Numerical Linear Algebra
where the algebra stops being the arithmetic
Generator

Chung's Laplacian against the undirected one, on four families

One function in the digraph library, called 9 times across 2 essays. Below: what it draws at its defaults, what it draws at every value an essay asks for, the 13 claims it put to the test while drawing them, and where it stands against the rule this site is named for.

At its defaults it draws chung's laplacian against the undirected one, on four families. The worst entrywise difference between the directed normalised Laplacian and the undirected normalised Laplacian of the same graph with its arrows removed. On the balanced and symmetric families the difference is at the rounding level at every size — 10⁻¹⁴ and below — because a balanced digraph's stationary distribution is its degree distribution and the construction then collapses to the undirected one. On the two-block family with a single arc back the difference is 0.163 at n = 30, which is the size of what the direction adds. The two curves are the same computation and the flat one is a closed form, so this figure is the field's second route rather than a comparison.

chung-reduction is one function in lib/figures/digraph.js — arrows — a laplacian that is not symmetric, and the walk that has to be computed before it can be. Everything below came out of it during this build, at arguments taken from the essays rather than invented for this page. A figure here is the figure a reader meets in an essay, and if the generator changes, this page changes with it.

At its defaults

Drawn even though every essay passes arguments — which on this site is every essay, at 100% of placements since the standard pass. A default nothing exercises is a trap for the next essay to call this with none, and this is the page where a default that has drifted from the figures around it becomes visible.

Chung's Laplacian against the undirected one, on four familiesThe worst entrywise difference between the directed normalised Laplacian and the undirected normalised Laplacian of the same graph with its arrows removed. On the balanced and symmetric families the difference is at the rounding level at every size — 10⁻¹⁴ and below — because a balanced digraph's stationary distribution is its degree distribution and the construction then collapses to the undirected one. On the two-block family with a single arc back the difference is 0.163 at n = 30, which is the size of what the direction adds. The two curves are the same computation and the flat one is a closed form, so this figure is the field's second route rather than a comparison.711151923273110⁻¹⁷10⁻¹⁴10⁻¹¹10⁻⁸10⁻⁵10⁻²verticesworst entrywise differencetwo blocks · random strongbalanced · symmetrica closed form, and what it excludesbalanced, worst3.9·10⁻¹⁴symmetric, worst1.1·10⁻¹⁶two blocks0.16random strong0.28balanced is undirected in disguiseand nothing else is

The worst entrywise difference between the directed normalised Laplacian and the undirected normalised Laplacian of the same graph with its arrows removed. On the balanced and symmetric families the difference is at the rounding level at every size — 10⁻¹⁴ and below — because a balanced digraph's stationary distribution is its degree distribution and the construction then collapses to the undirected one. On the two-block family with a single arc back the difference is 0.163 at n = 30, which is the size of what the direction adds. The two curves are the same computation and the flat one is a closed form, so this figure is the field's second route rather than a comparison.

upTo: 30

The arguments are the ones A conductance the arcs do not measure passes. A value drawn at the generator's defaults instead would be a picture no essay asked for and no assertion has been run against.

Chung's Laplacian against the undirected one, on four familiesThe worst entrywise difference between the directed normalised Laplacian and the undirected normalised Laplacian of the same graph with its arrows removed. On the balanced and symmetric families the difference is at the rounding level at every size — 10⁻¹⁴ and below — because a balanced digraph's stationary distribution is its degree distribution and the construction then collapses to the undirected one. On the two-block family with a single arc back the difference is 0.163 at n = 30, which is the size of what the direction adds. The two curves are the same computation and the flat one is a closed form, so this figure is the field's second route rather than a comparison.711151923273110⁻¹⁷10⁻¹⁴10⁻¹¹10⁻⁸10⁻⁵10⁻²verticesworst entrywise differencetwo blocks · random strongbalanced · symmetrica closed form, and what it excludesbalanced, worst3.9·10⁻¹⁴symmetric, worst1.1·10⁻¹⁶two blocks0.16random strong0.28balanced is undirected in disguiseand nothing else is

The worst entrywise difference between the directed normalised Laplacian and the undirected normalised Laplacian of the same graph with its arrows removed. On the balanced and symmetric families the difference is at the rounding level at every size — 10⁻¹⁴ and below — because a balanced digraph's stationary distribution is its degree distribution and the construction then collapses to the undirected one. On the two-block family with a single arc back the difference is 0.163 at n = 30, which is the size of what the direction adds. The two curves are the same computation and the flat one is a closed form, so this figure is the field's second route rather than a comparison.

upTo: 48

The arguments are the ones A conductance the arcs do not measure passes. A value drawn at the generator's defaults instead would be a picture no essay asked for and no assertion has been run against.

