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86 lines
3.1 KiB
Python
86 lines
3.1 KiB
Python
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"""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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