Random Walks

Part of the Undergraduate Texts in Mathematics book series (UTM)


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 of steps.


Random Walk Adjacency Matrix Regular Graph Access Time Edge Incident 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 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
  2. 58.
    R.A. Horn, C.R. Johnson, Matrix Analysis (Cambridge University Press, Cambridge, 1985)Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA

Personalised recommendations