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Node connectivity analysis in cloud-assisted IoT environments

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Abstract

Internet of Things (IoT) have gained much attention recently. However, the transmission behavior of nodes in such a cloud-assisted IoT environment is not well studied. In this work, we mainly address the problem of node connectivity analysis based on percolation theory. First, we investigate when it is possible for two nodes to communicate with each other. Based on the results from bond percolation in a two-dimensional lattice, as long as the probability is close less than 0.5 and each sub-square contains at least four nodes, percolation occurs. Then, the conditions for full connectivity in a network graph are established. How two adjacent sub-squares are connected differentiates this work from others. The full connectivity occurs almost surely if each sub-square contains at least one node and the probability of having an open sub-edge is no less than 0.3822. The proposed conditions for percolation and full connectivity are validated by simulations.

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Notes

  1. Hereafter, “almost surely” and “a.s.” will be used interchangeably.

  2. A path and a communicating path are used interchangeably in this work.

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Authors and Affiliations

  1. Department of Electrical Engineering, Graduate Institute of Communication Engineering, National Chung Hsing University, 250, Kuo-Kuang Road, South District, Taichung, Taiwan

    Min-Kuan Chang & Hsiao-Ping Tsai

  2. Department of Computer Science and Information Management, College of Computing and Informatics, Providence University, 200, Sec. 7, Taiwan Boulevard, Shalu Dist., Taichung, Taiwan

    Yu-Wei Chan

  3. Department of Electrical Engineering, National Chung Hsing University, Taiwan, 250, Kuo-Kuang Road, South District, Taichung, Taiwan

    Ting-Chen Chen & Min-Han Chuang

Authors
  1. Min-Kuan Chang
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  2. Yu-Wei Chan
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  3. Hsiao-Ping Tsai
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  4. Ting-Chen Chen
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Corresponding author

Correspondence to Yu-Wei Chan.

Appendix

Appendix

1.1 Derivations of (1)

  1. 1.

    For each vertex in the corner, the probability of having its two un-directional links facing toward the isolated island is equal to \(\frac{1}{C^4_2}=1/6\). There are \(m_1\) vertexes in the corner.

  2. 2.

    For a vertex in the perimeter of the isolated island other than the vertexes in the corner, the probability of having its two un-directional links facing toward the isolated island is upper bounded by to \(\frac{C^3_1}{C^4_2}=1/2\). There are \(m-m_1\) such vertexes.

  3. 3.

    For a vertex right adjacent to the vertexes in the perimeter of the isolated island, the probability of not having its two un-directional links facing toward the isolated island is upper bounded by \(\frac{C^3_1}{C^4_2}=1/2\). There are \(m+m_1\) such vertexes surrounding this island.

Therefore, the probability of having such an isolated island is bounded by (1).

1.2 Derivations of (2)

The upper bound given in (2) is derived when all the vertexes in the perimeter of an isolated island and all the vertexes surrounding this isolated island are those sub-square having only one un-directional link connecting one of its four adjacent sub-squares. In this case,

  1. 1.

    For each vertex in the corner, the probability of having its only one un-directional link facing toward the isolated island is equal to \(\frac{C^2_1}{C^4_3}=1/2\). There are \(m_1\) vertexes in the corner.

  2. 2.

    For a vertex in the perimeter of the isolated island other than the vertexes in the corner, the probability of having its only one un-directional link facing toward the isolated island is upper bounded by to \(\frac{C^3_1}{C^4_3}=3/4\). There are \(m-m_1\) such vertexes.

  3. 3.

    For a vertex right adjacent to the vertexes in the perimeter of the isolated island, the probability of not having its only one un-directional links facing toward the isolated island is upper bounded by \(\frac{C^3_1}{C^4_3}=3/4\). There are \(m+m_1\) such vertexes surrounding this island.

Also, the probability of having all these \(2(m-m_1)+m_1+2m+1=2m+m_1\) are those sub-square with only one un-directional link is \((1/2)^{2m+m_1}\). Based on these discussion, the probability of having an isolated island in this scenario is upper bounded by (2).

1.3 Derivations of (3)

The upper bound given in (3) is derived when all the vertexes in the isolated line and all the vertexes surrounding this isolated line are those sub-square having only one un-directional link connecting one of its four adjacent sub-squares. In this case,

  1. 1.

    For each vertex of two vertexes in the end of this isolated line, the probability of having its only one un-directional link facing toward the isolated line is equal to \(\frac{1}{C^4_3}=1/4\). There are \(m_1\) vertexes in the corner.

  2. 2.

    For each vertex other than two vertexes in the end of the isolated line, the probability of having its only one un-directional link facing toward the isolated line is upper bounded by to \(\frac{C^2_1}{C^4_3}=1/2\). There are \(m-2\) such vertexes.

  3. 3.

    For a vertex right adjacent to the vertexes in the perimeter of the isolated island, the probability of not having its only one un-directional links facing toward the isolated island is upper bounded by \(\frac{C^3_1}{C^4_3}=3/4\). There are \(2(m-2)+2\times 3=2m+2\) such vertexes surrounding this line.

Also, the probability of having all these \(m+2\times (m-2) + 6=3m+2\) are those sub-square with only one un-directional link is \((1/2)^{3m+2}\). Based on these discussion, the probability of having an isolated line in this scenario is upper-bounded by (3).

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Chang, MK., Chan, YW., Tsai, HP. et al. Node connectivity analysis in cloud-assisted IoT environments. J Supercomput 73, 2966–2986 (2017). https://doi.org/10.1007/s11227-016-1771-2

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