Chung's Laplacian against the undirected one, on four familiesThe worst entrywise difference between the directed normalised Laplacian and the undirected normalised Laplacian of the same graph with its arrows removed. On the balanced and symmetric families the difference is at the rounding level at every size — 10⁻¹⁴ and below — because a balanced digraph's stationary distribution is its degree distribution and the construction then collapses to the undirected one. On the two-block family with a single arc back the difference is 0.1902 at n = 48, which is the size of what the direction adds. The two curves are the same computation and the flat one is a closed form, so this figure is the field's second route rather than a comparison.71115192327313539434710⁻¹⁷10⁻¹⁴10⁻¹¹10⁻⁸10⁻⁵10⁻²verticesworst entrywise differencetwo blocks · random strongbalanced · symmetrica closed form, and what it excludesbalanced, worst1.5·10⁻¹³symmetric, worst1.1·10⁻¹⁶two blocks0.19random strong0.31balanced is undirected in disguiseand nothing else is

The worst entrywise difference between the directed normalised Laplacian and the undirected normalised Laplacian of the same graph with its arrows removed. On the balanced and symmetric families the difference is at the rounding level at every size — 10⁻¹⁴ and below — because a balanced digraph's stationary distribution is its degree distribution and the construction then collapses to the undirected one. On the two-block family with a single arc back the difference is 0.1902 at n = 48, which is the size of what the direction adds. The two curves are the same computation and the flat one is a closed form, so this figure is the field's second route rather than a comparison.

upTo: 12

The arguments are the ones A conductance the arcs do not measure passes. A value drawn at the generator's defaults instead would be a picture no essay asked for and no assertion has been run against.

Chung's Laplacian against the undirected one, on four familiesThe worst entrywise difference between the directed normalised Laplacian and the undirected normalised Laplacian of the same graph with its arrows removed. On the balanced and symmetric families the difference is at the rounding level at every size — 10⁻¹⁴ and below — because a balanced digraph's stationary distribution is its degree distribution and the construction then collapses to the undirected one. On the two-block family with a single arc back the difference is 0.08215 at n = 12, which is the size of what the direction adds. The two curves are the same computation and the flat one is a closed form, so this figure is the field's second route rather than a comparison.71110⁻¹⁷10⁻¹⁴10⁻¹¹10⁻⁸10⁻⁵10⁻²verticesworst entrywise differencetwo blocks · random strongbalanced · symmetrica closed form, and what it excludesbalanced, worst1.3·10⁻¹⁴symmetric, worst1.1·10⁻¹⁶two blocks0.082random strong0.21balanced is undirected in disguiseand nothing else is

The worst entrywise difference between the directed normalised Laplacian and the undirected normalised Laplacian of the same graph with its arrows removed. On the balanced and symmetric families the difference is at the rounding level at every size — 10⁻¹⁴ and below — because a balanced digraph's stationary distribution is its degree distribution and the construction then collapses to the undirected one. On the two-block family with a single arc back the difference is 0.08215 at n = 12, which is the size of what the direction adds. The two curves are the same computation and the flat one is a closed form, so this figure is the field's second route rather than a comparison.

upTo: 18

The arguments are the ones A conductance the arcs do not measure passes. A value drawn at the generator's defaults instead would be a picture no essay asked for and no assertion has been run against.

Chung's Laplacian against the undirected one, on four familiesThe worst entrywise difference between the directed normalised Laplacian and the undirected normalised Laplacian of the same graph with its arrows removed. On the balanced and symmetric families the difference is at the rounding level at every size — 10⁻¹⁴ and below — because a balanced digraph's stationary distribution is its degree distribution and the construction then collapses to the undirected one. On the two-block family with a single arc back the difference is 0.1459 at n = 18, which is the size of what the direction adds. The two curves are the same computation and the flat one is a closed form, so this figure is the field's second route rather than a comparison.711151910⁻¹⁷10⁻¹⁴10⁻¹¹10⁻⁸10⁻⁵10⁻²verticesworst entrywise differencetwo blocks · random strongbalanced · symmetrica closed form, and what it excludesbalanced, worst3.1·10⁻¹⁴symmetric, worst1.1·10⁻¹⁶two blocks0.15random strong0.22balanced is undirected in disguiseand nothing else is

