Robustness of networks against cascading failures

Physica A 389 (2010) 2310–2317

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Robustness of networks against cascading failures Bing-Lin Dou a,∗ , Xue-Guang Wang b , Shi-Yong Zhang a a

School of Computer Science, Fudan University, Shanghai 200433, People’s Republic of China

b

Department of Computer Science, East China University of Political Science and Law, Shanghai 201620, People’s Republic of China

article

info

Article history: Received 17 October 2009 Received in revised form 25 December 2009 Available online 7 February 2010 Keywords: Complex networks Cascading failures Robustness Load-capacity model

abstract Inspired by other related works, this paper proposes a non-linear load-capacity model against cascading failures, which is more suitable for real networks. The simulation was executed on the B-A scale-free network, E-R random network, Internet AS level network, and the power grid of the western United States. The results show that the model is feasible and effective. By studying the relationship between network cost and robustness, we find that the model can defend against cascading failures better and requires a lower investment cost when higher robustness is required. © 2010 Elsevier B.V. All rights reserved.

1. Introduction Research on network robustness or survivability, as an important aspect of complex networks, has been the concern of many scholars. It is the ability of a network to still provide the key services or functions in the situation of random failures or deliberate attacks. Network robustness can be divided into two categories [1]: static and dynamic robustness. The main difference between them is whether one will consider the redistribution of network traffic flows when they research the robustness problem. Static robustness estimates the ability of a network to maintain its functions according to numerical or analytic results [2–15]. Whereas dynamic robustness considers the chain reaction induced by a node’s failure or congestion [16–30]. Because of the limitation of node capacity, the fact that node load will surpass its capacity probably results in further malfunction when the traffic flows are redistributed. The mutual coupling relationship between nodes will ultimately bring about the collapse of the whole network. This dynamic process is called cascading failure. For studying cascading failures on complex networks, Motter and Lai [16] proposed a linear load-capacity model (called the ML model in this paper) and reported the damage assessment for network connectivity through simulating cascading failures. They also gave the conditions that global cascades occurred as follows: the network exhibits a highly heterogeneous distribution of loads; the removed node is among those with higher load. Based on the ML model, people have studied the features, control and defense policies of cascading failures and suggested some improved models [17,20,21,24,27,28]. However, Kim and Motter [29] found that there is no linear relationship between node’s load and capacity by analyzing four real networks. The real systems tend to have larger unoccupied portions of the capacities – smaller load-to-capacity ratios – on network elements with smaller capacities. This contrasts with the key assumption of a linear load-capacity relationship used in previous studies. Therefore, this paper proposes a non-linear load-capacity model which is more suitable for real networks and simulated on the B–A scale-free network, E–R random network, Internet AS level network, and the power grid of the western United States. The results show that the model is feasible and effective. The rest of this paper is organized as follows: in Section 2

∗ Corresponding address: Room 409, YiFu Building, Fudan University, No. 220, Handan Road, Shanghai 200433, People’s Republic of China. Tel.: +86 21 65643181; fax: +86 21 65643189. E-mail address: [email protected] (B.-L. Dou). 0378-4371/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2010.02.002

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a new model is suggested and a method described of how to simulate cascading failures on complex networks; in Section 3 the model is validated on four networks and the results discussed; finally, the paper’s conclusions are given. 2. Load-capacity model We firstly give some necessary definitions [31]. A graph is a pair (V , E ), with V the set of vertices {v1 , v2 , . . . , vN }, and E ⊆ V × V the set of edges {e1 , e2 , . . . , eM }. Note that this paper only considers unweighted, undirected, single-edge, selfloop-free graphs. A path P (v, w) between vertex v and w is a subset of consecutive edges P (v, w) = {e1 , e2 , . . . , ek } ⊆ E with e1 = (v, v1 ), ek = (vk−1 , w), and for all other edges ei = (vi−1 , vi ). The length |P (v, w)| of a path is given by the number of edges in it. The distance d(v, w) between any two vertices v, w is given by the minimal length of any path between them (the paths with d(v, w) are called the shortest path), and is ∞ by definition if there is no path between them. A subgraph G0 = (V 0 , E 0 ) contains a subset of vertices V 0 of V and all edges e = (v, w) with v, w ∈ V 0 . A component is a maximal subgraph when the distance between any two vertices is finite. The size of a component is defined as the number of vertices in it. The largest connected component is defined as the component with the highest number of vertices in it. 2.1. The model Given a network G = (V , E ), we suppose that at each time step, one unit of information is transmitted along the shortest path between every pair of nodes. Thus, we take betweenness centrality as each node’s initial load, which describes the number of all shortest paths through the node. Let xi,j denote the total number of shortest paths between nodes vi and vj , and xi,j;n denote the number of shortest paths between nodes vi and vj that pass through node vn . Define Ln =

