- Email: [email protected]
Critical thresholds for scale-free networks against cascading failures
Physica A 416 (2014) 252–258
Contents lists available at ScienceDirect
Physica A journal homepage: www.elsevier.com/locate/physa
Critical thresholds for scale-free networks against cascading failures Duan Dong-Li a,∗ , Ling Xiao-Dong b , Wu Xiao-Yue c , OuYang Di-Hua a , Zhong Bin a a
Engineering University of Armed Police Force, Xi’an, 710086, China
b
China Satellite Maritime Tracking and Control Department, Jiangyin, 214431, China
c
College of Information Systems and Management, National University of Defense Technology, Changsha, 410073, China
highlights • • • •
A load redistribution model under single node loss in networks is proposed. Thresholds can determine if a network is susceptible to breakdown, or is immune. Changing trends for two critical thresholds are in an opposite way with τ and β . γ can affect the cascading process of scale-free network within LLPSR and GLPSR.
article
info
Article history: Received 11 December 2013 Received in revised form 11 August 2014 Available online 6 September 2014 Keywords: Scale-free networks Cascading failures Robustness Critical thresholds
abstract We explore the critical thresholds of scale-free networks against cascading failures with a tunable load redistribution model which can tune the load redistribution range and heterogeneity of the broken node. Research suggests that the critical behavior belongs to the universality class of global load preferential sharing rule (GLPSR) and local load preferential sharing rule (LLPSR) in networks. Networks collapse completely when α < αc1 and are immune to single failure when α > αc2 . The changing trends for the critical thresholds of αc1 and αc2 are in an opposite way with initial load distribution coefficient and redistribution heterogeneity coefficient. It means networks may show different properties in the middle ground between total robustness and total collapse. Another striking finding is that the decrease of the exponent γ (γ > 1) of scale-free networks would make the system stronger against cascading failure within LLPSR and GLPSR. © 2014 Elsevier B.V. All rights reserved.
1. Introduction In recent years, major disasters occur frequently on the Internet, power grids as well as other infrastructure networks, attracting more and more experts in various fields to study these catastrophic events [1–8]. From the current study results, most of these disasters can be considered, to some extent, to be triggered by minor events. The network in the real world should carry materials, energy, information, date, or some form of loads in its evolutionary process. All these networks have some dynamic behaviors, such as the network topologies have to undertake all forms of flows. Hence, the overload mechanism of cascading failure (OMCF) is a simplified model that is often used to represent real dynamics. It has been successfully
∗
Corresponding author. Tel.: +86 02984563712. E-mail address: [email protected] (D.-L. Duan).
http://dx.doi.org/10.1016/j.physa.2014.08.040 0378-4371/© 2014 Elsevier B.V. All rights reserved.
D.-L. Duan et al. / Physica A 416 (2014) 252–258
253
used to probe the behavior of catastrophic accidents on networks. This paper focuses on some of the basic properties of this dynamic behavior. The OMCF was introduced into complex networks by Motter and Lai [9] with an ML model, in which the initial load Fi of node i is assumed to be its betweenness, its capacity Ci is proportional to its initial load: Ci = (1 + α)Fi , and the damage caused by a cascade is quantified in terms of the size of the largest connected component after the cascade relative to its original size. Here, the constant α is a capacity coefficient representing the ability of the node to afford extra load, which actually characterizes the fault tolerance of networks. Networks can handle better with the cascade if the value of α is designed higher. Intuitively, for a given network, there always are two thresholds of α , where cascading failures can cause the network to disintegrate almost entirely when α < αc1 , and the network is immune to cascading breakdown when α > αc2 . αc1 and αc2 respectively are the first and the second phase-transition points. The simulation results in Ref. [9] show that the two phase-transition points do exist. Zhao et al. uncovered the phase-transition phenomenon of scale-free networks with the ML model, and the analytic formulas of αc1 and αc2 have been determined [10,11]. The load redistribution mechanism of the ML model is based on the concept of betweenness, which means that the betweenness of the new network formed by removal of one node represents the updated load. The dynamic process of this mechanism is based on a fundamental assumption that all nodes in the network can master the global information to fit the new network instantly. Compared with the global redistribution mechanism of the ML model, Wang and Chen [12] proposed a nearest neighbor load redistribution model, in which the load of the broken node is carried by its nearest neighbors, and Wu et al. [13] examined the impact of the initial load distribution on the cascading process of the system. Wang and Rong [14,15] improved the model by adjusting the preferential weight, in which the degree of the broken node as well as its neighbor’s degree is considered. It assumes that the initial load distribution strength be equal to the load redistribution homogeneity parameter. Using the normalized avalanche size to quantify the robustness of the network based on the improved model, an estimate of lower bound of the second phase-transition point is given. The load redistribution mechanism of these neighbor sharing rules is based on a typical concept that the node in the system can only master its neighbor’s information. However, the node in the real system is always an agent, which may not only obtain its neighbor’s surrounding information but also it can grasp some global information through certain means. When congestion occurs in the traffic network, the information conveyed by the traffic police in congested intersection and real-time traffic broadcasts would affect the distribution of traffic flow to some extent, which would help the traffic flow intending to cross the congestion road reselect a more appropriate path. In addition, there is a similar situation on the Internet. Once one