Phase transition properties in K-connected wireless multi-hop networks

Abstract

Consider a wireless multi-hop network formed by distributing a total of n nodes randomly and uniformly in the unit cube [0,1]d (d = 1, 2, 3) and connecting any two distinct nodes directly iff (if and only if) their Euclidean distance is not greater than a given threshold r(n). We study the phase transition phenomenon of a κ-connected (κ ε) multi-hop network in this paper. We show that the phase transition of κ-connectivity becomes sharper as n increases. We derive a generic analytical formula for the phase transition width for large n and for any fixed κ ε in d-dimensional space. The result in this paper is important for understanding phase transition behavior, and it provides valuable insight into the design and implementation of wireless multi-hop networks.