Five systems, one universal crossover
Across wildly different systems, a pattern repeats: detection becomes possible when the log-likelihood ratio (LLR) per unit of evidence crosses approximately 1 nat. Below this threshold the signal drowns in noise; above it, structure becomes extractable. Drag the sliders to watch each system cross the line.
Two communities of size n/2. Edges within communities have probability
a/n, across have b/n. The signal-to-noise ratio
c = (a-b)/sqrt(2(a+b)) controls detectability. At c = 1,
the phase transition occurs and communities become invisible.
Detect a constant signal μ buried in Gaussian noise with std
σ. The LLR per sample is μ²/(2σ²).
The crossover happens at SNR = μ/σ = √2 ≈ 1.41,
where LLR per sample hits 1 nat.
A coin has bias p = 0.5 + ε. Each flip gives
DKL(p || 0.5) nats of evidence. How many flips until
the total LLR reaches 1? For small ε, you need roughly
1/(2ε²) flips.
In an election with K constituencies, a small bias δ
favors one of 2 candidates. The LLR per constituency is approximately
(1 - ln 2)δ². Detection requires
K ≈ 1/[(1 - ln 2)δ²] constituencies.
In a population of size N, selection coefficient s is nearly
neutral when Ns ≈ 1. The LLR per fixation event is
2s/(e2Ns - 1) - ln(2Ns/(e2Ns - 1)).
Over many fixations, the signal accumulates.