critical router failed, the control system has to reselect routers to transfer data. The routers redistributed by the control system may not only include its neighbor routers. According to the redistribution rule of real networks always lies between the global preferential rule and the local preferential rule or between even the shared rule and the extremely heterogeneous rule, a new cascading model is proposed based on a tunable load redistribution model. It can tune the load redistribution range and the redistribution heterogeneity of extra load respectively by a redistribution range coefficient and a redistribution heterogeneity coefficient. In fact, this model can reflect the management effect of the network administrator. In this paper, we investigate a model for load redistribution under single node loss in scale-free networks and characterize the critical thresholds of a robustness parameter that determine if a network is susceptible to total disintegration or is immune. From the physical sense, the thresholds correspond to two types of phase-transition phenomenon on networks. Single threshold is difficult to fully and accurately reflect the network’s robustness against cascading failure, as it just means that there is a sharp transition from total robustness to susceptibility to total collapse. In theory, we can easily understand that the overall resilience of the network is improved as both of αc1 and αc2 decrease. But, do the thresholds decrease as we desire when something about the network changes? Based on the tunable load distribution model, this paper mainly focuses on the evolution mechanism of the critical thresholds on scale-free networks and discusses the properties of the middle ground between total robustness and total susceptibility with an analytical method and numerical simulation. 2. Cascading failure model For simplicity, assume that initial load Fi of node i be a function of its degree ki and defined as Fi = ρ kτi . Here, ρ is a constant value. τ is a tunable parameter, which controls the strength of the initial load. This assumption of non-dimensional structure load is reasonable and valid when it is hard to determine the real load of the system, e.g., data flow of every node on Internet usually has a certain correlation with its degree. Following the previous ML model, each node i in the network has a capacity coefficient, which is the maximum flow that the node can carry. Since the node capacity on real-life networks is generally limited by cost and technique, it is natural to assume that the capacity Ci of the node i be proportional to its initial load for simplicity: Ci = (1 + α)Fi [9,15]. The load of the broken node i will be redistributed to some or all of the intact nodes, which would cause one update of Fj Fj → Fj′ = Fj + 1Fj .
(1)
Assume that the extra load 1Fj of node j be proportional to the broken load Fi
1Fj = Fi · p(lij , θ , kj , β).
(2)
Here, p(lij , θ , kj , β) is the preferential probability, lij represents the distance from node i to node j, if they are 1-order neighbors, then lij = 1. θ and β are redistribution policy parameters respectively to control the redistribution range and
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Fig. 1. Simulation results of CFN within LLPSR and GLPSR for BA networks (N = 100, m = 2, and m0 = 2 [1]). (a) CFN within LLPSR; (b) CFN within GLPSR.
β
homogeneity. The weight of node j to share the broken load of node i is one function of lij and kj as ηij = cl−θ ij kj . Whereby, the preferential probability function is β
β
cl−θ l−θ ij kj ij kj p(lij , θ , kj , β) = = . β −θ β clim km l−θ im km m∈Ωi
(3)
m∈Ωi
Here, c is a constant value, Ωi represents the set of the intact nodes, θ ∈ [0, ∞), β ∈ [0, ∞). When θ = 0, l−θ im = 1 for all intact nodes, the redistribution policy is a global load preferential sharing rule (GLPSR), the load is distributed to all the intact nodes in some way based on degree. If β = 0 when θ = 0, it is a global load evenly sharing rule (GLESR), the load is distributed to all the intact nodes evenly. When θ = ∞, l−θ im = 1 only for lim = 1, the redistribution policy is a local load preferential sharing rule (LLPSR), in which the load of the broken node is transferred only to its neighbor nodes in some way based on degree. If β = 0 when θ = ∞, it is a local load evenly sharing rule (LLESR). The situation of 0 < θ < ∞ is actually an intermediate redistribution rule between GLPSR and LLPSR, which is not considered in this paper. We will discuss it in future work. In our numerical simulation, when θ ≤ 0.01, we consider the rule as the global sharing rule, and when θ > 106 we consider the rule as the local sharing rule. Following the previous measure index of networks’ robustness against cascading failures [16–18], we adopt the normalized avalanche size to quantify the robustness of the whole network N
CFN =
CFi
i =1
N (N − 1)
.
(4)
Here, CFi denotes the number of broken nodes induced by removing node i. Clearly, 0 ≤ CFi ≤ (N − 1). In the numerical experiment, the node is removed one by one, and CFi is calculated following the order. We mainly focus on the properties of the critical thresholds and the relationship between them and network robustness against cascading failure. The simulation way and results of the above cascading model can be referred to in our previous works; see Refs. [16–18]. Fig. 1 shows the CFN simulation results of BA networks, where the data points represent the normalized avalanche size by attacking one node, calculated by the mean value of 30 networks with N = 100, m = 2(⟨k⟩ = 4), and m0 = 2: here, Fig. 1(a) is within the rule of LLPSR and Fig. 1(b) is within the rule of GLPSR. It can be seen that there are two phase-transition points at some parameter region, in which CFN = 1 when α < αc1 and CFN = 0 when α > αc2 . The results in Fig. 1(a) and (b) also indicate that different trends exist for αc1 and αc2 within the rule of LLPSR and GLPSR. 3. Critical thresholds for scale-free networks To avoid the breakdown by failed node i, the node j among the intact nodes should satisfy: Fj + 1Fj < Cj .
(5)
According to the definition of the initial load in Section 2 and the load redistribution mechanism of the failed node from (1)–(3), the above condition formula can be simplified as β
τ
τ
Fj + Fi · p(lij , θ , kj , β) < (1 + α)Fj ⇒ ρ kj + ρ ki
β−τ
cl−θ ij kj
m∈Ωm
β
cl−θ im km
τ
τ l−θ ij ki kj
< (1 + α)ρ kj ⇒
m∈Ωm
β
l−θ im km
< α.
