Numerical Linear Algebra
where the algebra stops being the arithmetic
Ladder

Spectral partition — the ladder

3 distinct arguments against one idea, from the one that introduces it to the one that assumes the rest.
  1. 0481216202410⁻²10⁻¹1vertices on the smaller sideconductance of the prefix cut0.00752, the best prefixthe rounding stepλ₂0.14cuts considered23best conductance0.0075at k =12worst prefix1the dashed curve is the eigenvectorthe solid one is what it costs

    The vector that has to be rounded

    A spectral partition is an eigenvector, and an eigenvector is a real vector. The answer wanted is a subset. Something has to turn one into the other, and the something is a heuristic applied after the linear algebra has finished.

    rung 1 · graph
  2. 10⁻³10⁻²10⁻¹110¹12345678910conductance, and the two bounds on itpathcyclegridbarbelltwo blockshypercubepreferentialstarcompletethe bar is the inequalitythe dot is the graph

    A bound with a square root in it

    Cheeger's inequality brackets a graph's best cut between λ₂/2 and √(2λ₂). The lower bound is attained exactly. The upper one is loose by a factor of fourteen — on the one graph in the census with a real bottleneck, which is the shape it is always quoted about.

    rung 2 · graph
  3. 14710131619110¹indexeigenvalue of Lan eigenvalue with no vectorλ₂0.35λ₃0.35gap1.7·10⁻¹⁵partitions found3best conductance0.045worst0.045two eigenvalues, one valueand the answer is not a function of the graph

    A partition decided in the last digit

    On a graph with a symmetry there is no Fiedler vector — there is a plane, and every vector in it is an exact eigenvector. Twenty-four runs with the edge weights nudged by 10⁻¹² return ten different partitions of a cycle and, on a hypercube, two different qualities of answer.

    rung 3 · graph

All ladders