 
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 # n 2
# steps; for 3SAT, we were able to reduce the number of steps from O (2 n ) 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 M uv > 0. M uv 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 #
v#N(u)
M uv = 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 M uv . We are
interested in the long term behavior of traversing like this on the graph, i.e., the probability