(6)
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Fig. 2. Comparison of simulation results and analytic results of critical thresholds within LLPSR for the BA network with N = 1000, m = 3, and m0 = 2. (a) Error bar and analytic results of αc1 (τ = 0.5); (b) error bar and analytic results of αc2 (τ = 1.5).
According to the above conditional formula, we can see that the constants c and ρ cancel out in the final form. Hence, l
−θ β−τ τ kj ki
the second critical threshold can be defined as αc2 = max ij
−θ β
m∈Ω lim km
. It is a lower bound that the network is immune to
cascading breakdown. Namely, CFN = 0 when α > αc2 . Similarly, the first critical threshold can be defined as αc1 = min −θ β−τ τ lij kj ki
−θ β
m∈Ω lim km
. It is an upper bound that single failure can cause the network to disintegrate almost entirely. Namely, CFN = 1
when α < αc1 . 3.1. Critical thresholds within LLPSR When θ → +∞, only the neighboring nodes of the failed one share the load. So, the conditional formula (6) of cascading β−τ τ k ki
< α , where Γi represents the set of the neighboring nodes of the failed node i. kmax β We can easily obtain m∈Γi km = k P (k′ |ki )k′β , where P (k′ |ki ) is the conditional probability that a node of ki has k′ =kmin i ′ a neighbor of k . In networks without degree–degree correlations, it can be obtained that P (k′ |ki ) = k′ P (k′ )/⟨k⟩. Therefore,
failure can be rewritten as j
β
m∈Γi km
the condition formula within LLPSR can be simplified as β−τ τ −1
kj
ki
⟨k⟩
⟨kβ+1 ⟩
< α.
(7)
Considering the degree distribution of scale-free networks, we can obtain γ −1
⟨kβ ⟩ = (γ − 1)kmin
kmax
kβ−γ dk =
kmin
kmax (γ − 1) γ −1 . kmin kβ−γ +1 k min (β − γ + 1)
(8)
Here, γ is the exponent of scale-free networks, and kmin and kmax are the minimum and maximum node degrees, respectively. 1
In scale-free networks, ⟨k⟩ ≈ 2kmin and kmax ≈ kmin N γ −1 . According to the definition of αc1 and αc2 , and the formula of (7), the critical thresholds within LLPSR are
αc1 =
∇(β, γ , ⟨k⟩, N )N ϑ(τγ −)−1 1 , (τ ) ∇(β, γ , ⟨k⟩, N )N β−ξ γ −1 ,
Here, ∇(β, γ , ⟨k⟩, N ) =
) ∇(β, γ , ⟨k⟩, N )N β−ϑ(τ γ −1 ,
β≥τ
∇(β, γ , ⟨k⟩, N )N ξ (τγ −)−1 1 , β<τ τ, τ ≥ 1 1, τ ≥ 1 , ξ (τ ) = 1, τ < 1 , ϑ(τ ) = τ , τ < 1 .
β < τ.
β≥τ
4(β−γ +2) β−γ +2 (γ −1)⟨k⟩(N γ −1 −1)
αc2 =
(9)
From the above formula, we conclude that: (1) fix the value of τ , αc1 reaches its local maximum and αc2 reaches its local minimum when β = τ . When β = τ = 1, αc1 reaches its global maximum and αc2 reaches its global minimum; (2) the increase of ⟨k⟩ can lower the αc1 and αc2 , which makes it easier for the network to be immune to disintegration, in the sense that the capacity parameter α does not have to be as large; (3) the decrease of γ (γ > 1) will make both the αc1 and αc2 smaller. Comparison of simulation results and analytic results of critical thresholds within LLPSR for the BA network with N = 1000, m = 3, and m0 = 2 is shown in Figs. 2 and 3. It should be noted that the expectation and variance in simulation are calculated by generating 30 networks with the BA algorithm in Ref. [1]. As can be seen from Fig. 2(a), the first phase-transition point reaches its maximum value at β = 0.5 taking the initial load intensity parameter τ = 0.5 in simulation. The system collapses entirely once one node breaks when the capacity coefficient is below the first phase-transition point. From the subgraph of Fig. 2(a), αc1 reaches its global maximum at β = τ = 1. It also can be seen from Fig. 2(b) that the second phasetransition point reaches its minimum value at β = 1.5 taking the initial load intensity parameter τ = 1.5 in simulation.
