We now return to the digraph representation of a Markov chain that corresponds to the DPS pathway ensemble discussed in Section 4.1.4. A problem with (partially) overlapping sets of sources and sinks can easily be converted into an equivalent problem where there is no overlap, and then the GT method discussed in Section 4.4 and Section 4.5 can be applied as normal.
As discussed above, solving a problem with sources reduces to solving single-source problems. We can therefore explain how to deal with a problem of overlapping sets of sinks and sources for a simple example where there is a single source-sink and, optionally, a number of sink nodes.
First, a new node is added to the set of sinks and its adjacency lists are initialised to and . Then, for every node we update its waiting time as and add node to the set of sources with probabilistic weight initialised to , where is the original probabilistic weight of source (the probability of choosing source from the set of sources). Afterwards, the node is deleted.