 
Summary: Com S 633: Randomness in Computation
Lecture 7 Scribe: Ankit Agrawal
In the last lecture, we looked at random walks on line and used them to devise randomized
algorithms for 2SAT and 3SAT. For 2SAT we could design a randomized algorithm taking
O n2 steps; for 3SAT, we were able to reduce the number of steps from O (2n) to O 4
3
n
.
Today we will extend the concept of random walks to graphs.
1 Random Walks on Graphs
Consider a directed graph G = (V, E), V  = n, E = m. Each edge (u, v) of the graph has
a weight Muv > 0. Muv denotes the probability to reach v from u in one step. A natural
restriction, therefore is that for each vertex, the sum of the weights on outgoing edges is 1,
i.e.,
u
vN(u)
Muv = 1 (1.1)
where N(u) is the set of neighbor vertices of u. A random walk on graph, therefore implies
starting at some vertex, and traversing the graph according to the probabilities Muv. We are
interested in the long term behavior of traversing like this on the graph, i.e., the probability
