Author: Aanavi Dalmia

Summary

  1. Log-normal link latency and its parameters
  2. The single-path mix under the log-normal law and its Bayesian attribution results
  3. Dandelion++ as a second, structurally different anonymity layer: the geometric stem, its closed-form model, and its own results
  4. Single-path mix vs Dandelion++ at matched design budget

Parameters

group parameter symbol value
consensus nodes N 1000
active-slots coefficient f 1/30
Pareto shape (stake inequality) $k_{stake}$ 4/3(G=0.60, Cardano-calibrated)
epoch length T 388,800
network / links connectivity (random regular graph) C 8
link law (every AC-path leg and every gossip edge) d 7.2 ms+LogNormal
broadcast latency (multi-source Dijkstra) $E[D_{br}]$ ≈0.251 s
single-path mix mix hops / layers k 3
per-hop mix delay (mean) 1/$λ_M$ swept, 0.001→1.0 s
sender hold (mean) $1/λ_S$ coupled: $(1/λ_M)/ρ$
delay ratio $ρ=λ_S/λ_M$ 4
receiver delay — none (k stages, not k+1)
Dandelion++ stem-forwarding probability p 0.9 (q=1−p=0.1)
attacker's assumed hop count k 10 (=1/(1−p))
per-relay-hop hold (mean) $1/λ_M$ swept, 0.001→1.00 s, same grid
sender hold $1/λ_S$ ≈0 (no deliberate protocol stage)
cover / sender set cover count (fixed) count 19
statistics seed / repetitions 0 / 50 per cell

1. The single-path mix under the log-normal link law

Every AC-path leg — sender-to-first-mixer, each inter-mix hop, the last-mixer-to-receiver leg, and every edge of the gossip/broadcast graph — is modelled as $d=L+X,$ $X∼LogNormal(μ,σ)$, the same shifted log-normal law used across the rest of the project's latency modelling. Three independent justifications support it over the project's own "uniform" and "geographic" alternatives:

  1. Mechanistic. Link latency is the outcome of several multiplicative effects stacked along a path (queueing, routing detours, per-hop processing); a product of positive random factors is log-normal in the limit, by the same argument that makes sums of independent effects tend to Normal — the standard treatment is Limpert, Stahel & Abbt, "Log-normal Distributions across the Sciences: Keys and Clues," BioScience 51(5), 341–352 (2001).
  2. Empirically mixed, not a clean win for log-normal. Karakaş (2003, METU MS thesis) fit gamma, log-normal, and Weibull to real delay traces on two measured paths; none passes a Kolmogorov–Smirnov test at either (large sample sizes make the critical threshold too tight for any of the three to survive). Log-normal has the best fit on most measurements on one path (GIT–METU, post firewall-fix); Weibull wins on the other (METU–HU) — so this is a ranking on the better-behaved dataset, not a validated ground truth.
  3. Maximum-entropy. The precise version of this argument constrains the mean and variance of ln ⁡d, not of d itself — i.e. a typical multiplicative scale and a multiplicative spread, both strictly positive. Under exactly those two constraints, the least-committal (maximum-entropy) distribution for ln ⁡d is Gaussian, making d log-normal, by the standard maximum-entropy argument (Jaynes, "Information Theory and Statistical Mechanics,"Physical Review 106, 620–630, 1957).

Unlike the project's discrete "geographic" law (three continents, exact ties possible), the log-normal law is continuous, so no candidate can tie exactly on $d_i^S$ and no free ceiling is handed to the defender by construction — any residual Top-1 accuracy has to come from the attack itself, not from an artefact of the latency model.

1.1 Calibration

The floor, mean, and standard deviation are taken as given inputs — moment-matched to real one-way ping data (WonderNetwork, 12 cities across 3 continents, 66 city pairs) — rather than re-derived here:

value what it is
floor L 7.2 ms smallest measured one-way city-pair latency
mean 62.9 ms population mean of the measured one-way latencies
sd 33.3 ms population standard deviation

Because the floor is a pure shift, $E[d]=L+E[X]$ and $Var(d)=Var(X)$, so it is subtracted before solving for the underlying Normal's own parameters:

$m=mean−L=55.7 ms,σ_{log}⁡=ln⁡ ⁣(1+sd^2/m^2)=0.553,μ_{log}⁡=ln⁡(m)−σ_{log}⁡^2/2=−3.041$

1.2 Bayesian sender attribution