Huge Simplification

We can represent any Markov chain using a directed graph or digraph. The vertices are the states, and the edges tell if we can move from one vertex to another vertex in one minute. We draw an edge only if the probability of traversing that edge is not zero. We can also construct the graph from the matrix if we replace each non-zero entry of the matrix with 1 or certainty. The graph shows we can move from vertex to vertex only if a path between these vertices exists, but, most importantly, the shortest edge gives the quickest possible time to make the transition. The edges are strong connections between the vertices. Edges tell us which vertices we return to over and over, and which ones we visit a few times and do not return to.

More formally: if we start at E1 and remain there, the probability of returning to E1 later is 1, and we call E1, our midgut source, a persistent state. States between source and sink are transient states. Digraphs avoid difficult calculations. We see that Ei persists if and only if a bidirected edge connects vertex i to vertex j. If a path exists from vertex i to vertex j but not back from vertex j to vertex i, then vertex i is transient. Like the midgut source and the thorax sink, any vertex from which we cannot get to another state is an absorbing state.

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