The worst entrywise difference between the directed normalised Laplacian and the undirected normalised Laplacian of the same graph with its arrows removed. On the balanced and symmetric families the difference is at the rounding level at every size — 10⁻¹⁴ and below — because a balanced digraph's stationary distribution is its degree distribution and the construction then collapses to the undirected one. On the two-block family with a single arc back the difference is 0.1459 at n = 18, which is the size of what the direction adds. The two curves are the same computation and the flat one is a closed form, so this figure is the field's second route rather than a comparison.

upTo: 38

The arguments are the ones A conductance the arcs do not measure passes. A value drawn at the generator's defaults instead would be a picture no essay asked for and no assertion has been run against.

Chung's Laplacian against the undirected one, on four familiesThe worst entrywise difference between the directed normalised Laplacian and the undirected normalised Laplacian of the same graph with its arrows removed. On the balanced and symmetric families the difference is at the rounding level at every size — 10⁻¹⁴ and below — because a balanced digraph's stationary distribution is its degree distribution and the construction then collapses to the undirected one. On the two-block family with a single arc back the difference is 0.1683 at n = 38, which is the size of what the direction adds. The two curves are the same computation and the flat one is a closed form, so this figure is the field's second route rather than a comparison.7111519232731353910⁻¹⁷10⁻¹⁴10⁻¹¹10⁻⁸10⁻⁵10⁻²verticesworst entrywise differencetwo blocks · random strongbalanced · symmetrica closed form, and what it excludesbalanced, worst9.7·10⁻¹⁴symmetric, worst1.1·10⁻¹⁶two blocks0.17random strong0.28balanced is undirected in disguiseand nothing else is

The worst entrywise difference between the directed normalised Laplacian and the undirected normalised Laplacian of the same graph with its arrows removed. On the balanced and symmetric families the difference is at the rounding level at every size — 10⁻¹⁴ and below — because a balanced digraph's stationary distribution is its degree distribution and the construction then collapses to the undirected one. On the two-block family with a single arc back the difference is 0.1683 at n = 38, which is the size of what the direction adds. The two curves are the same computation and the flat one is a closed form, so this figure is the field's second route rather than a comparison.

What it checked while drawing

Every figure above asserted its own claims on the way to being drawn, and a claim that failed would have failed the build rather than drawn a wrong picture. Those assertions used to leave no trace at all: a passing one returned true and the only evidence the figure had checked anything was that nothing crashed. The list below is what they actually said, collected by running this generator with an observer installed — not a description of what it is believed to check.

13 distinct claims across 6 sets of arguments, grouped below by shape — because most of them are one sentence with a different number in it, and how many separate times that sentence was put to the test is the informative part.

a digraph with at least two vertices

a range of sizes

and an unbalanced family does not

and every entry of it is nonnegative, as a probability must be

arcs inside the vertex set

Chung's Laplacian is symmetric to the rounding level

no repeated arc

no self-loops

positive arc weights

the assertion refuses a counterexample

the balanced family reduces to the undirected normalised Laplacian

the power iteration reached a stationary vector

the symmetric family reduces to the undirected normalised Laplacian

Against the rule

The rule does not apply to it. It factorises nothing, so there is no residual it could be withholding. That is worth stating rather than leaving blank: a site that reported the rule as satisfied by every generator would be counting mostly generators the rule never reached.

Across the library: the rule bites on 214 of 397 generators — 194 print a residual and 20 are exempt with a published reason; 183 factorise nothing. Read from lib/residual-rule.js, which is the same body the gate enforces from, and the gate's last check fails the build if this page and it disagree about any generator.

Where it is called

Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.

The matrix that is a graph

A conductance the arcs do not measure

Symmetrising a directed Laplacian with respect to its walk recovers everything the arrows took — a real spectrum, a sweep cut, a Cheeger inequality. What it does not recover is the quantity: the inequality bounds the probability that a step of the walk crosses the cut, which on one graph here is three times the weight of the arcs that do.

 The matrix that is a graph

A Laplacian that is not symmetric

Point the edges and the matrix stops being symmetric. Its row sums are still exactly zero, so the null vector survives; everything built on the quadratic form does not, and the eigenvalues need a plane rather than a line. Asymmetry permits that and does not force it, which the smallest case here demonstrates by being asymmetric and real.

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