This paper studies networks where all nodes are distributed on a unit square A=? [- [1/2], [1/2]]2 following a Poisson distribution with known density ? and a pair of nodes separated by an Euclidean distance x are directly connected with probability gr?(x)=?g(x/r?), independent of the event that any other pair of nodes are directly connected. Here, g:[0,8)? [0,1] satisfies the conditions of rotational invariance, nonincreasing monotonicity, integral boundedness, and g(x)=o(1/(x2log2x)) ; further, r?=v{(log?+b)/(C?)} where C=?R2g(||x||)dx and b is a constant. Denote the aforementioned network by G(X?,gr?,A). We show that as ?? 8, 1) the distribution of the number of isolated nodes in G(X?,gr?,A) converges to a Poisson distribution with mean e-b ; 2) asymptotically almost surely (a.a.s.) there is no component in G(X?,gr?,A) of fixed and finite order k >; 1; c) a.a.s. the number of components with an unbounded order is one. Therefore, as ?? 8, the network a.a.s. contains a unique unbounded component and isolated nodes only; a sufficient and necessary condition for G(X?,gr?,A) to be a.a.s. connected is that there is no isolated node in the network, which occurs when b? 8 as ?? 8. These results expand recent results obtained for- connectivity of random geometric graphs from the unit disk model and the fewer results from the log-normal model to the more general and more practical random connection model.