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Fig. 3. Critical thresholds of αc1 and αc2 vs. τ and β within LLPSR for the BA network with N = 1000, ⟨k⟩ = 6 (a) αc1 vs. τ and β ; (b) αc2 vs. τ and β . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
The system is immune to single failure when the capacity coefficient is above the second phase-transition point. From the subgraph of Fig. 2(b), αc2 reaches its global minimum at β = τ = 1. Fig. 3 is the pseudo-color image of the critical thresholds with τ and β . It can be seen that the changing trends for αc1 and αc2 are in an opposite way with the two parameters. The impacts of topology parameters on cascading failures for more general scale-free networks are shown in Fig. 4. As can be seen from Fig. 4(a), αc1 and αc2 increase with γ . It means that the robustness of scale-free networks against cascading failure tends to be stronger as the degree distribution gets more homogeneous. From the formula of (7) and the definition of αc1 and αc2 , we can see that αc1 = αc2 when β = τ = 1. This value is denoted as αc , namely αc = αc1 = αc2 when β = τ = 1. It was used to describe the universal robustness of networks in Refs. [14,15], independent of the initial load distribution strength and load redistribution homogeneity. However, this single threshold also means that there is no middle ground between total robustness and total susceptibility. We check the value with power-law degree distributions (γ = 2.5, 3.0, and 3.5) for different ⟨k⟩. The results can be seen in Fig. 4(b). It shows that the increase of ⟨k⟩ will make the value smaller, but the trend is in the opposite way for the increase of γ . 3.2. Critical thresholds within GLPSR When θ = 0, all the intact nodes share the load. So, the conditional formula (6) of cascading failure can be rewritten as
β−τ τ kj ki
β
m∈Ωi km
< α , where Ωi represents the set of the intact nodes once node i failed.
According to the node degree and the probability theory in the networks without degree–degree correlations, it can be β obtained that m∈Ωi km = (N − 1)⟨kβ+1 ⟩/⟨k⟩. Therefore, the condition formula within GLPSR can be simplified as β−τ τ
kj
ki ⟨k⟩
(N − 1)⟨kβ+1 ⟩
< α.
(10)
According to the definition of αc1 and αc2 , the critical thresholds within GLPSR are
β−τ γ −1 , β<τ αc1 = ∆(β, γ , N )N ∆(β, γ , N ), β ≥ τ Here, ∆(β, γ , N ) =
2(β−γ +2) . β−γ +2 (N −1)(γ −1)(N γ −1 −1)
αc2 =
τ
∆(β, γ , N )N γ −1 , ∆(β, γ , N )N
β γ −1
,
β<τ β ≥ τ.
(11)
From the formula of (11), we conclude that: (1) fix the value of τ , αc1 reaches its local maximum and αc2 reaches its local minimum when β = τ . When β = τ = 0, αc1 reaches its global maximum and αc2 reaches its global minimum; (2) the average degree of ⟨k⟩ has no impact on the robustness against cascading failure within GLPSR; (3) the decrease of γ will make the system stronger as the two thresholds become smaller. Comparison of simulation results and analytic results of critical thresholds within GLPSR for the BA network with N = 1000, m = 3, and m0 = 2 is shown in Fig. 5. The expectation and variance in simulation are calculated by generating 30 networks with the BA algorithm in Ref. [1]. As can be seen from Fig. 5(a), the first phase-transition point reaches its maximum value and the second phase-transition point reaches its minimum value at β = τ taking the initial load intensity parameter τ = 0.1, 0.5, 1.0, and 1.5 in simulation. The system collapses entirely once one node breaks when the capacity coefficient is below the first phase-transition point. But, the system is immune to single failure when the capacity coefficient is above the second phase-transition point. Fig. 5(b) shows the error bar of the two critical thresholds αc1 and αc2 , which is the average results of 30 simulations. As can be seen that the changing trends for αc1 and αc2 are in an opposite way with β = τ .
D.-L. Duan et al. / Physica A 416 (2014) 252–258
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Fig. 4. Critical thresholds of αc1 and αc2 vs. topology parameters of γ and ⟨k⟩. (a) αc1 and αc2 vs. γ ; (b) αc1 and αc2 vs. ⟨k⟩ (β = τ = 1).
Fig. 5. Comparison of simulation results and analytic results of critical thresholds within GLPSR for the BA network with N = 1000, m = 3, and m0 = 2. (a) Simulation results of αc1 and αc2 ; (b) error bar and analytic results of αc1 and αc2 (β = τ ).
4. Conclusions and discussion In conclusion, we propose a tunable load redistribution model and investigate the critical thresholds of scale-free networks against cascading failures within LLPSR and GLPSR. We find that various parameters play an important role in robustness against cascading failure. By the analysis of the critical thresholds αc1 and αc2 , we obtain the theoretical estimate for the two phase-transition points and provide a numerical check. In addition, we draw some interesting and counterintuitive conclusions: (1) Impacts of load redistribution rang: the increase of load redistribution rang (θ from ∞ to 0) would minify both the critical thresholds. It is easy to understand, as the more global information obtained, the easier the system fits the new network instantly. (2) Impacts of initial load distribution strength or load redistribution homogeneity parameter: the simulation and analytic results both show that the changing trends for the critical thresholds of αc1 and αc2 are in an opposite way with β or τ . There is a contradiction between fault tolerance against single failure and robustness against complete breakdown to control τ and β . This might explain why blackout of current power grid occurs frequently even satisfying N − 1 rule strictly. (3) Impacts of topology parameters: the decrease of γ (γ > 1) and the increase of ⟨k⟩ will make both the αc1 and αc2 smaller. But it should be noted that the impact of ⟨k⟩ will fade away with the increase of load redistribution rang. In addition, the simulation results shown in Figs. 2 and 5 do not match exactly with the theory, which mainly results from some approximation in the analysis. For example, to calculate ⟨kβ ⟩ in Eq. (7), we use the approximation of ⟨k⟩ ≈ 2kmin and 1
kmax ≈ kmin N γ −1 . Another underlying reason may be the generating mechanism of scale-free networks in the algorithm. We will mainly focus on this problem in future work. Acknowledgments This work was supported in part by the open funding programme of joint laboratory of flight vehicle ocean-based measurement and control under Grant No. FOM2014OF016 and in part by the National Natural Science Foundation of China under Grant No. 71401178.
