Robustness of complex networks with the local protection strategy against cascading failure

Robustness of complex networks with the local protection strategy against cascading failure

Presenter : Yen Fen Kao

Advisor : Yeong Sung Lin

2013 Safety Science

2 Agenda

Introduction The mitigation strategy Simulation analysis of mitigation strategy Theoretical analysis of cascading model Conclusions

3 Introduction

Cascading failures induced by targeted attacks and random failures can occur in many natural and man- made systems, and frequently trigger many catastrophic events.

4 Introduction

Earlier studies on cascading failures pay close attention to analyzing the evolving characteristics of cascading failures and mainly focus on the cascading modeling, the cascade control and defense, the attack strategies, the cascading phenomena in the diverse networks, and so on.

Later, many researchers find that catastrophic events induced by cascading failures can also occur in coupled networks

5 Introduction

Motivated by that fact, Buldyrev et al. (2010) developed a framework for understanding the robustness of interacting networks against cascading failures and present exact analytical solutions for the critical fraction of nodes.

After that, cascading failures on coupled networks have started to be studied actively and a number of important aspects of cascading failures in coupled networks have been discussed.

6 Introduction

Most works on cascading failures from the single network to coupled networks have focused only on the modeling of cascading failure, the cascade control and defense strategies, or a fundamental property of coupled networks without considering the load.

In many infrastructure networks, when a node overloads, its neighboring nodes can provide some protection resources to avoid its failure.

7 Introduction

This paper propose a new mitigation method. Without changing the total protection

capacities of the whole network, this paper numerically investigate its effectiveness on improving the robustness of BA scale-free networks against cascading failures and find that the simple local protection method can dramatically optimize the resilience of BA networks.

8 Introduction

Adopting the mitigation strategy, this paper numerically discuss how BA networks can reach the strongest robust level against cascading failures and find, compared with previous results, the optimal value of the parameter decreases. We verify this result by the theoretical analysis.

9 Agenda


10 The mitigation strategy

In previous studies, the rule that the overload nodes are immediately removed from the network is widely adopted. However, few works discuss this problem: when the load on a node exceeds its capacity, whether there exist some strategies to maintain its normal and efficient functioning to avoid the cascading propagation or not?

Wang and Rong (2009)constructed a cascading model of an overload node with the breakdown probability. However, this mechanism with the breakdown probability increases the total price of the whole network.

Therefore, without changing the total price of the whole network, how to protect the overload nodes has become one of the key issues in the control and the defense of cascading failures.

Considering that the neighboring nodes of an overload node may provide some protection resources to this overload node to maintain its normal and efficient functioning, we propose a local protection method.


：the set of the neighboring nodes of node iCm：the capacity of node m to handle the load Lm： the initial load of node m

The parameter p：a random number in 0 and 1, which decides to the strength of the protection resources provided by the neighboring nodes.

p= 0, the neighboring nodes of the overload node cannot provide the extra capacity p= 1, the strongest protection is provided by the neighboring nodes

i


For simplicity, this paper apply this strategy to a simple cascading model.

Our purpose of this paper is to analyze to what extent this mitigation strategy can enhance the network robustness against cascading failures and what is the optimal strategy of the parameter selection after adopting this protection method.

13 The cascading model

The initial load Li on node i in a network

is defined as

ki ： the degree of node I

a ： a tunable parameter The capacity Ci of node i is defined as

β ： an adjustable parameter characterizing the

tolerance of the network against cascading failures F After node i fails, the additional load Lj received by node j

is proportional to its initial load Lj .

14 The cascading model

When the load on node I exceeds its capacity, the capacity Ci,t of node i can dynamically be adjusted to

The capacities of the neighboring nodes of node i can simultaneously be adjusted to

15

This paper quantify the effectiveness of the mitigation strategy on improving the network robustness against cascading failures by two measures ：

1. the avalanche size S ,

2. the proportion P(S)

The mitigation strategy

16 Agenda


17 Simulation analysis of mitigation strategy

The degree distribution of the scale-free networks satisfies a power law form:

k ： the number of links of a randomly chosen node in the network

： the scaling exponent.




The role of the mitigation strategy on improving the network robustness when the diverse types of nodes are attacked.

This paper simply apply two attacks

1. attacking the nodes with the highest load (HL)

2. attacking the ones with the lowest load (LL) Normalized average avalanche size ,we

quantify the network robustness after and before adopting the mitigation strategy

A ： the set of nodes attacked

nA ： the number of nodes attacked

n ： the number of all nodes in a network



23 Agenda


24 Theoretical analysis of cascading model

A

f


By the probability theory and the network structure, this paper focus on the mathematical expectations of the expressions.

and Taking into account the characteristic of the no degree–

degree correlation in BA networks, we have


Where is the conditional probability that node i with the degree ki has a neighbor node with the degree k’ . Similarly, we have

The above inequality (6) is simplified to


Analyzing the above inequality (9), we can obtain the critical threshold βc in three ranges of a < 1, a = 1, and a > 1

kmax and kmin represent the minimum and maximum degree of nodes in a network, respectively. When the value p is equal to 0, the equation has been verified by Wang et al. (2008).


When a > 1, we have

According the Eq. (10) and the inequality (12) and the correctness of the Eq. (12) when p = 0, we can get βc(a > 1) > βc (a = 1).

In the case of a < 1, we can also obtain βc (a < 1) > βc (a = 1).


However, in Fig. 3, we observe that the optimal value a is about 0.8.

The smaller deviation between computer simulations and theoretical analysis mainly derives from the above approximation in the analytical predictions. i.e., two nodes i and j connected by other simultaneously have the minimum degree.

In fact, in BA networks generated by the BA model, according to the evolving mechanism of the BA model, two nodes with the lowest degree can be not connected by each other.


D

For a BA network with the finite size, its degree distribution is , where m is equal to kmin (here ). Thus, in a BA network can be calculated by


In BA networks, kmax can be calculated by


In computer simulations, the parameter p, the average degree <k>, and the network size N is 0.5, 4, and 5000, respectively. So, only in the case of p = 0.5, we can obtain , which shows that the network robustness can dramatically be improved.

This paper also find that the bigger the value p, the smaller the value βc, the stronger the network robustness.

33 Agenda


34 Conclusions

By the local dynamical adjustment of the capacity of an overload node, without changing the total price of the whole network, this paper propose a method to effectively protect the overload node to avoid its breakdown Characterized a fundamental property, by which the links that affect the total survivability level of the optimal routing paths belong to a typically small subset.

Applying a simple cascading model, this paper numerically investigate the effectiveness of the mitigation strategy on improving the robustness against cascading failures in BA scale-free networks.

35 Conclusions

According to the average avalanche size and the proportion between the protected nodes by the mitigation strategy and the failed nodes before adopting the mitigation strategy, this paper find that the mitigation strategy can effectively enhance the network robustness and obtain the correlation between the network robustness and some parameters.

After applying the mitigation strategy, we obtain the optimal value a, at which BA networks can reach the strongest robust level against cascading failures

Thanks for your attention

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Robustness of complex networks with the local protection strategy against cascading failure