Research on ad-hoc network connectivity has mainly focused on asymptotic results in the number of nodes in the network. For a one-dimensional ad-hoc network G , assuming all the nodes are independently uniform distributed in a closed interval [0, Z](z ∞ ℝ ), we derive a generic formula for the probability that the network is connected. The finite connected ad-hoc networks is analyzed. And we separately suggest necessary conditions to make the ad-hoc network to be connected in one and two dimensional cases, facing possible failed nodes (f-nodes). Based on the necessary condition and unit-disk assumption for the node transmission, we prove that the nodes of the connected two-dimensional ad-hoc networks (G ) can be divided into at most five different groups. For an f-node n in either of the five groups, we derive a close formula for the probability that there is at least one route between a pair of nodes in G - {n }. © 2008 Science in China Press.