N X xi,j;n i6=j=1

(1)

xi,j

as the initial load of node vn . A node’s capacity means the ability to handle the load on it, which is strictly limited by cost. In the ML model, the capacity Cn of node vn is proportional to its initial load Ln , that is, Cn = (1 + α)Ln ,

n = 1, 2, . . . , N .

(2)

Here, α ≥ 0 is tolerance parameter which denotes the additional capacity of node vn . As mentioned earlier, the relationship between capacity and load shows a nonlinear behavior in real systems. Therefore, we give a new load-capacity model which is more close to real networks and adopts two parameters for a certain degree of flexibility, as follows: Cn = Ln + β Lαn ,

n = 1, 2, . . . , N ,

(3)

where α ≥ 0 and β ≥ 0. When α = 1, this model degenerates to the ML model. 2.2. Cascading process Crucitti et al. found that [32] the breakdown of a single node is sufficient to affect the efficiency of a network up to the collapse of the entire system if the node is among the ones with the largest load, which is particularly important for networks with a highly heterogeneous distribution of node loads such as B–A scale-free networks, but also real-world networks such as the Internet and electrical power grids. So, we remove the node with the largest load in the initial network, which will result in redistribution of network traffic flows and a change of the load on each node. Because of the limited node capacity, it is probable that some nodes collapse when these nodes’ loads exceed their capacity. We remove those failed nodes from network, which will induce another traffic redistribution until each remaining node’s load does not surpass its capacity. In previous studies, the network robustness was described as follows: N 00

. (4) N0 Here, N 0 and N 00 respectively denote the size of largest connected component in the network before and after cascading failures occur. When g ≈ 1, the whole network is almost connected; g ≈ 0 means that network completely collapses. In addition, the increase of network capacity depends on its cost. In this paper, let g =

I0 =

X v∈V

Lv ,

I =

X

Cv

v∈V

respectively denote the network’s initial cost and current cost. Let I (5) I0 denote the investment cost. We will discuss the relationship between g and e in the following section in order to maximize the network robustness and minimize the network cost. e=

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1 0.8 α=0.70

0.6 g

α=0.75 α=0.80

0.4

α=0.85 α=0.90

0.2

α=0.95 α=1.00

0 0

5

10 β

15

20

Fig. 1. Cascading failures on the B–A scale-free network with node number N = 5000 and average degree hki ≈ 4.

1

a 1

0.8

b

0.98

0.6 g

g

0.96 0.94

0.4 ML Model

0.92

Current Model

0.2 0

0.9

1

1.5

1

2

3 e

2 e

4

2.5

5

3

Fig. 2. Cost-robustness relationship between the current model and the ML model on the B–A scale-free network. Fig. 2(b) is a detail of Fig. 2(a) when g > 0.90. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

