Editing Resistance distance (section)

===Relationship to random walks===

The resistance distance between vertices <math>u</math> and <math>v</math> is proportional to the '''commute time''' <math>C_{u,v}</math> of a [[random walk]] between <math>u</math> and <math>v</math>. The commute time is the expected number of steps in a random walk that starts at <math>u</math>, visits <math>v</math>, and returns to <math>u</math>. For a graph with <math>m</math> edges, the resistance distance and commute time are related as <math>C_{u,v}=2m\Omega_{u,v}</math>.<ref>{{cite book |last1=Chandra, Ashok K and Raghavan, Prabhakar and Ruzzo, Walter L and Smolensky, Roman |title=Proceedings of the twenty-first annual ACM symposium on Theory of computing - STOC '89 |chapter=The electrical resistance of a graph captures its commute and cover times |date=1989 |issue=21 |pages=574–685 |doi=10.1145/73007.73062 |isbn=0897913078 |chapter-url=https://dl.acm.org/doi/abs/10.1145/73007.73062}}</ref>


Resistance distance is also related to the '''escape probability''' between two vertices. The escape probability <math>P_{u,v}</math> between <math>u</math> and <math>v</math> is the probability that a random walk starting at <math>u</math> visits <math>v</math> before returning to <math>u</math>. The escape probability equals 
:<math>
P_{u,v} = \frac{1}{\deg(u)\Omega_{u,v}},
</math>
where <math>\deg(u)</math> is the [[Degree (graph theory)|degree]] of <math>u</math>.<ref>{{cite book |last1=Doyle |first1=Peter |last2=Snell |first2=J. Laurie |title=Random Walks and Electric Networks |date=1984 |publisher=American Mathematical Society |isbn=9781614440222}}</ref> Unlike the commute time, the escape probability is not symmetric in general, i.e., <math>P_{u,v}\neq P_{v,u}</math>.