To complete the post on Random walk for developers as discussed in the previous two, we briefly summarize the goal and the process.
Random walk is a special case of a process called Markov Chain where the future events are conditionally dependent on the present and not the past events. It is based on random numbers that assign a real number to the events based on probabilities. The probabilities to move between a finite set of states (sometimes two - forward and backward as in the case of a drunkard walking in a straight line) is called transition probabilities. Random walk leads to diffusion.
The iterations in a random walk are done based on the equation px,y = px+z,y+z for any translation z where px,y is the transition probability in space S.
Random walks possesses certain other properties
- time homogeneity
 - space homogeneity
and sometime skip-free transitions which we have already discussed
The transition probabilities are the main thing to be worked out in a random walk.
Usually this is expressed in terms of ranges such as
p(x) = pj if xj = ej  where j = 1 to d in a d-dimensional vector defined space.
       = qj if xj = -ej
       = 0
We have since how this simplifies to a forward and backward motion in  the case of a drunkard's linear walk.
The walk is just iterative calculating a new position based on the outcome of transition probabilites.
The walk itself may be performed k times to average out the findings (hitting time) from each walk.
Each step traverses the bipartite graph.
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