3. Simulation and discussion This section will simulate and validate our model on two model networks and two real networks, which are the B–A scale-free network, E–R random network, Internet AS level network and the power grid of the western United States. At the same time, we discuss the cascading processes on these four networks from the point of view of degree–degree correlations. 3.1. Simulation on the B–A scale-free network First, we construct the B–A scale-free network according to the literature [33], whose degree distribution decays as a power-law, that is p(k) ∼ k−γ , γ = 3. At the same time, Ref. [34] pointed out that the load on scale-free network also obeys a power-law distribution, i.e., there are a small number of nodes with a large load in such a network. Next, we discuss the relationship between model parameters α , β and g on the scale-free network with node number N = 5000 and average degree hki ≈ 4 (see Fig. 1). As a function of α and β , g is obtained by simulating cascading failures on scale-free network with α = 0.70, 0.75, 0.80, 0.85, 0.90, 0.95, 1.00 (from right to left) and β ∈ [0, 20]. From Fig. 1, the network connectivity suddenly emerges in a small range of β with given α . For instance, β = 12.5, g = 0.0188 and β = 13.0, g = 0.9672 when α = 0.70. Another example is that β = 0, g = 0.0062 and β = 0.5, g = 0.8421 when α = 0.95. This means that we can effectively defend against cascading failures and get better network robustness through increasing by a relatively small capacity. However, g will be gradually and slowly close to 1, even though a large value of β is given after the critical phenomenon occurs. We can see that β = 3.5, g = 0.9446 when α = 0.80; g only reaches 0.9768 when we double the β value. We have similar results for other values of α . That is, the failure of the node with the largest load will not result in a widespread malfunction under these circumstances and the network has a good connectivity. Thus, we can select appropriate model parameters according to robustness requirements. We have also validated the cost-robustness relationship between the current model and the ML model based on the B–A scale-free network (see Fig. 2). We set the parameter of the ML model α ∈ [0, 4] and the current model’s parameters α = 0.95 and β ∈ [0, 7]. In Fig. 2(a), when e < 1.207, the ML model (blue curve) is better than the current model (red curve), but the network robustness g is less than 0.7. When the increment of network cost is more than 20%, the current model can get better robustness than the ML model. In order to facilitate the discussion, a detail of Fig. 2(a) when g > 0.90 is given in Fig. 2(b). When g = 0.99, e ≈ 2.562 in the current model; for the same e, the ML model’s robustness g ≈ 0.98. So, we can obtain the optimal value g ∗ (e∗ ) from Fig. 2 according to the requirements of network robustness and cost.

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1 0.8 α=0.70

0.6 g

α=0.75 α=0.80

0.4

α=0.85 α=0.90

0.2

α=0.95 α=1.00

0 0

0.5

1

1.5

2

2.5

3

3.5

4

β Fig. 3. Cascading failures on the E–R random network with node number N = 5000 and average degree hki ≈ 4.

1

ML Model Current Model 1

0.8

0.98

0.6 g

g

0.96 0.94

0.4 0.2

b

0.92

a

0.9 1.08

1.1

1.14

1.12

e

0

1

1.02

1.04

1.06

1.08

1.1 e

1.12

1.14

1.16

1.18

1.2

Fig. 4. Cost-robustness relationship between the current model and the ML model on the E–R random network. Fig. 4(b) is a detail of Fig. 4(a) in the dashed box.

3.2. Simulation on the E–R random network In last section, we discussed the problem of cascading failures on the heterogeneous B–A scale-free network. Here, a homogeneous network will be considered. We use the method in literature [35] for generating the E–R random network, which has the same node number and average degree as the B–A network. The degree and load of the E–R network approximately obey the Poisson distribution. Because the E–R random network is homogeneous, the dynamic behavior of cascading failures on it is different from the heterogeneous network. From Fig. 3, the transition feature of g is very obvious through adjusting parameters α and β . For example, when α = 0.80, β = 0.7, g = 0.0145 and β = 0.8, g = 0.9998. The rate of change of g versus β is about 9.85. With the same value of α , β = 1.5, g = 0.0158 and β = 3.5, g = 0.9446 on the B–A network (see Fig. 1). Here, the rate of change of g versus β is only 0.46. We also find that there is a smaller value of β when the current model is used on the E–R random network than on the B–A scale-free network with the same α and g. The reason is a result of the difference in topological features between the B–A and E–R networks. There exists a few ‘‘hub’’ nodes with large degree and a great number of nodes with small degree in the B–A network. When the nodes with the largest load are removed (because of taking the betweenness centrality of nodes as their loads, these nodes also have large degree), the paths of network traffic will be reselected. When the cascading process occurs, excessive traffic will pass those nodes with small degree which also have relatively little capacity. This further destroys network connectivity and one needs to give them larger capacity to prevent cascading failures. For the homogeneous E–R network, network traffic will not be delivered through a few nodes, and as a result, cascading failures will not bring about large-scale network collapse. Therefore, a small increment of node capacity can get better network robustness. Fig. 4 shows the cost-robustness relationship between the current model and the ML model based on the E–R network, and sets the ML model’s parameter α ∈ [0, 0.2] and the current model’s parameters α = 0.90, β ∈ [0, 0.8]. In contrast with Fig. 2, we find that the investment costs of these two models on the E–R random network are 1.08 and 1.09, whereas their costs on the B–A scale-free network respectively reach 1.35 and 1.45 when g ≈ 0.90. From Fig. 4(b), the investment cost of the current model has a certain advantage when g > 0.90. On the contrary, the ML model can get a better network robustness when e ∈ [1.05, 1.07] (see Fig. 4(a)).

