k hk0 −k k

k0 ≥k

(12)

So the average degree k and its second moment k2  for the new distribution p˜ (k ) can be expressed by ∞ 

k =

p˜ (k )k = k0  × (1 − h )

(13)

k=0

and

 2 k

=

∞ 





p˜ (k )k2 = k0 (1 − h )2 + k0 h(1 − h ) 2

(14)

k=0

Here, k0  and k20  can be calculated by the original distribution p(k0 ). Then the fault-tolerance strength criterion for criticality is obtained by

 2 k0 k2  = (1 − hr ) + hr = 2 k k0 

(15)

It can be rearranged to yield the critical threshold for percolation

hr = 1 −

1 k0 − 1

(16)

where k0 ≡ k20 /k0 . Since the scale-free topology has a power-law distribution

p(k ) = Ck−λ (k = m, m + 1, ..., Kmax )

(17)

where m and Kmax are the minimum degree and the maximum degree, respectively. The coefficient C is obtained from



+∞ m

p(k )dk =



+∞

m

Ck−λ dk = 1

(18)

That is

C = (λ − 1 )mλ−1

(19)

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Therefore, in the scale-free topology, the k0 can be approximated by



k0 =

k0

2



 Kmax

k0 

m

Due to p(k0 ) = Ck0

 Kmax k0 = mK max m

−λ

k0 p(k0 )dk0 2

= mK max

(20)

k0 p(k0 )dk0

= (λ − 1 )mλ−1 k0

k0 · (λ − 1 )mλ−1 k0 2

k0 · (λ − 1 )mλ−1 k0

−λ

−λ

−λ

dk0

dk0

, we have

=

2 − λ Kmax 3−λ − m3−λ · 3 − λ Kmax 2−λ − m2−λ

(21)

Using a continuum approximation valid in the limit of Kmax m > 1, k0 is then simply given by

 m |2 − λ| k0 = → × mλ−2 k3−λ k0  |3 − λ| k  2 k0

λ>3 2<λ<3 1<λ<2

(22)

Considering that the degree distribution of BA-E scale-free topology meets the case λ > 3, we have finally

k0 =

λ−2 ×m λ−3

(23)

Now, substituting Eq. (23) into Eq. (16), the fault-tolerance strength criterion of BA-E scale-free topology is a function of variables m and λ

1 hr = 1 − λ−2 × m−1 λ−3

(24)

For the purpose of constructing the BA-E scale-free topology with stronger fault-tolerance, Eq. (24) can be used as the topological fault-tolerance strength criterion. When m and λ are constant, the smaller hr is, the less removal ratio of random nodes that topology can tolerate is, and the worse fault-tolerance the topology is. 4.2. BA-E intrusion-tolerance index The heterogeneity of scale-free topology leads to paralysis when the topology is confronted with the selective remove attacks. In this section, we use the uniformity to measure the topological intrusion-tolerance. The authors in [22] proposed the topology structure entropy which can well measure the heterogeneity of topologies. The topology structure entropy corresponds to the uniform topology when it is maximized and corresponds to the star topology when it is minimized. So, the optimization of topological intrusion-tolerance is equivalent to maximize the topology structure entropy of scale-free topology. The topology structure entropy is defined as follows:

E=−

N−1 

N Ii ln Ii = −

i=1

N  i=1 ki ln ki + ln ki N i=1 ki i=1

(25)

 where Ii = ki / N−1 i=1 ki . When the topology structure entropy is used to evaluate a specific scale-free topology, it can be described as follows: the greater E is, the more uniform the scale-free topology is, and the stronger intrusion-tolerance the scale-free topology is. Set K (G ) = {k 1 , k 2 , . . . k N } as the degree sequence for scale-free topology G, where k1 ≥ k2 ≥ · · · ≥ kN . Assuming τ i represents the serial number of the node vi , the relationship between the node degree ki and the serial number τ i is expressed  T∗ τ and N is the total number of nodes, the by the function ki = f (τi ). For k = f (τ ) = N − T ∗ , where 1 ρ (k = N − s )ds = N degree-rank function is given by

 k = f (τ ) =

1   −λ+2   λ−1 1  λ−1 N C λ+ NC λ−1

λ−1

(26)

Substituting Eq. (19) into Eq. (26), we have

 f (τ ) =

λ−1

 1 − λ−1

NC

 λ+

N−λ+2C

λ−1

1  − λ−1 1

= kmin N λ−1



λ−1 λ + N−λ+2 kmin

1 − λ−1

Considering that the degree distribution of BA-E scale-free topology meets λ > 3, that is N −λ+2 kmin 1

f (τ ) = kmin (N/λ ) λ−1

(27) λ−1

≈ 0 and we get (28)

