There has been some advice on how to ask questions. So I reask my question.I have a problem related to graph theory, random graphs or random geometric graphs in special that really confuses me. Assuming Graph G with node set V={x1,x2,…,xn}, all the nodes are moving in R2 space. There is an edge between xi and xj if and only if the Euclidean distance is less than r(r is a constant). If the distance is larger than r due to the movement, the edge breaks. Edges emerge and break as a result of nodes movement. Assuming the movement is modeled by Brownian motion, I would like to analysis the property of reachability. Given a time threshold T and the speed of node movement , how can I analysis the probability of that one node can find a path to any node during T from time dimension? The key is the movement of nodes and edges break and emerge with time passing. How to compute the orobability of reachiabiliy?Thanks very much!

independentBrownian motion? Are you looking for the probability at a fixed time $t$ you have a connection between $x_1$ and $x_2$? Or the probability that $x_1$ and $x_2$ were ever connected during the interval $[0,T]$? What are the initial conditions of the positions? – Bati Jul 4 '13 at 13:02