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Contents lists available at ScienceDirect
Physica A journal homepage: www.elsevier.com/locate/physa
Critical thresholds for scale-free networks against cascading failures Duan Dong-Li a,∗ , Ling Xiao-Dong b , Wu Xiao-Yue c , OuYang Di-Hua a , Zhong Bin a a
Engineering University of Armed Police Force, Xi’an, 710086, China
b
China Satellite Maritime Tracking and Control Department, Jiangyin, 214431, China
c
College of Information Systems and Management, National University of Defense Technology, Changsha, 410073, China
highlights • • • •
A load redistribution model under single node loss in networks is proposed. Thresholds can determine if a network is susceptible to breakdown, or is immune. Changing trends for two critical thresholds are in an opposite way with τ and β . γ can affect the cascading process of scale-free network within LLPSR and GLPSR.
article
info
Article history: Received 11 December 2013 Received in revised form 11 August 2014 Available online 6 September 2014 Keywords: Scale-free networks Cascading failures Robustness Critical thresholds
abstract We explore the critical thresholds of scale-free networks against cascading failures with a tunable load redistribution model which can tune the load redistribution range and heterogeneity of the broken node. Research suggests that the critical behavior belongs to the universality class of global load preferential sharing rule (GLPSR) and local load preferential sharing rule (LLPSR) in networks. Networks collapse completely when α < αc1 and are immune to single failure when α > αc2 . The changing trends for the critical thresholds of αc1 and αc2 are in an opposite way with initial load distribution coefficient and redistribution heterogeneity coefficient. It means networks may show different properties in the middle ground between total robustness and total collapse. Another striking finding is that the decrease of the exponent γ (γ > 1) of scale-free networks would make the system stronger against cascading failure within LLPSR and GLPSR. © 2014 Elsevier B.V. All rights reserved.
1. Introduction In recent years, major disasters occur frequently on the Internet, power grids as well as other infrastructure networks, attracting more and more experts in various fields to study these catastrophic events [1–8]. From the current study results, most of these disasters can be considered, to some extent, to be triggered by minor events. The network in the real world should carry materials, energy, information, date, or some form of loads in its evolutionary process. All these networks have some dynamic behaviors, such as the network topologies have to undertake all forms of flows. Hence, the overload mechanism of cascading failure (OMCF) is a simplified model that is often used to represent real dynamics. It has been successfully
∗
Corresponding author. Tel.: +86 02984563712. E-mail address: [email protected] (D.-L. Duan).
http://dx.doi.org/10.1016/j.physa.2014.08.040 0378-4371/© 2014 Elsevier B.V. All rights reserved.
D.-L. Duan et al. / Physica A 416 (2014) 252–258
253
used to probe the behavior of catastrophic accidents on networks. This paper focuses on some of the basic properties of this dynamic behavior. The OMCF was introduced into complex networks by Motter and Lai [9] with an ML model, in which the initial load Fi of node i is assumed to be its betweenness, its capacity Ci is proportional to its initial load: Ci = (1 + α)Fi , and the damage caused by a cascade is quantified in terms of the size of the largest connected component after the cascade relative to its original size. Here, the constant α is a capacity coefficient representing the ability of the node to afford extra load, which actually characterizes the fault tolerance of networks. Networks can handle better with the cascade if the value of α is designed higher. Intuitively, for a given network, there always are two thresholds of α , where cascading failures can cause the network to disintegrate almost entirely when α < αc1 , and the network is immune to cascading breakdown when α > αc2 . αc1 and αc2 respectively are the first and the second phase-transition points. The simulation results in Ref. [9] show that the two phase-transition points do exist. Zhao et al. uncovered the phase-transition phenomenon of scale-free networks with the ML model, and the analytic formulas of αc1 and αc2 have been determined [10,11]. The load redistribution mechanism of the ML model is based on the concept of betweenness, which means that the betweenness of the new network formed by removal of one node represents the updated load. The dynamic process of this mechanism is based on a fundamental assumption that all nodes in the network can master the global information to fit the new network instantly. Compared with the global redistribution mechanism of the ML model, Wang and Chen [12] proposed a nearest neighbor load redistribution model, in which the load of the broken node is carried by its nearest neighbors, and Wu et al. [13] examined the impact of the initial load distribution on the cascading process of the system. Wang and Rong [14,15] improved the model by adjusting the preferential weight, in which the degree of the broken node as well as its neighbor’s degree is considered. It assumes that the initial load distribution strength be equal to the load redistribution homogeneity parameter. Using the normalized avalanche size to quantify the robustness of the network based on the improved model, an estimate of lower bound of the second phase-transition point is given. The load redistribution mechanism of these neighbor sharing rules is based on a typical concept that the node in the system can only master its neighbor’s information. However, the node in the real system is always an agent, which may not only obtain its neighbor’s surrounding information but also it can grasp some global information through certain means. When congestion occurs in the traffic network, the information conveyed by the traffic police in congested intersection and real-time traffic broadcasts would affect the distribution of traffic flow to some extent, which would help the traffic flow intending to cross the congestion road reselect a more appropriate path. In addition, there is a similar situation on the Internet. Once one critical router failed, the control