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With continuous approximation, for the BA-E scale-free topology, the topology structure entropy E can be expressed as follows

N

E=−

1

f (τ ) ln f (τ )dr + ln N 1 f (τ )dr



1

N



f (τ )dr

(29)

Now, substituting Eq. (28) into Eq. (29), we have then

E=

σ N1−σ lnN N1−σ

−1

+ ln

1 N1−σ − 1 − ,σ = 1−σ 1−σ

1

λ−1

(30)

In Eq. (30), the topology structure entropy E of BA-E scale-free topology can be expressed as the function of variables N and λ. When N and λ are constant, the topology structure entropy E can be used as the topological intrusion-tolerance strength criterion. The greater E is, the more uniform the topology is, and the stronger intrusion-tolerance the topology is. This means topology structure entropy E should be maximized if the stronger intrusion-tolerance of BA-E scale-free topology is needed. 4.3. Mathematical optimization model of BA-E for fault-tolerance and intrusion-tolerance This section wants to construct an optimal intrusion-tolerance topology on the basis of strong fault-tolerance, we use hr as the measure of the topological fault-tolerance strength criterion, and use E as the measure of the topological intrusiontolerance strength criterion. For hr >0.5, it means that the topology can tolerate more than half of all the nodes to randomly fail, and the topology has stronger fault-tolerance. So this optimization problem of intrusion-tolerance based on strong faulttolerance can be transformed into the following mathematical model

max E s.t ht >0.5

(31)

λ> 3

In this expression, combined with the analysis of Section 4.1 and according to Eq. (24), Eq. (32) is obtain by

1 >0.5 hr = 1 − λ−2 λ−3 × m − 1

(32)

The optimization model is then simply given by

max E s.t 3 <λ< 3.5

(33)

In order to achieve the optimal intrusion-tolerance without losing the strong fault-tolerance, we need to find an optimal

λ to make the E maximize under the condition of 3 < λ < 3.5. So combined with the analysis of Section 4.2, when m = 1, N=200 and according to Eq. (30), we draw the topology structure entropy E, as shown in Fig. 2. p In Fig. 2, the topology structure entropy E achieves its maximum when λ = 3.37. Combining λ = 3 + m(1−p ) with m = 1, we can get p ≈ 0.27. Under the condition of m = 1, N = 200, p ≈ 0.27, we implement the BA-E model, the environment parameters are shown in Table 1, then we can evolve the optimal BA-E topology, as shown in Fig. 3. Fig. 3 represents the contrast of the theoretical value and actual value in the degree distribution of BA-E topology, and the straight line is the theoretical degree distribution

Fig. 2. The topology structure entropy function E.

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H. Liu et al. / Computers and Electrical Engineering 000 (2016) 1–11 Table 1 Environment parameters. Parameters

Value

Amplify circuit loss ε amp (pJ/bit/m2 ) Transmit/receive circuit loss Eelec (nJ/bit) Maximum transmission range dmax (m)

100 50 150

Parameters

Value

Deployment area A (m2 ) Transmission data l(bits) Energy distribution of nodes e (J)

500 × 500 100 (0.8,0.9)

Fig. 3. The degree distribution of BA-E topology.

Table 2 The degree distribution deviation of BA-E topology. Node degree Degree distribution

