Which room does the wanderer favor?

Question

Five rooms are connected by doorways as follows: $A$ connects to $B$ , $C$ , $D$ ; $B$ connects to $A$ , $C$ ; $C$ connects to $A$ , $B$ , $D$ , $E$ ; $D$ connects to $A$ , $C$ ; $E$ connects to $C$ . A wanderer moves each minute by walking through one of the doorways of its current room, chosen uniformly at random.

In the long run, what fraction of time does the wanderer spend in room $C$ ?

Accepted Answer

For a random walk that always steps to a uniformly random neighbor, there is a clean rule: the long-run share of a room is proportional to how many doorways it has (its number of connections, called its degree). Rooms with more doorways get visited more, exactly in proportion to their doorway count. Count the doorways (degrees): - A: 3 doors, B: 2, C: 4, D: 2, E: 1. Total over all rooms: 3 + 2 + 4 + 2 + 1 = 12. (This equals twice the number of doorways, since each door is shared by two rooms.) The long-run share of a room is its degree divided by this total: \piC = \frac{\deg(C)}{12} = \frac{…

Which room does the wanderer favor?

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