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On the Phase Transition Width of K-Connectivity in Wireless Multihop Networks

机译：无线多跳网络中K连通性的相变宽度

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摘要

In this paper, we study the phase transition behavior of k-connectivity (k=1,2,ldots) in wireless multihop networks where a total of n nodes are randomly and independently distributed following a uniform distribution in the unit cube [0,1]^{d} (d=1,2,3), and each node has a uniform transmission range r(n). It has been shown that the phase transition of k-connectivity becomes sharper as the total number of nodes n increases. In this paper, we investigate how fast such phase transition happens and derive a generic analytical formula for the phase transition width of k-connectivity for large enough n and for any fixed positive integer k in d-dimensional space by resorting to a Poisson approximation for the node placement. This result also applies to mobile networks where nodes always move randomly and independently. Our simulations show that to achieve a good accuracy, n should be larger than 200 when k=1 and d=1; and n should be larger than 600 when kle 3 and d=2, 3. The results in this paper are important for understanding the phase transition phenomenon; and it also provides valuable insight into the design of wireless multihop networks and the understanding of its characteristics.
机译：在本文中，我们研究了无线多跳网络中k连接性（k = 1,2，ldots）的相变行为，该无线多跳网络中的n个节点按照单位立方体中的均匀分布随机且独立地分布[0,1 ] ^ {d}（d = 1,2,3），并且每个节点具有统一的传输范围r（n）。已经表明，随着节点总数n的增加，k连接的相变变得更加尖锐。在本文中，我们研究了这种相变发生的速度，并借助Poisson近似，推导了足够大的n和d维空间中任何固定的正整数k的k连接性的相变宽度的通用解析公式。节点放置。此结果也适用于节点始终随机且独立移动的移动网络。我们的仿真表明，要获得良好的精度，当k = 1和d = 1时，n应当大于200。当kle 3且d = 2，3时，n应当大于600。本文的结果对于理解相变现象很重要。它还为无线多跳网络的设计及其特性的理解提供了宝贵的见识。

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