1

2

3

4

5

7

Theoretical value Actual value

0.7299 0.7900

0.1151 0.0900

0.0351 0.0200

0.0146 0.0200

0.0073 0.0200

0.0025 0.0100

p(k ) ∼ k−3.37 . In actual degree distribution, the majority of the nodes have one or two links but a few nodes have a large number of links. As can be seen from the Fig. 3, the actual degree distribution of BA-E topology is in accord with the theoretical power-law distribution with scaling exponent λ = 3.37, especially when the node degree is small. Table 2 represents the corresponding deviation to the degree distribution of BA-E topology, the node degree are from range of one to seven. The result shows that the deviation of topological degree distribution for BA-E topology is in low values. 5. Simulation and analysis In WSNs, the nodes which are deployed in harsh environments usually face random faults and selective remove attacks. In order to simulate the random faults of nodes, we assume that each node has the same removal probability. Because the selective remove attacks are malicious and selective, we remove nodes according to the node degree from maximum to minimum in experiment. Finally, we use the number of nodes in the maximum connected component to measure the topological fault-tolerance and topological intrusion-tolerance. Moreover, EAEM model possesses energy-saving advantage; CHMM model constructs a useful robustness topology against energy exhaustion and random faults. According to the optimization model in Section 4, the BA-E model is compared with the EAEM model, the CHMM model and the classic BA model by MATLAB simulation experiments. The every experimental result is an average of 50 times in the environment parameters (see Table 1). (a) Experiment 1: comparison of fault-tolerance and intrusion-tolerance. In order to measure the fault-tolerance and intrusion-tolerance of BA-E topology, the BA-E model is compared with the EAEM model, the CHMM model and the BA model. We remove the nodes randomly or selectively from maximum to minimum degree, and the contrast diagrams are shown in Figs. 4 and 5. Fig. 4 represents the fault-tolerance comparison of topologies after the random removes. According to the simulation diagram, the BA-E model almost has the same fault-tolerance with the EAEM model, the CHMM model and the BA model. This reflects that the BA-E model inherits the strong fault-tolerance of the scale-free topology. Fig. 5 represents the intrusiontolerance comparison of topologies after the selective removes. According to the simulation diagram, the BA model and the Please cite this article as: H. Liu et al., A scale-free topology model with fault-tolerance and intrusion-tolerance in wireless sensor networks, Computers and Electrical Engineering (2016), http://dx.doi.org/10.1016/j.compeleceng.2016.01.003

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Fig. 4. Comparison of fault-tolerance.

Fig. 5. Comparison of intrusion-tolerance.

EAEM model are paralyzed when their topologies are suffered from three to four nodes to remove. The CHMM model can only tolerate 5 nodes in the selective remove attacks, while the BA-E model has more surviving nodes in the maximum connected component, the BA-E model can tolerate 7 nodes. Therefore, the BA-E model has the strong robustness against random faults and the better capability of intrusion-tolerance against selective remove attacks. That is to say, the BA-E topology is still able to complete the detection task when a part of the nodes are failed for random faults and selective remove attacks. (b) Experiment 2: comparison of network lifetime. Because of the limited energy of the nodes, prolonging the network lifetime becomes another goal in the design of the topology. According to the radio frequency model [25], every node exchanges data with its all neighbor nodes in each round, and we record the network lifetime under different proportions of failure nodes. We implement the BA-E model, the EAEM model, the CHMM model and the BA model. Fig. 6 represents the network lifetime comparison of topologies under the proportions of failure nodes (Primary node, 10%, 30%, 50%, 70% and 80%), respectively. As is shown in Fig. 6, by comparing with the other three models, the BA-E model make the topology has a much longer network lifetime, and the network lifetime is prolonged when the failure of primary node appears. Additionally, the network lifetime is increased by 7.8% approximately when the proportion of failure nodes are 10%; and it is also better than the other three models when the proportion of failure nodes are 50%. Moreover, the BA-E model increases the network lifetime more with the ascension of the proportion of failure nodes. This is mainly because the BA-E model considers the fault-tolerance and the intrusion-tolerance in the design of the scale-free topology. So the degree distribution of scale-free topology is more uniform and the node’s energy consumption is more balanced. That is to say, the BA-E topology also improves the network lifetime. Please cite this article as: H. Liu et al., A scale-free topology model with fault-tolerance and intrusion-tolerance in wireless sensor networks, Computers and Electrical Engineering (2016), http://dx.doi.org/10.1016/j.compeleceng.2016.01.003

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Fig. 6. Comparison of network lifetime.

