Numerical Linear Algebra
where the algebra stops being the arithmetic
Ladder

Random walk — the ladder

3 distinct arguments against one idea, from the one that introduces it to the one that assumes the rest.
  1. 01428425670849810⁻¹⁶10⁻¹⁴10⁻¹²10⁻¹⁰10⁻⁸10⁻⁶10⁻⁴10⁻²iterationchange between iteratestwo rates, one curveα0.85λ₂(P)0.9predicted rate0.77measured0.77iterations108against the solve10⁻¹⁶the upper dashed line is αᵏthe curve is on the other one

    A ranking that is an eigenvector

    PageRank is the stationary vector of a walk that follows links with probability α and jumps at random otherwise. The iteration and the elimination agree to 4·10⁻¹⁷. What α is set to changes which pages come third, fourth and fifth.

    rung 1 · graph
  2. 0173451688510211910⁻⁶10⁻⁵10⁻⁴10⁻³10⁻²10⁻¹1steps of the walkdistance from stationarya rate read as a timeλ₂ of the walk0.9steps measured122steps predicted131ratio0.93the dashed line is the eigenvaluethe curve is the walk

    The rate is the second eigenvalue

    A walk forgets where it started at a rate the graph's second eigenvalue names exactly. Across three orders of magnitude in the step count the prediction is five per cent high — and the published rate for PageRank is right for a reason nobody states, which is that a link graph is in pieces.

    rung 2 · graph
  3. 02244668811013215417610⁻¹⁶10⁻¹⁴10⁻¹²10⁻¹⁰10⁻⁸10⁻⁶10⁻⁴10⁻²iterationchange between iteratestwo rates, one curveα0.85λ₂(P)1predicted rate0.85measured0.84iterations176against the solve1.4·10⁻¹⁶the upper dashed line is αᵏthe curve is on the other one

    A chain with no stationary vector

    A page with no outgoing links loses forty per cent of the walker's probability in six hundred steps. A directed cycle never converges at all. And on a graph whose links only run one way, the entire rank of half the vertices is exactly one minus the teleportation parameter.

    rung 3 · graph

All ladders