system has to reselect routers to transfer data. The routers redistributed by the control system may not only include its neighbor routers. According to the redistribution rule of real networks always lies between the global preferential rule and the local preferential rule or between even the shared rule and the extremely heterogeneous rule, a new cascading model is proposed based on a tunable load redistribution model. It can tune the load redistribution range and the redistribution heterogeneity of extra load respectively by a redistribution range coefficient and a redistribution heterogeneity coefficient. In fact, this model can reflect the management effect of the network administrator. In this paper, we investigate a model for load redistribution under single node loss in scale-free networks and characterize the critical thresholds of a robustness parameter that determine if a network is susceptible to total disintegration or is immune. From the physical sense, the thresholds correspond to two types of phase-transition phenomenon on networks. Single threshold is difficult to fully and accurately reflect the network’s robustness against cascading failure, as it just means that there is a sharp transition from total robustness to susceptibility to total collapse. In theory, we can easily understand that the overall resilience of the network is improved as both of αc1 and αc2 decrease. But, do the thresholds decrease as we desire when something about the network changes? Based on the tunable load distribution model, this paper mainly focuses on the evolution mechanism of the critical thresholds on scale-free networks and discusses the properties of the middle ground between total robustness and total susceptibility with an analytical method and numerical simulation. 2. Cascading failure model For simplicity, assume that initial load Fi of node i be a function of its degree ki and defined as Fi = ρ kτi . Here, ρ is a constant value. τ is a tunable parameter, which controls the strength of the initial load. This assumption of non-dimensional structure load is reasonable and valid when it is hard to determine the real load of the system, e.g., data flow of every node on Internet usually has a certain correlation with its degree. Following the previous ML model, each node i in the network has a capacity coefficient, which is the maximum flow that the node can carry. Since the node capacity on real-life networks is generally limited by cost and technique, it is natural to assume that the capacity Ci of the node i be proportional to its initial load for simplicity: Ci = (1 + α)Fi [9,15]. The load of the broken node i will be redistributed to some or all of the intact nodes, which would cause one update of Fj Fj → Fj′ = Fj + 1Fj .
(1)
Assume that the extra load 1Fj of node j be proportional to the broken load Fi
1Fj = Fi · p(lij , θ , kj , β).
(2)
Here, p(lij , θ , kj , β) is the preferential probability, lij represents the distance from node i to node j, if they are 1-order neighbors, then lij = 1. θ and β are redistribution policy parameters respectively to control the redistribution range and
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Fig. 1. Simulation results of CFN within LLPSR and GLPSR for BA networks (N = 100, m = 2, and m0 = 2 [1]). (a) CFN within LLPSR; (b) CFN within GLPSR.
β
homogeneity. The weight of node j to share the broken load of node i is one function of lij and kj as ηij = cl−θ ij kj . Whereby, the preferential probability function is β
β
cl−θ l−θ ij kj ij kj p(lij , θ , kj , β) = = . β −θ β clim km l−θ im km m∈Ωi
(3)
m∈Ωi
Here, c is a constant value, Ωi represents the set of the intact nodes, θ ∈ [0, ∞), β ∈ [0, ∞). When θ = 0, l−θ im = 1 for all intact nodes, the redistribution policy is a global load preferential sharing rule (GLPSR), the load is distributed to all the intact nodes in some way based on degree. If β = 0 when θ = 0, it is a global load evenly sharing rule (GLESR), the load is distributed to all the intact nodes evenly. When θ = ∞, l−θ im = 1 only for lim = 1, the redistribution policy is a local load preferential sharing rule (LLPSR), in which the load of the broken node is transferred only to its neighbor nodes in some way based on degree. If β = 0 when θ = ∞, it is a local load evenly sharing rule (LLESR). The situation of 0 < θ < ∞ is actually an intermediate redistribution rule between GLPSR and LLPSR, which is not considered in this paper. We will discuss it in future work. In our numerical simulation, when θ ≤ 0.01, we consider the rule as the global sharing rule, and when θ > 106 we consider the rule as the local sharing rule. Following the previous measure index of networks’ robustness against cascading failures [16–18], we adopt the normalized avalanche size to quantify the robustness of the whole network N
CFN =
CFi
i =1
N (N − 1)
.
(4)
Here, CFi denotes the number of broken nodes induced by removing node i. Clearly, 0 ≤ CFi ≤ (N − 1). In the numerical experiment, the node is removed one by one, and CFi is calculated following the order. We mainly focus on the properties of the critical thresholds and the relationship between them and network robustness against cascading failure. The simulation way and results of the above cascading model can be referred to in our previous works; see Refs. [16–18]. Fig. 1 shows the CFN simulation results of BA networks, where the data points represent the normalized avalanche size by attacking one node, calculated by the mean value of 30 networks with N = 100, m = 2(⟨k⟩ = 4), and m0 = 2: here, Fig. 1(a) is within the rule of LLPSR and Fig. 1(b) is within the rule of GLPSR. It can be seen that there are two phase-transition points at some parameter region, in which CFN = 1 when α < αc1 and CFN = 0 when α > αc2 . The results in Fig. 1(a) and (b) also indicate that different trends exist for αc1 and αc2 within the rule of LLPSR and GLPSR. 3. Critical thresholds for scale-free networks To avoid the breakdown by failed node i, the node j among the intact nodes should satisfy: Fj + 1Fj < Cj .
(5)
According to the definition of the initial load in Section 2 and the load redistribution mechanism of the failed node from (1)–(3), the above condition formula can be simplified as β
τ
τ
Fj + Fi · p(lij , θ , kj , β) < (1 + α)Fj ⇒ ρ kj + ρ ki
β−τ
cl−θ ij kj
m∈Ωm
β
cl−θ im km
τ
τ l−θ ij ki kj
< (1 + α)ρ kj ⇒
m∈Ωm
β
l−θ im km
< α.