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Fig. 5. Internet AS level network with node number N = 16 493 and average degree hki = 4.0468. (a) Degree distribution, (b) load distribution.

1 0.8

g

0.6 α=0.80

0.4

α=0.85 α=0.90

0.2

α=0.95 α=1.00

0

0

2

4

6

8

10 β

12

14

16

18

20

Fig. 6. Cascading failures on the Internet AS level network.

3.3. Simulation on the Internet AS level network We study cascading failures on a real-world network in this section. The Internet AS level network [36] is adopted, which has 16 493 nodes and 33 372 undirect links. The topological structure composed by autonomous systems (AS) on the Internet is heterogeneous, of which the node degree and load basically obey a power-law distribution (see Fig. 5). The results of cascading failures on the Internet AS level network are shown in Fig. 6. We set the current model’s parameters α = 0.80, 0.85, 0.90, 0.95, 1 (from right to left) and β ∈ [0, 20]. With an increase of α , the network robustness g can easily be more than 0.9 when β lies in a small range. In addition, the curve with α = 0.80 changes unevenly when β < 7.5 and obviously decreases at β = 2, 4. This is possibly related to the order of removing failed nodes when failures occur and traffic flows are redistributed. However, the curve of network robustness will tend to a steady state when β is greater than a special value and cascading failures will not result in a severe influence on network connectivity. Fig. 7 shows the cost-robustness relationship between the ML model and the current model. Every curve of the current model is obtained through changing β for given α , the ML model’s parameter α ∈ [0, 4]. The inset Fig. 7(b) displays the detail of Fig. 7(a) when e ∈ [1, 1.5]. The current model’s cost (four curves with α = 0.80, 0.85, 0.90, 0.95) for withstanding cascading failures is better than the ML model’s (black curve) when g > 0.85. Although the ML model has an advantage for low cost, the poor network robustness is not acceptable in this case. Thus, we can select model parameters corresponding to the optimal cost-robustness point through multi-curve comparison. In addition, the B–A scale-free network will collapse (g ≈ 0) when cascading failures occur with e = 1 (see Fig. 2). However, the robustness of the Internet AS level network g ≈ 0.1 under the same condition. The reason is that the Internet AS level network is not completely heterogeneous. 3.4. Simulation on the power grid of the western United States The power grid of the western US [37] has 4941 nodes and 6594 undirect links, whose average degree is 2.67. Its degree distribution is consistent with an exponential and is thus relatively homogeneous. The distribution of loads, however, is more skewed than that displayed by semirandom networks with the same distribution of links, indicating that the power grid has structures that are not captured by the B–A or E–R model [16] (see Fig. 8). We simulate cascading failures on the power grid and obtain its robustness by setting the current model’s parameters α = 0.80, 0.85, 0.90, 0.95, 1 (from right to left) and β ∈ [0, 100] (see Fig. 9). Because the power grid has a certain degree of

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b

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0.6 g

g

0.6 ML Model

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Current Model, α=0.95

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e

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Fig. 7. Cost-robustness relationship between the current model and the ML model on the Internet AS level network. Fig. 7(b) is a detail of Fig. 7(a) when e ∈ [1, 1.5].

Fig. 8. Power grid of the western US with node number N = 4941 and average degree hki = 2.67. (a) Degree distribution, (b) load distribution.

1

g

0.8

0.6

α=0.80 α=0.85

0.4

α=0.90 α=0.95 α=1.00

0.2 0

20

40

60

80

100

β Fig. 9. Cascading failures on the power grid of the western US.

homogeneity, g = 0.2145 when β = 0. However, the emergence of its robustness is a continuous transition in comparison with the E–R random network, which results from the particularity of load distribution of the power grid. In the power grid, there are some key nodes which are important for network connectivity. So, it is necessary to let their capacity satisfy load requirements for better network robustness. Thus, there is a larger β for given α . From Fig. 10, we find the conclusion is consistent with the one from Fig. 4. When g > 0.90, the current model’s robustness is better than the ML model’s (black dashed) with the same investment cost. The current model needs more investment cost in order to obtain the same robustness as the ML model when 0.45 < g ≤ 0.9. So, we need to balance network robustness and cost according to the actual situation when we select one model and design the network system.

