Terminal-state theorems for a two-stage Maki--Thompson rumor model: final size, stifler composition, and non-identifiability from pooled statistics
Abstract
Rumor models of the Daley--Kendall and Maki--Thompson family anchor the statistical mechanics of social contagion, yet the stifler composition a rumor leaves behind has resisted closed-form analysis. Inspired by the asymmetric two-step pairing (A2SP) process of step-growth polymerization, we import its common-factor cancellation into a minimal two-stage MT variant in which re-contacted spreaders are re-activated into a senior class at rate αk1 rather than stifled, and obtain exact terminal-state theorems. (i) Saturation: the rumor exhausts the ignorant pool for every stifling-to-recruitment ratio μ>0, with a three-way tail classification and a logarithmically critical line at μ=1. (ii) Final size: the terminal stifler pool splits as (x1,x2)=(μ/(α+μ),α/(α+μ)), independent of seed size; the fingerprint inverts linearly (α=μx2/x1) and matches Gillespie simulations within 0.95%. (iii) Pooled versus history-resolved: pooled statistics are exactly α-free on any static graph; only the history-resolved composition carries α. Restoring the classic teller-retires channel breaks the exact closure beyond quantified gates (contact rates of $1−−3\%$ of k1 for spreader channels at μ≤1.5, up to 32% at μ=2), and on configuration-model networks the composition bias decays as ⟨ k⟩−0.71. An exploratory inversion of 354,930 Higgs--Twitter retweet episodes, screened by an identity check and four negative controls, yields α̂=2.21% (joint cluster-bootstrap CI [1.6%,3.0%]; ≈3.7% after calibration at the matched threshold). Terminal compositions identify this effective re-activation rate, an upper bound on contact-coupled re-activation; the mechanism itself requires timing information beyond the terminal state.
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BibTeX
@misc{tang2026terminalstate,
title = {{Terminal-state theorems for a two-stage Maki--Thompson rumor model: final size, stifler composition, and non-identifiability from pooled statistics}},
author = {Qingquan Tang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.41206},
eprint = {2610.41206},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.41206},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Terminal-state theorems for a two-stage Maki--Thompson rumor model: final size, stifler composition, and non-identifiability from pooled statistics
AU - Qingquan Tang
PY - 2026
DA - 2026/10/30
PB - arXiv
DO - 10.48550/arxiv.2610.41206
UR - https://arxiv.org/abs/2610.41206
AB - Rumor models of the Daley--Kendall and Maki--Thompson family anchor the statistical mechanics of social contagion, yet the stifler composition a rumor leaves behind has resisted closed-form analysis. Inspired by the asymmetric two-step pairing (A2SP) process of step-growth polymerization, we import its common-factor cancellation into a minimal two-stage MT variant in which re-contacted spreaders are re-activated into a senior class at rate αk_1 rather than stifled, and obtain exact terminal-state theorems. (i) Saturation: the rumor exhausts the ignorant pool for every stifling-to-recruitment ratio μ>0, with a three-way tail classification and a logarithmically critical line at μ=1. (ii) Final size: the terminal stifler pool splits as (x_1,x_2)=(μ/(α+μ),α/(α+μ)), independent of seed size; the fingerprint inverts linearly (α=μx_2/x_1) and matches Gillespie simulations within 0.95%. (iii) Pooled versus history-resolved: pooled statistics are exactly α-free on any static graph; only the history-resolved composition carries α. Restoring the classic teller-retires channel breaks the exact closure beyond quantified gates (contact rates of $1−−3\%$ of k_1 for spreader channels at μ≤1.5, up to 32% at μ=2), and on configuration-model networks the composition bias decays as ⟨ k⟩^−0.71. An exploratory inversion of 354,930 Higgs--Twitter retweet episodes, screened by an identity check and four negative controls, yields α̂=2.21% (joint cluster-bootstrap CI [1.6%,3.0%]; ≈3.7% after calibration at the matched threshold). Terminal compositions identify this effective re-activation rate, an upper bound on contact-coupled re-activation; the mechanism itself requires timing information beyond the terminal state.
ER -