(6)
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Fig. 2. Comparison of simulation results and analytic results of critical thresholds within LLPSR for the BA network with N = 1000, m = 3, and m0 = 2. (a) Error bar and analytic results of αc1 (τ = 0.5); (b) error bar and analytic results of αc2 (τ = 1.5).
According to the above conditional formula, we can see that the constants c and ρ cancel out in the final form. Hence, l
−θ β−τ τ kj ki
the second critical threshold can be defined as αc2 = max ij
−θ β
m∈Ω lim km
. It is a lower bound that the network is immune to
cascading breakdown. Namely, CFN = 0 when α > αc2 . Similarly, the first critical threshold can be defined as αc1 = min −θ β−τ τ lij kj ki
−θ β
m∈Ω lim km
. It is an upper bound that single failure can cause the network to disintegrate almost entirely. Namely, CFN = 1
when α < αc1 . 3.1. Critical thresholds within LLPSR When θ → +∞, only the neighboring nodes of the failed one share the load. So, the conditional formula (6) of cascading β−τ τ k ki
< α , where Γi represents the set of the neighboring nodes of the failed node i. kmax β We can easily obtain m∈Γi km = k P (k′ |ki )k′β , where P (k′ |ki ) is the conditional probability that a node of ki has k′ =kmin i ′ a neighbor of k . In networks without degree–degree correlations, it can be obtained that P (k′ |ki ) = k′ P (k′ )/⟨k⟩. Therefore,
failure can be rewritten as j
β
m∈Γi km
the condition formula within LLPSR can be simplified as β−τ τ −1
kj
ki
⟨k⟩
⟨kβ+1 ⟩
< α.
(7)
Considering the degree distribution of scale-free networks, we can obtain γ −1
⟨kβ ⟩ = (γ − 1)kmin
kmax
kβ−γ dk =
kmin
kmax (γ − 1) γ −1 . kmin kβ−γ +1 k min (β − γ + 1)
(8)
Here, γ is the exponent of scale-free networks, and kmin and kmax are the minimum and maximum node degrees, respectively. 1
In scale-free networks, ⟨k⟩ ≈ 2kmin and kmax ≈ kmin N γ −1 . According to the definition of αc1 and αc2 , and the formula of (7), the critical thresholds within LLPSR are
αc1 =
∇(β, γ , ⟨k⟩, N )N ϑ(τγ −)−1 1 , (τ ) ∇(β, γ , ⟨k⟩, N )N β−ξ γ −1 ,
Here, ∇(β, γ , ⟨k⟩, N ) =
) ∇(β, γ , ⟨k⟩, N )N β−ϑ(τ γ −1 ,
β≥τ
∇(β, γ , ⟨k⟩, N )N ξ (τγ −)−1 1 , β<τ τ, τ ≥ 1 1, τ ≥ 1 , ξ (τ ) = 1, τ < 1 , ϑ(τ ) = τ , τ < 1 .
β < τ.
β≥τ
4(β−γ +2) β−γ +2 (γ −1)⟨k⟩(N γ −1 −1)
αc2 =
(9)
From the above formula, we conclude that: (1) fix the value of τ , αc1 reaches its local maximum and αc2 reaches its local minimum when β = τ . When β = τ = 1, αc1 reaches its global maximum and αc2 reaches its global minimum; (2) the increase of ⟨k⟩ can lower the αc1 and αc2 , which makes it easier for the network to be immune to disintegration, in the sense that the capacity parameter α does not have to be as large; (3) the decrease of γ (γ > 1) will make both the αc1 and αc2 smaller. Comparison of simulation results and analytic results of critical thresholds within LLPSR for the BA network with N = 1000, m = 3, and m0 = 2 is shown in Figs. 2 and 3. It should be noted that the expectation and variance in simulation are calculated by generating 30 networks with the BA algorithm in Ref. [1]. As can be seen from Fig. 2(a), the first phase-transition point reaches its maximum value at β = 0.5 taking the initial load intensity parameter τ = 0.5 in simulation. The system collapses entirely once one node breaks when the capacity coefficient is below the first phase-transition point. From the subgraph of Fig. 2(a), αc1 reaches its global maximum at β = τ = 1. It also can be seen from Fig. 2(b) that the second phasetransition point reaches its minimum value at β = 1.5 taking the initial load intensity parameter τ = 1.5 in simulation.
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Fig. 3. Critical thresholds of αc1 and αc2 vs. τ and β within LLPSR for the BA network with N = 1000, ⟨k⟩ = 6 (a) αc1 vs. τ and β ; (b) αc2 vs. τ and β . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
The system is immune to single failure when the capacity coefficient is above the second phase-transition point. From the subgraph of Fig. 2(b), αc2 reaches its global minimum at β = τ = 1. Fig. 3 is the pseudo-color image of the critical thresholds with τ and β . It can be seen that the changing trends for αc1 and αc2 are in an opposite way with the two parameters. The impacts of topology parameters on cascading failures for more general scale-free networks are shown in Fig. 4. As can be seen from Fig. 4(a), αc1 and αc2 increase with γ . It means that the robustness of scale-free networks against cascading failure tends to be stronger as the degree distribution gets more homogeneous. From the formula of (7) and the definition of αc1 and αc2 , we can see that αc1 = αc2 when β = τ = 1. This value is denoted as αc , namely αc = αc1 = αc2 when β = τ = 1. It was used to describe the universal robustness of networks in Refs. [14,15], independent of the initial load distribution strength and load redistribution homogeneity. However, this single threshold also means that there is no middle ground between total robustness and total susceptibility. We check the value with power-law degree distributions (γ = 2.5, 3.0, and 3.5) for different ⟨k⟩. The results can be seen in Fig. 4(b). It shows that the increase of ⟨k⟩ will make the value smaller, but the trend is in the opposite way for the increase of γ . 3.2. Critical thresholds within GLPSR When θ = 0, all the intact nodes share the load. So, the conditional formula (6) of cascading failure can be rewritten as
β−τ τ kj ki
β
m∈Ωi km
< α , where Ωi represents the set of the intact nodes once node i failed.
