Let G be an undirected graph consisting of N nodes. Let each node in G be connected with two other nodes selected randomly from the remaining nodes with equal probability. What is the probability that the graph G has more than one connected components?
I hope the question well formed.
EDIT. I understand, I didn't form the question properly. Let me give some background from where the problem is arising.
Say N vectors are chosen uniformly over unit circle. Each vector can be expressed as a linear combination of two other vectors from the set. Suppose the two vectors are chosen simply by picking those which have largest inner product with a vector. Now form an $N \times N$ coefficient matrix where each column contains the representation of a vector in terms of others. Naturally the diagonal is all 0. Let $C$ be this coefficient matrix. Now, compute $W = |C| + |C^T|$ as sum of absolute values of $C$ and it's transpose. Think of $W$ as the adjacency matrix of a graph of $N$ nodes corresponding to the vectors. Each node is connected to at least two other nodes. But it may be connected to more nodes also. Is it highly probable that this graph has more than one connected components? A closed form expression for the probability is not required as long as some bound can be obtained.