Random walk along a line of stations

Question

Four stations sit in a line: 1 - 2 - 3 - 4. Each is connected only to its immediate neighbors. A walker moves each step to a uniformly random neighbor of its current station. In the long run, what fraction of time does the walker spend at station 1 (an end of the line)?

Accepted Answer

For a uniform random walk on a network, the long-run share of a node is proportional to its degree, the number of connections it has. Count connections: - Station 1: 1 neighbor (just station 2). - Station 2: 2 neighbors (1 and 3). - Station 3: 2 neighbors (2 and 4). - Station 4: 1 neighbor (just station 3). Total of the degrees: 1 + 2 + 2 + 1 = 6. The share of station 1 is its degree over this total: \pi1 = \frac{1}{6}. So the walker spends one sixth of its time at station 1. Each middle station, with two connections, gets 	frac26 = 	frac13, twice as much as an end. Takeaway. End nodes of a…

Random walk along a line of stations

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