6. Conclusions Designing the topology which can tolerate both random faults and selective remove attacks is one of the most focused research issues in WSNs. It has been shown that the power-law scaling exponent can affect network fault-tolerance against random faults and network intrusion-tolerance against selective remove attacks. To the best of our knowledge, we are the first that propose the BA-E scale-free topology model for WSNs that it has the ability to change the power-law scaling exponent within [3, +∞ ) by adjusting its parameter. The adjustable power-law degree distribution improves the weak intrusiontolerance of scale-free topology without breaking the strong fault-tolerance. The required time of the BA-E model is theoretically analyzed. According to the analysis result, BA-E model has O((N − m0 − 1 )(1 + m )) time complexity for N sensor nodes. The performance of the BA-E model is compared with three existing models in simulation experiments. The BA-E model has been shown to outperform all these contrast models. It has the stronger network fault-tolerance, and it also can maximize the network intrusion-tolerance and prolong the network lifetime. In the future, this work deserves to be considered for studying the interrelation between the fault-tolerance and the intrusion-tolerance with node faults or link faults coexistence. Acknowledgments The authors are thankful to the anonymous reviewers for their valuable comments. This work is supported by the Natural Science Foundation of Hebei Province of China under Grant nos. F2015203091 and F2015203212. The Scientific and Technological Research and Development Planning Projects of Qinhuangdao City of China under Grant no. 201502A216. The Independent Research Project Topics B Category for Young Teacher of Yanshan University of China under Grant no. 14LGB017, and Yanshan University Doctoral Foundation under Grant no. B867. References [1] Younis M, Senturk I, Akkaya K, Lee S, Senel F. Topology management techniques for tolerating node failures in wireless sensor networks: a survey. Comput Netw 2014;58:254–83. [2] Albert R, Jeong H, Barabasi AL. Error and attack tolerance of complex network. Nature 2000;406:378–82. [3] Yang LX, Yang XF. The spread of computer viruses over a reduced scale-free network. Physica A 2014;396:173–84. [4] Momhammad ASM, Mahdi J, Zohreh A. Topology and vulnerability of the Iranian power grid. Physica A 2014;406:24–33. [5] Peng GS, Wu J. Optimal network topology for structural robustness based on natural connectivity. Physica A 2016;443:212–20. [6] Bari A, Jaekel A, Jiang J, Xu YF. Design of fault tolerant wireless sensor networks satisfying survivability and lifetime requirements. Comput Commun 2012;35:320–33. [7] Barabasi AL, Ravasz E, Vicsek T. Deterministic scale-free networks. Physica A 2001;3-4:559–64. [8] Zhu HL, Luo H, Peng HP, Li LX, Luo Q. Complex networks-based energy-efficient evolution model for wireless sensor networks. Chaos Solitons Fractals 2009;41:1828–35. [9] Qi XG, Ma SQ, Zheng GZ. Topology evolution of wireless sensor networks based on adaptive free-scale networks. J Iran Chem Soc 2011;8:467–75. [10] Chen LJ, Liu M, Chen DX, Xie L. Topology evolution of wireless sensor networks among cluster heads by random walkers. Chin J Comput 2009;32:69– 76. [11] Saffre F, Jovanovic H, Hoile C, Nicolas S. Scale-free topology for pervasive networks. BT Technol J 2004;22:200–8. [12] Liu LF, Qi XG, Xue JL, Xie M. A topology construct and control model with small-world and scale-free concepts for heterogeneous sensor networks. Int J Distrib Sens Netw 2014;1:1–8. [13] Holme P, kim BJ. Growing scale-free networks with tunable clustering. Phys Rev E 2002;65:026107. [14] Zheng GZ, Liu SY, Qi XG. Scale-free topology evolution for wireless sensor networks with reconstruction mechanism. Comput Electr Eng 2012;38:643– 51. [15] Yin RR, Liu B, Liu HR, Li YQ. The critical load of scale-free fault-tolerant topology in wireless sensor networks for cascading failures. Physica A 2014;409:8–16. [16] Xiao S, Xiao G, Cheng TH. Tolerance of international attacks in complex communication networks. IEEE Commun Mag 2008;46:146–52.

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Haoran Liu is an assistant professor in the college of information science and technology from the Yanshan University, Qinhuangdao, China. He has received his Ph.D. degree in the institute national des sciences appliqués de Rouen in 2009. His current research interests lie in the area of wireless sensor networks and fault-tolerant topology control. Rongrong Yin is a lecturer in the college of information science and technology from the Yanshan University, Qinhuangdao, China. She has received her Ph.D. degree in control theory and engineering from the institute of electrical engineering in 2014. Her current research interests lie in the area of wireless sensor networks and fault-tolerant topology control. Bin Liu is a professor in the college of information science and technology from the Yanshan University, Qinhuangdao, China. He has received his Ph.D. degree in the institute electrical engineering from the Hebei University of Technology, Tianjin, P.R. China in 2009. His current research interests lie in the area of low power wireless sensor networks and topology control. Yaqian Li is an assistant professor in the institute electrical engineering from the Yanshan University, Qinhuangdao, China. She has received her Ph.D. degree in the institute national des sciences appliqués de Rouen in 2010. Her current research interests lie in the area of wireless sensor networks and fault-tolerant topology control.

Please cite this article as: H. Liu et al., A scale-free topology model with fault-tolerance and intrusion-tolerance in wireless sensor networks, Computers and Electrical Engineering (2016), http://dx.doi.org/10.1016/j.compeleceng.2016.01.003