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In the research on network security, distinguishing the vulnerable components of networks is very important for protecting infrastructures systems. Here, we probe how to identify the vulnerable nodes of complex networks in cascading failures, which was ignored before. Concerned with random attack (RA) and highest load attack (HL) on nodes, we model cascading dynamics of complex networks. Then, we introduce four kinds of weighting methods to characterize the nodes of networks including Barabási-Albert scale-free networks (SF), Watts-Strogatz small-world networks (WS), Erdos-Renyi random networks (ER), and two real-world networks. The simulations show that, for SF networks under HL attack, the nodes with small value of the fourth kind of weight are the most vulnerable and the ones with small value of the third weight are also vulnerable. Also, the real-world autonomous system with power-law distribution verifies these findings. Moreover, for WS and ER networks under both RA and HL attack, when the nodes have low tolerant ability, the ones with small value of the fourth kind of weight are more vulnerable and also the ones with high degree are easier to break down. The results give us important theoretical basis for digging the potential safety loophole and making protection strategy.

In modern society, people’s life depends on the infrastructure networks more and more, such as the power grid, Internet, transportation networks and the financial networks, and so forth. The overall efficiency of these network systems is being increased, while the internal connections and the dynamical characteristics within the networks are becoming more close and complex, respectively. These behaviours make the networks more vulnerable and increase the possibility of system crash. Especially, with the improvement of netware-based degree, a small incident, through a cascade of reaction, can lead to the collapse of the whole network systems and a great number of economic loss. The typical example is the accident that emerged in the power grid of North America in 2003 [

These large-scale accidents have threatened the network safety and attracted considerable attentions of scientific researchers [

On the other hand, there exist the physical flows (also defined as load) in real-world networks, such as the electric stream in power grids, the data transmitted in communication networks, and the cars in transportation networks. Also, the physical load is dynamical. The fault of some local components (nodes or edges) always leads to the redistribution of load over the whole networks. Then, the overloaded components will fail as the new load on them exceeds their capacity (the maximum load). Therefore, the new redistribution of load over the whole networks will begin and lead to the cascade over the networks. This evolving procedure is called “cascading failure” which emerged locally and always resulted in the whole collapse of networks. Therefore, the cascading failures of complex networks have been one of the hottest topics in network safety. Induced by random breakdown and intentional attack, Motter explored the condition of cascading failures occurring in complex networks [

All the previous researches mainly focus on the structural vulnerability or the cascading dynamics of the integral complex networks. However, in the cascading failures caused by overloaded breakdown, the following important problems have not been considered. What features do the nodes easy to break down or crash have? How should we describe the characteristics of the congested node in complex networks? These problems are the correlations between the vulnerability of nodes and the cascading failures in complex networks. We argue that, by exploring this problem, we are able to identify the vulnerable nodes, analyze the potential safety hazard in networks, and find the bottlenecks in the dynamical change of network “flows”. Finally, it can provide the important theoretical basis for protecting complex networks and improving the robustness of real-world networks.

Here, this paper explores the correlations between the vulnerability of nodes and the cascading failures in complex networks. Firstly, by assigning the load on nodes, we model the cascading dynamics of complex networks induced by random attack and intentional attack on nodes. Secondly, we introduce four kinds of weighting methods to describe the characteristics of the nodes in complex networks including BA scale-free networks (SF), WS small-world networks (WS), ER random networks (ER), and two real-world networks (autonomous system network and US airport network). Finally, in order to identify the features of the vulnerable nodes in complex networks, we numerically computed the ratio of the failed nodes with four kinds of weights less than their respective average weights to the total failed ones in networks. As a result, we find that, for SF network under intentional attack, the ratio of the failed nodes with small value of the fourth kind of weight defined here is the highest. The autonomous system network with power-law distribution also shows similarity to SF network. It reveals that the nodes with small value of the fourth kind of weight defined in this work are more vulnerable. Moreover, for the WS small-world network and ER random network, under both random and intentional attack, when the tolerance ability of nodes is low, the ratio of the failed nodes with small value of the fourth kind of weight is higher, and, at the same time, the ratio of the failed ones with high degree is also higher. It means that the nodes with smaller value of the fourth kind of weight and large degree are vulnerable and the nodes with large degree are also vulnerable.

The rest of this paper is organized as follows. Section

In this section, we will model the cascading dynamics of complex networks under node-based attacks. For a general undirected network comprising of

The definition of load on node: usually, the physical flows (data packets, energy, etc.) are transmitted in many networks according to the shortest path routing strategy [

The relationship between the load and the capacity: usually, there is some maximum load (the maximum capacity) that node

The evolving procedure of cascading failures: beginning with the removal of some nodes in networks, the load on other nodes will change and be redistributed over the whole networks. At some time

The iterative process of cascading failures in complex networks.