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a

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g

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ML Model Current Model, α=0.85 Current Model, α=0.90 Current Model, α=0.95 4

5

6

0.2 0

2

4

6

7 e

8

8

9

10

10

e Fig. 10. Cost-robustness relationship between the current model and the ML model on the power grid of the western US. Fig. 10(b) is a detail of Fig. 10(a) when g > 0.90.

3.5. Degree–degree correlations and cascading processes We discuss cascading processes on these four networks from the point of view of degree–degree correlations. Newman described networks with different degree–degree correlations by an assortativity coefficient [38,39] as follows, M −1



P

ji ki − M −1

i

r = M −1

P1 i

2

i






2 ( j + k ) i 2 i

P1

j2i + k2i − M −1

2 , ( j + k ) i 2 i

(6)

P1 i

where ji , ki are the degrees of the vertices at the ends of the ith edge, with i = 1, . . . , M. When r > 0, the degree–degree correlation is positive, or assortative mixing; when r < 0, it is negative, or disassortative mixing. The networks generated by the B–A model or E–R model are uncorrelated, i.e., r = 0. According to the formula (6), r = −0.189 for the Internet AS level network and r = −0.003 for the power grid. So, they are disassortative mixing. The cascading processes on these two types of networks are different. We explain the differences between Figs. 3 and 6 as an example. In Fig. 3, there are discontinuous transitions for all values of α and the connectivity of the network suddenly emerges. However, the transitions seem to change from continuous to discontinuous as α increases in Fig. 6. The edges are placed at random in the E–R model resulting in no degree–degree correlations between vertices. The distributions of vertex degree and betweenness centrality are relatively even in E–R networks and the capacity of every vertex is also similar. Therefore, when β is less than the critical point for given any α in Fig. 3, to remove nodes with the highest load induces the overload of the other nodes very easily and large-scale cascades occur; when β reaches or exceeds the critical point, a great number of nodes have capacities enough for redistributed traffic flows which can guarantee the network’s connectivity. In the disassortative Internet AS level network, a few nodes with large degree are preferably connected to ones with small degree; those nodes with medial degree are connected with each other and some shortest paths between them do not include nodes with the largest degree. For small α , the connectivity among nodes with medial degree is hardly destroyed. So, the largest cluster size in the network will gradually increase with β , which is very different from the case of the E–R network. Fig. 6 shows the discontinuous transition just occurs when α > 0.95. Although the B–A model can explain the power-law distribution in the Internet, the networks generated by it are different from the Internet AS level network. In terms of degree–degree correlations, the B–A network is uncorrelated and the Internet AS level network is disassortative. From Fig. 5, the load on the Internet AS level network does not completely obey a powerlaw distribution, which means that nodes with medial degree bear a certain amount of load. This is the root of the difference between these two networks (see Figs. 1 and 6). As mentioned above, the power grid has a complex network structure. It has a homogeneous feature, which is similar to the E–R network, and it is disassortative, which is same as the Internet AS level network. However, from the experiments (see Fig. 9), we think that the cascading process on the power grid is more similar to one on the Internet AS level network. In addition, the effect of degree–degree correlations on the cascading process has been reported by some authors. Noh [40] suggested that the percolation transition in a disassortative network belongs to the same universality class as that in an uncorrelated network, but that in the assortative network it belongs to a distinct universality class. Assortativity is an essential ingredient for the universality class of the percolation transition. Payne et al. [41] showed that the class of networks for which global cascades occur generally expands as degree–degree correlations become increasingly positive. But, under certain conditions, large-scale cascades can paradoxically occur when degree–degree correlations are sufficiently positive or negative, but not when correlations are relatively small. Sun et al. [42] demonstrated that both assortative mixing and disassortative mixing can enhance the transport capacity of networks. In this paper, according to the experimental results, we conclude that the Internet AS level network and the power grid have smaller cascading size than the other two networks before each of them is completely connected. The reason is just disassortative mixing, which induces a group of nodes in

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those networks to prevent cascading propagation. The conclusion may not be general because of the limitation of network size and load-capacity model in this paper. 4. Conclusions This paper proposes a non-linear load-capacity model against cascading failures based on related works, which is more suitable for real networks. The model adopts two parameters for better flexibility. Its feasibility is validated on the B–A scalefree network, E–R random network, Internet AS level network, and the power grid of the western United States. By studying the relationship between network cost and robustness, we find that our model can defend against cascading failure better than the ML model, and it has less investment cost when higher robustness is required. In short, our model is very effective. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42]

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