Algebraic Combinatorics pp 21-30 | Cite as

# Random Walks

- 5.6k Downloads

## Abstract

Let *G* be a finite graph. We consider a random walk on the vertices of *G* of the following type. Start at a vertex *u*. (The vertex *u* could be chosen randomly according to some probability distribution or could be specified in advance.) Among all the edges incident to *u*, choose one uniformly at random (i.e., if there are *k* edges incident to *u*, then each of these edges is chosen with probability 1∕*k*). Travel to the vertex *v* at the other end of the chosen edge and continue as before from *v*. Readers with some familiarity with probability theory will recognize this random walk as a special case of a finite-state Markov chain. Many interesting questions may be asked about such walks; the basic one is to determine the probability of being at a given vertex after a given number *ℓ* of steps.

## Keywords

Random Walk Adjacency Matrix Regular Graph Access Time Edge Incident## References

- 71.L. Lovász, Random walks on graphs: a survey, in
*Combinatorics. Paul Erdǒs is Eighty*, vol. 2, Bolyai Society Mathematical Studies, vol. 2 (Keszthely, Hungary, 1993), pp. 1–46Google Scholar - 58.R.A. Horn, C.R. Johnson,
*Matrix Analysis*(Cambridge University Press, Cambridge, 1985)Google Scholar