Here, we consider two kinds of attack strategies.

Random attack (RA): we choose some proportion of nodes randomly and then remove them from the networks; here we assume the proportion

Highest load attack (HL): first, we descend the order of nodes according to the initial load

In order to identify the vulnerable nodes of complex networks in cascading failures, in this paper, we introduce four kinds of weighting methods

where

Furthermore, according to the theory of the degree of networks and probability [

Since the BA scale-free networks, WS small-world networks, and ER random networks have no degree-degree correlation [

Now, finally (

One can see that the weighting method

Therefore, we define the fourth kind of weighting method

Using Bayes’ rules [

Considering that it has been shown that small-world networks do not show betweenness-betweenness correlations [

Now, it is obvious that the four kinds of weighting methods of node introduced here can describe the characteristics of nodes and distinguish the nodes in network.

In this paper, to investigate the vulnerable nodes in networks subject to cascading failures, we mainly take the following typical complex networks into account: Barabasi-Albert scale-free networks (SF), Watts-Strogatz small-world networks (WS), and ER random networks (ER).

Scale-free networks (SF): SF network model in this paper is generated according to the two rules: growth and preferential attachment [

WS small-world networks (WS): here, according to Watts-Strogatz model [

ER random networks (ER): the random network model studied is generated according to the rules in [

Also, in order to compare with the network models, we consider two real-world networks: the autonomous system network (AS) and US airport network (US airport).

The autonomous system network (AS): from the AS level topology of Internet, the Internet can be seen as a network comprising of routers. Usually, the data is transmitted between routers according to BGP protocols. Thus, the routers (as nodes) and links construct the autonomous system network (AS) [

US airport network: as an example of transportation networks, we study the famous USA airport network with 500 airports and 2980 links [

Now, in this section, concerned with two kinds of node-based attacks, we will investigate how to identify the vulnerable nodes of complex networks subject to cascading failures. The studied networks include SF, WS, and ER complex networks models and two real-world networks.

Firstly, we use the relative size of nodes in the largest component of network (

Obviously, the metric

Secondly, to identify the features of the vulnerable nodes in complex networks, we numerically computed the ratio of the failed nodes with four kinds of weights less than their respective average weights to the total of the failed ones in networks when the iterative process of complex networks in Figure

One can see that the ratio in (

In this part, induced by random attack (RA) and the highest-load attack (HL), we mainly focus on analyzing the integral robustness and identifying the vulnerable nodes of three kinds of typical complex networks models: scale-free networks (SF), WS small-world networks (WS), and ER random networks (ER).

From the relative size of nodes in the largest component

From the ratio in (

For WS small-world network, from Figure

For ER random network under HL attack, as

Under RA attack, there are the failed nodes only for small

Under RA and HL attacks, the relative size of nodes in the largest component

Under HL attack, the ratio for different weighting methods as a function of

Under RA attack, the ratio for different weighting methods as a function of

In order to compare with the simulations of the network models, we also analyze two real-world networks: the autonomous system network (AS) and US airport network.

As shown in Figure

Under RA and HL attacks, the relative size of nodes in the largest component

From Figure

Under HL attack, the ratio for different weighting methods as a function of

For US airport network, as shown in Figure

In the research on cascading dynamics, finding and distinguishing the vulnerable nodes of networks are very important for the protection of infrastructures systems, but the traditional research on the vulnerability of complex networks has not considered this. This paper mainly probes the question of how to identify the vulnerable nodes of complex networks in cascading failures caused by the overload on nodes. We model the cascading dynamics of complex networks induced by deleting some proportion of nodes that are chosen randomly or intentionally. Then, four kinds of weighting methods of node are introduced to distinguish the failed nodes of complex networks, including BA scale-free networks, WS small-world networks, ER random networks, and two real-world networks. The main contributions of this paper are as follows.

For SF networks, under HL attack, the nodes with small

For WS small-world networks and ER random network, when the tolerance ability of node is low, no matter under RA attack or HL attack, the nodes with small

The findings of this paper provide important theory basis for analyzing network security, mining the hidden potential risk of networks, and protecting various real-world networks with load assigned to nodes.

This work is supported by the National Natural Science Foundation of China (Grant nos. 61202362, 61121061, 61262057, 61372191, and 91124002), the 863 programs (Grant nos. 2012AA01A401, 2010AA-012505, and 2011AA010702), the State Key Development Program of Basic Research of China 973 (Grant nos. 2013CB329601 and 2011CB302600), the China Postdoctoral Science Foundation Programs (BS2013SF009, 2012m520114, and 2013T60037), and the Beijing Higher Education Young Elite Teacher Project.