Phase Transition Properties in K-Connected Wireless Multi-Hop Networks

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 k-connected (k isin Nopf) multi-hop network in this paper. We show that the phase transition of k-connectivity becomes sharper as n increases. We derive a generic analytical formula for the phase transition width for large n and for any fixed k isin Nopf 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.