According to the node degree and the probability theory in the networks without degree–degree correlations, it can be β obtained that m∈Ωi km = (N − 1)⟨kβ+1 ⟩/⟨k⟩. Therefore, the condition formula within GLPSR can be simplified as β−τ τ
kj
ki ⟨k⟩
(N − 1)⟨kβ+1 ⟩
< α.
(10)
According to the definition of αc1 and αc2 , the critical thresholds within GLPSR are
β−τ γ −1 , β<τ αc1 = ∆(β, γ , N )N ∆(β, γ , N ), β ≥ τ Here, ∆(β, γ , N ) =
2(β−γ +2) . β−γ +2 (N −1)(γ −1)(N γ −1 −1)
αc2 =
τ
∆(β, γ , N )N γ −1 , ∆(β, γ , N )N
β γ −1
,
β<τ β ≥ τ.
(11)
From the formula of (11), we conclude that: (1) fix the value of τ , αc1 reaches its local maximum and αc2 reaches its local minimum when β = τ . When β = τ = 0, αc1 reaches its global maximum and αc2 reaches its global minimum; (2) the average degree of ⟨k⟩ has no impact on the robustness against cascading failure within GLPSR; (3) the decrease of γ will make the system stronger as the two thresholds become smaller. Comparison of simulation results and analytic results of critical thresholds within GLPSR for the BA network with N = 1000, m = 3, and m0 = 2 is shown in Fig. 5. The expectation and variance in simulation are calculated by generating 30 networks with the BA algorithm in Ref. [1]. As can be seen from Fig. 5(a), the first phase-transition point reaches its maximum value and the second phase-transition point reaches its minimum value at β = τ taking the initial load intensity parameter τ = 0.1, 0.5, 1.0, and 1.5 in simulation. The system collapses entirely once one node breaks when the capacity coefficient is below the first phase-transition point. But, the system is immune to single failure when the capacity coefficient is above the second phase-transition point. Fig. 5(b) shows the error bar of the two critical thresholds αc1 and αc2 , which is the average results of 30 simulations. As can be seen that the changing trends for αc1 and αc2 are in an opposite way with β = τ .
D.-L. Duan et al. / Physica A 416 (2014) 252–258
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Fig. 4. Critical thresholds of αc1 and αc2 vs. topology parameters of γ and ⟨k⟩. (a) αc1 and αc2 vs. γ ; (b) αc1 and αc2 vs. ⟨k⟩ (β = τ = 1).
Fig. 5. Comparison of simulation results and analytic results of critical thresholds within GLPSR for the BA network with N = 1000, m = 3, and m0 = 2. (a) Simulation results of αc1 and αc2 ; (b) error bar and analytic results of αc1 and αc2 (β = τ ).
4. Conclusions and discussion In conclusion, we propose a tunable load redistribution model and investigate the critical thresholds of scale-free networks against cascading failures within LLPSR and GLPSR. We find that various parameters play an important role in robustness against cascading failure. By the analysis of the critical thresholds αc1 and αc2 , we obtain the theoretical estimate for the two phase-transition points and provide a numerical check. In addition, we draw some interesting and counterintuitive conclusions: (1) Impacts of load redistribution rang: the increase of load redistribution rang (θ from ∞ to 0) would minify both the critical thresholds. It is easy to understand, as the more global information obtained, the easier the system fits the new network instantly. (2) Impacts of initial load distribution strength or load redistribution homogeneity parameter: the simulation and analytic results both show that the changing trends for the critical thresholds of αc1 and αc2 are in an opposite way with β or τ . There is a contradiction between fault tolerance against single failure and robustness against complete breakdown to control τ and β . This might explain why blackout of current power grid occurs frequently even satisfying N − 1 rule strictly. (3) Impacts of topology parameters: the decrease of γ (γ > 1) and the increase of ⟨k⟩ will make both the αc1 and αc2 smaller. But it should be noted that the impact of ⟨k⟩ will fade away with the increase of load redistribution rang. In addition, the simulation results shown in Figs. 2 and 5 do not match exactly with the theory, which mainly results from some approximation in the analysis. For example, to calculate ⟨kβ ⟩ in Eq. (7), we use the approximation of ⟨k⟩ ≈ 2kmin and 1
kmax ≈ kmin N γ −1 . Another underlying reason may be the generating mechanism of scale-free networks in the algorithm. We will mainly focus on this problem in future work. Acknowledgments This work was supported in part by the open funding programme of joint laboratory of flight vehicle ocean-based measurement and control under Grant No. FOM2014OF016 and in part by the National Natural Science Foundation of China under Grant No. 71401178.
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