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!

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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