  In practice, since many communication networks are huge in scale, or
complicated in structure, or even dynamic, the predesigned linear network codes
based on the network topology is impossible even if the topological structure
is known. Therefore, random linear network coding has been proposed as an
acceptable coding technique for the case that the network topology cannot be
utilized completely. Motivated by the fact that different network topological
information can be obtained for different practical applications, we study the
performance analysis of random linear network coding by analyzing some failure
probabilities depending on these different topological information of networks.
We obtain some tight or asymptotically tight upper bounds on these failure
probabilities and indicate the worst cases for these bounds, i.e., the networks
meeting the upper bounds with equality. In addition, if the more topological
information of the network is utilized, the better upper bounds are obtained.
On the other hand, we also discuss the lower bounds on the failure
probabilities.
