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The Research on Cascading Failure of Farey Network

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Data Science (ICPCSEE 2017)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 727))

Abstract

Cascading failure is an important part of the dynamics in complex network. In this paper, we research the cascading failure of Farey network which is scale-free network with fractal properties by data analysis. According to the analyses, we obtain an iterative expression of the failure nodes’ number at the certain time step, anther iterative expression of the time which is needed for the network being global collapse, and the sequence of nodes failure in the Farey network. By theoretical derivation, we get an approximate solutions of perturbation threshold R to make the network achieve the global collapse. If the nodes number of the Farey network is lager, simulate results are closer to theoretical value. By the simulation, we obtain the cascading failure process of the Farey network after suffering deliberate attack and random attack. Simulation results show that, the failure nodes of Farey network increase gradually as R raises, until the network is global collapse. Moreover, Farey network shows stronger robustness for random attack. abstract environment.

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Acknowledgments

This research is supported by the National Natural Science Foundation of China (No. 61603206) and the Nature Science Foundation from Qinghai Province (No. 2017-ZJ-949Q).

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Authors and Affiliations

  1. School of Computer Science, Qinghai Normal University, Xining, 810008, China

    Xiujuan Ma & Fuxiang Ma

Authors
  1. Xiujuan Ma
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  2. Fuxiang Ma
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Corresponding author

Correspondence to Xiujuan Ma .

Editor information

Editors and Affiliations

  1. Central South University, Changsha, China

    Beiji Zou

  2. Central South University, Changsha, China

    Min Li

  3. Harbin Institute of Technology, Harbin, China

    Hongzhi Wang

  4. Harbin University of Science and Technology, Harbin, China

    Xianhua Song

  5. Harbin University of Science and Technology, Harbin, China

    Wei Xie

  6. Harbin Sea of Clouds and Computer Technology, Harbin, China

    Zeguang Lu

Appendix

Appendix

In order to explain the recursive expressions in Subsect. 3.1, we give the following example. In Fig. 4, there are 17 nodes in the network \((N=17)\). We add a perturbation to node 6 at time step 5, i.e. \(t_{start}=5\). Start the recursive as follows:

At initial moment, there is only one failure node at time step 5, and it is node 6 , i.e. \(f_5=\{6\}\), \(N_t=\{6\}\), \(I_5=|N_5|=1\).

At time step 6, we have \(f_6=\bigcup \limits _{i\in f_5}n_5(i)=n_5(6)=\{3, 4, 10, 11\}\). \(N_6=N_5\cup f_6=\{3, 4, 6, 10, 11\}\), \(I_6=|N_6|=5\).

At time step 7, we get

\(f_7=\bigcup \limits _{i\in f_6}n_6(i)=n_6(3)\cup n_6(4)\cup n_6(10)\cup n_6(11)\)

\(=\{1, 2, 5, 8, 15\}\cup \{1, 7, 12\}\cup \emptyset \cup \emptyset \)

\(=\{1, 2, 5, 7, 8, 12, 15\}\)

\(N_7=N_6\cup f_7=\{1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 15\}\), \(I_7=|N_7|=12\).

At time step 8, we obtain

\(f_8=\bigcup \limits _{i\in f_7}n_7(i)=n_7(1)\cup n_7(2)\cup n_7(5)\cup n_7(7)\cup n_7(8)\cup n_7(12)\cup n_7(15)\)

\(=\{13\}\cup \{9, 17\}\cup \{9, 14, 16\}\cup \{13\}\cup \{14\}\cup \emptyset \cup \emptyset \)

\(=\{9, 13, 14, 16, 17\}\)

\(N_8\!=\!N_7\cup f_8\!=\!\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17\}\), \(I_8\!=\!|N_8|\!=\!17\).

If \(I_t=N\), then the recursive is over.

Fig. 4.
figure 4

The farey network of T = 4, the number of nodes is 17. The nodes are ordered with their order joined in the network.

Full size image
  1. (2)

    In the front example, suppose the attacked node is node 6. According to Fig. 4, obviously we have

\(d(6, 3)=d(6, 4)=d(6, 10)=d(6, 11)=1\);

\(d(6, 1)=d(6, 2)=d(6, 5)=d(6, 7)=d(6, 8)=d(6, 12)=d(6, 15)=2\);

\(d(6, 9)=d(6, 13)=d(6, 14)=d(6, 17)=3\).

So, \(t_{end}=t_{start}+max\{d(6, i),\ i\ne 6,\ i=1, 2, \dots N\}=t_{start}+max\{1, 2, 3\}=5+3=8\). It means that the network \(F_4\) is complete collapse at time step 8.

  1. (3)

    The sequence of failure nodes

In the above example, when attack node 6, the first group of failure nodes are 3, 4, 10, 11. All the length of the shortest pathes between node 6 and nodes 3, 4, 10, 11 are 1. All the length of the shortest pathes between node 6 and nodes 9, 13, 14, 17 are 3, thus nodes 9, 13, 14, 17 are the last group of failure nodes.

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Ma, X., Ma, F. (2017). The Research on Cascading Failure of Farey Network. In: Zou, B., Li, M., Wang, H., Song, X., Xie, W., Lu, Z. (eds) Data Science. ICPCSEE 2017. Communications in Computer and Information Science, vol 727. Springer, Singapore. https://doi.org/10.1007/978-981-10-6385-5_34

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