How do I convert a list to a Graph with directed edges?

data =
  {{1, 3, 2, 1, 3, 1, 3, 1, 2, 1, 2}, {1, 2, 1, 2, 1, 2, 3, 2, 3, 2, 3}, {1, 3, 2, 
    1, 2, 3, 1, 2, 1, 3, 2}, {1, 3, 2, 1, 2, 1, 2, 1, 3, 1, 3}, {1, 2, 3, 1, 3, 1, 
    3, 2, 1, 2, 1}};

DirectedEdge @@@ Partition[#, 2, 1] & /@ data

{{1 \[DirectedEdge] 3, 3 \[DirectedEdge] 2, 2 \[DirectedEdge] 1, 
  1 \[DirectedEdge] 3, 3 \[DirectedEdge] 1, 1 \[DirectedEdge] 3, 
  3 \[DirectedEdge] 1, 1 \[DirectedEdge] 2, 2 \[DirectedEdge] 1, 
  1 \[DirectedEdge] 2}, {1 \[DirectedEdge] 2, 2 \[DirectedEdge] 1, 
  1 \[DirectedEdge] 2, 2 \[DirectedEdge] 1, 1 \[DirectedEdge] 2, 
  2 \[DirectedEdge] 3, 3 \[DirectedEdge] 2, 2 \[DirectedEdge] 3, 
  3 \[DirectedEdge] 2, 2 \[DirectedEdge] 3}, {1 \[DirectedEdge] 3, 
  3 \[DirectedEdge] 2, 2 \[DirectedEdge] 1, 1 \[DirectedEdge] 2, 
  2 \[DirectedEdge] 3, 3 \[DirectedEdge] 1, 1 \[DirectedEdge] 2, 
  2 \[DirectedEdge] 1, 1 \[DirectedEdge] 3, 
  3 \[DirectedEdge] 2}, {1 \[DirectedEdge] 3, 3 \[DirectedEdge] 2, 
  2 \[DirectedEdge] 1, 1 \[DirectedEdge] 2, 2 \[DirectedEdge] 1, 
  1 \[DirectedEdge] 2, 2 \[DirectedEdge] 1, 1 \[DirectedEdge] 3, 
  3 \[DirectedEdge] 1, 1 \[DirectedEdge] 3}, {1 \[DirectedEdge] 2, 
  2 \[DirectedEdge] 3, 3 \[DirectedEdge] 1, 1 \[DirectedEdge] 3, 
  3 \[DirectedEdge] 1, 1 \[DirectedEdge] 3, 3 \[DirectedEdge] 2, 
  2 \[DirectedEdge] 1, 1 \[DirectedEdge] 2, 2 \[DirectedEdge] 1}}

If you have a list of vertices representing such a path, you can use

Graph[DirectedEdge @@@ Partition[list, 2, 1]]

If you just want the graph fort his Markov process, use Graph[p].

You can also use BlockMap

toEdges1 = BlockMap[Apply @ DirectedEdge, #, 2, 1] &;

or the (undocumented) 6-argument form of Partition:

toEdges2 = Partition[#, 2, 1, {1, -1}, {}, DirectedEdge] &;

Using it with data:

toEdges1 /@ data

Row[Graph[#, ImageSize -> 200] & /@ toEdges1 /@ data]

toEdges1 /@ data == 
    toEdges2 /@ data == 
     ( DirectedEdge @@@ Partition[#, 2, 1] & /@ data)

 True