BMultiplicative smoothing on a graph

Smooth positive data on a graph the multiplicative way: replace each value by a weighted geometric mean of its neighbours. The result can never go negative, which reads like a stability guarantee. It is not one.

Taking logs turns the multiplicative scheme into ordinary explicit Euler for the heat equation, on log u. So it has exactly the ordinary stability cliff, at step h = 2/lambda_max. Push the step past the tick and watch the red copy stay positive while it diverges.

initial data
graph

step 0

ordinary Euler: u <- u - h L u
arithmetic mean (kept)
0.778294
converges to
0.778294
nodes <= 0
0
multiplicative: u <- u exp(-h L log u)
geometric mean (kept)
0.643738
converges to
0.643738
max |log u - mean|
1.248
lambda_2 (closed form)
0.152241
lambda_max (closed form)
3.84776
contraction per step max|1 - h lambda|
0.928781
arithmetic mean under the multiplicative scheme
0.778294 (not kept)
all multiplicative values positive
yes
best fixed step 2/(lambda_2 + lambda_max)
0.5

Colour is log u on one scale for both copies, centred on the initial mean of log u. A red ring marks a value outside the initial colour range; an empty red ring marks a value at or below zero, which only ordinary Euler can produce.

The precise statement

G connected and undirected, Laplacian L with eigenvalues 0 = lambda_1 < lambda_2 <= ... <= lambda_max, u in (0, oo)^n. The flow u' = -u o (L log u) is v' = -L v for v = log u: the geometric mean is invariant and u converges to it at rate lambda_2. Ordinary heat keeps the arithmetic mean instead.

Multiplicative Euler u <- u o exp(-h L log u) is v <- (I - h L) v. It is positive for every h and converges for all data iff 0 < h < 2/lambda_max; the contraction per step is max(|1 - h lambda_2|, |1 - h lambda_max|).

Where it breaks

At h = 2/lambda_max the scheme neither converges nor diverges. On the 4-cycle (lambda_max = 4, h = 1/2) with log u = (1, 0, 0, 0), exact steps give (0, 0.5, 0, 0.5), then (0.5, 0, 0.5, 0), then (0, 0.5, 0, 0.5): a period-2 orbit. Above the cliff the top mode grows like |1 - h lambda_max|^n while every value stays positive.

On images the geometric-mean limit is biased low for mean-1 speckle unless corrected, ordinary smoothing wins on PSNR, and the log-ratio edge detector loses on raw 1-look speckle (X1, section B3). Directed graphs were not tested.

Prior art and sources

Classical results it reduces to

  • Explicit-Euler limit h < 2/lambda_max for the heat equation
  • Consensus on graphs
  • Homomorphic speckle filtering
  • SAR ratio edge detectors (Touzi et al. 1988)

NNC anchors

  • A non-Newtonian gradient for contour detection in images with multiplicative noise (mora_2012_nngradient)
  • Continuous analog of multiplicative algebraic reconstruction technique for computed tomography (tateishi_2016_continuous_mart)

OpenAI families named (context only)

  • Deterministic nonbipartite Ramanujan graphs in every fixed degree (family 178)

Model-written and unreviewed. No result on this page depends on one.

Write-up and checks: docs/research/nnc-openai-crossanalysis-2026-10/X1/FINDINGS.md. Every number on this page is computed in your browser by lib/crossovers/, tested against independent closed forms in lib/crossovers.test.ts.