The Distribution of Path Lengths On Directed Weighted Graphs
  We consider directed weighted graphs and define various families of path counting functions. Our main results are explicit formulas for the main term of the asymptotic growth rate of these counting functions, under some irrationality assumptions on the lengths of all closed orbits on the graph. In addition we assign transition probabilities to such graphs and compute statistics of the corresponding random walks. Some examples and applications are reviewed. 
Avner Kiro, Yotam Smilansky, Uzy Smilansky
math.CO math.DS