∇ λ ∫ ∑
Discrete maths · Where probability walks
Mass on a graph A three-state chain. Watch a blob of mass mix — or fail to.
Ask about this picture What this instrument is doing
You’re looking at Three nodes, every pair joined. Circle size is the probability sitting on that node. Each tick, mass stays put with probability stay and splits the rest evenly along the edges.
When you change it Stay is how sticky a node is. Play runs the clock; Pause then Step walks one tick at a time. Start the pile on A, B or C. Sticky C gives one node its own stay of 0.95 — watch where the pile settles.
Try this Start on A, set stay ~0.9, and wait. Mixing is slow. Drop stay to 0.4 — the three sizes equalise faster. Stay = 0 sloshes but still settles. Stay = 1 freezes. Then Sticky C: the stationary pile is no longer thirds.
A 100% B 0% C 0%
Pause Step tick 0
Stay probability 0.70 Start on A Start on B Start on C
Equal nodes Sticky C (stay 0.95)
stationary pile π 33% · 33% · 33%
distance to π 0.667 Each tick, a fraction 0.30 of every pile leaves along the edges. The gap to π shrinks by ×0.55 per tick — that number is the second eigenvalue of the transition matrix.
Ask the studio
Mass on a graph
This chat can see the instrument. Questions here are about this picture.
Exclusive to this lab — pick one, or type your own about the controls.
When does this mix? What is a stationary pile of mass? If I start on C instead of A, does the long run care? What would a stay of 1 do to the picture?