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9b03a68a84
Extends the pd Blend simulator along two axes the deanonymization model opened up, adds the reports/blend/pd report of record, and fixes three correctness defects found while reviewing the result. Linkability over time (pd.linkability): - time to link an emitter ~ 30s*ln(1/(1-alpha))/(stake*q): inversely proportional to stake, so a 5% staker is linked in ~2 days and a 0.001% staker only after ~27 years; - time to certify a node's stake >= theta from the count of attributable observations (relative precision ~1/sqrt(N)): sizing a node costs 100-400x more than identifying it, and sub-0.1% stake is practically unlearnable. Both are closed forms over the exact deanonymization rates and a stake-proportional 30 s emission cadence, checked against a Monte-Carlo of the emission process in verify. Messaging redundancy (R independent cascades per emission, R = 1..4): - `redundancy` knob threaded through config/rng/propagation/engine/metrics/ sweep; a node receives from whichever cascade reaches it first, so arrival times combine element-wise. Delivery and capture both follow 1-(1-x)^R, so redundancy trades reliability against anonymity and divides time-to-link by ~R. Measured: delivery 0.34 -> 0.81 at 30% churn for R = 1 -> 4, while a 1%-staker's time to link falls 10 d -> 2.5 d. - Redundancy buys NO coverage: a cascade only delivers if the sender could already route to its relay, so every delivered cascade floods the sender's own component. Coverage is flat in R to four decimals at every degree. - Near the percolation threshold the cascades fail together rather than independently, so redundancy under-delivers against 1-(1-p1)^R there. Churn percolation (configs/percolation.yaml, verify check 7): - the flood only crosses responsive nodes, so it lives on the responsive sub-graph -- site percolation on a d-regular graph. A network survives churn only up to u_c = 1 - 1/(degree-1); measured collapse lands on the predicted threshold for every degree (3 -> 0.50, 6 -> 0.80, 16 -> 0.93), which inverts into the sizing rule degree > 1 + 1/(1-u). Correctness fixes: - redundancy delay used the fastest cascade's own full delay, which over-states it (min-max vs max-min); now the element-wise earliest arrival, reducing exactly to the single-cascade model at R = 1 (test); - the "redundancy improves coverage" claim was false in both the report and the simulator README -- removed and replaced with the measured result; - per-hop latency is degree-dependent (1.5 s at degree 16 to 2.7 s at degree 3), not a flat 1.6 s; and the worst-case observation figure was averaged over degrees -- at degree 8 and f_adv = 0.2 it is 0.83 -> 1.000. Statistics: round counts raised for resolution rather than speed -- 8000 rounds per cell in the main sweep, 9600 in the redundancy study, 6400 in the percolation study, giving SEM <= 0.009 on every delivery rate and <= 0.04 s on every delay mean. The previous redundancy grid (144 rounds/cell) produced a non-monotonic delivery curve; it is now monotonic and within 0.015 of theory. Adversary and deanonymization metrics remain closed-form and exact. reports/blend/pd: the report of record -- peering-degree trade-offs across speed, observation, eclipse, deanonymization and reliability, plus the time-to-link, stake-inference, redundancy and churn-threshold sections, with 21 figures of record and an explicit sampling-error statement. Co-Authored-By: Claude Opus 5 (1M context) <noreply@anthropic.com>
86 lines
3.1 KiB
Python
86 lines
3.1 KiB
Python
"""Time-to-link and stake-inference laws, and a Monte-Carlo check of the emission process."""
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import math
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import numpy as np
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from pd.linkability import (
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capture_prob,
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obs_for_precision,
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redundant,
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stake_rel_precision,
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time_to_link_seconds,
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time_to_stake_seconds,
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)
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def test_redundant_values_and_bounds():
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assert abs(redundant(0.1, 1) - 0.1) < 1e-12
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assert abs(redundant(0.1, 2) - 0.19) < 1e-12
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assert redundant(0.0, 4) == 0.0
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assert redundant(1.0, 3) == 1.0
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# strictly increasing in R for 0 < x < 1
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vals = [redundant(0.2, R) for R in (1, 2, 3, 4)]
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assert all(b > a for a, b in zip(vals, vals[1:], strict=False))
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def test_capture_prob_linkable_vs_population():
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d1 = 0.2 ** 3
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assert abs(capture_prob(d1, 1.0, 1) - d1) < 1e-12 # linkable, single cascade
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assert abs(capture_prob(d1, 0.5, 1) - 0.5 * d1) < 1e-12
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assert abs(capture_prob(d1, 1.0, 2) - (1 - (1 - d1) ** 2)) < 1e-12
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def test_time_to_link_matches_geometric_definition():
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p, alpha, slot = 0.02, 0.9, 30.0
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q = p / 0.01 # stake=0.01 -> s*q = p
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t = time_to_link_seconds(0.01, q, alpha, slot)
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n = round(t / slot)
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assert 1 - (1 - p) ** n >= alpha - 1e-12
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assert 1 - (1 - p) ** (n - 1) < alpha
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def test_time_to_link_scales_inverse_stake():
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q, alpha = 0.01, 0.9
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t1 = time_to_link_seconds(0.01, q, alpha)
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t2 = time_to_link_seconds(0.005, q, alpha)
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assert abs(t2 / t1 - 2.0) < 0.02 # halving stake ~doubles the time
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def test_time_to_link_unlinkable_is_infinite():
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assert time_to_link_seconds(0.05, 0.0, 0.9) == math.inf
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assert time_to_stake_seconds(0.01, 0.0, 100) == math.inf
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def test_redundancy_cuts_time_by_about_R():
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d1, s, alpha = 0.2 ** 3, 0.01, 0.9 # small d1 -> q_R ~ R*d1
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t1 = time_to_link_seconds(s, capture_prob(d1, 1.0, 1), alpha)
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t4 = time_to_link_seconds(s, capture_prob(d1, 1.0, 4), alpha)
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assert 3.5 < t1 / t4 < 4.0 # ~4x faster with R=4
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def test_time_to_stake_scaling():
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q = 0.008
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lin = time_to_stake_seconds(0.01, q, 200) / time_to_stake_seconds(0.01, q, 100)
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assert abs(lin - 2) < 1e-9 # linear in n_obs
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inv = time_to_stake_seconds(0.001, q, 100) / time_to_stake_seconds(0.01, q, 100)
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assert abs(inv - 10) < 1e-9 # inverse in threshold
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assert abs(time_to_stake_seconds(0.05, q, 100) - 100 / (0.05 * q) * 30) < 1e-6
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def test_obs_for_precision_and_precision():
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assert obs_for_precision(0.1) == 100
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assert obs_for_precision(0.05) == 400
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assert obs_for_precision(0.5) == 4
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assert abs(stake_rel_precision(100) - 0.1) < 1e-12
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def test_time_to_link_matches_simulation():
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"""Empirical alpha-quantile of the first-observation slot matches the closed form."""
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s, q, alpha = 0.02, 0.05, 0.9 # p = s*q = 1e-3
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rng = np.random.default_rng(7)
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first = rng.geometric(s * q, size=300_000) # slots until first success, support {1,2,...}
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emp_slots = float(np.quantile(first, alpha))
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closed_slots = time_to_link_seconds(s, q, alpha) / 30.0
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assert abs(emp_slots - closed_slots) / closed_slots < 0.02
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