Social Propagation: Contagion, Thresholds, and Policy Friction

Abstract

This playground models how content spreads across a large social network and how platform policy can suppress coordinated political manipulation. It treats sharing as a branching contagion process governed by a reproduction number, applies the epidemic threshold to decide whether a cascade self-sustains, and accumulates an expected count of successful manipulations as content ages and friction is applied. The companion below sets out the model, its grounding in epidemiology and the sociology of diffusion, the closed-form results used to calibrate it, and where it should not be trusted.

Background: two traditions of contagion

Two research traditions describe how things spread through populations.

The first is mathematical epidemiology. The compartmental SIR model of Kermack and McKendrick (1927) partitions a population into Susceptible, Infected, and Recovered and tracks flows between them. In its simplest deterministic form the infected fraction grows according to dS/dt = minus beta S I, dI/dt = beta S I minus gamma I, and dR/dt = gamma I, with the basic reproduction number R0 = beta over gamma. The single most important result is the threshold theorem: an outbreak grows only when R0 exceeds 1. Below 1 each infection produces on average fewer than one further infection and the chain dies out. Above 1 a finite fraction of the population is eventually reached, given by the final-size relation, which for a well-mixed population settles near 1 minus 1 over R0.

The second tradition is the sociology of diffusion. Granovetter's threshold model (1978) and Watts's cascade model (2002) describe adoption that requires social proof: a person acts only when the fraction of their contacts who have already acted crosses a personal threshold theta. Centola and Macy (2007) sharpened the distinction between simple contagion, where a single exposure can transmit (like a virus), and complex contagion, where multiple reinforcing exposures are needed (like risky or costly behaviour). Political persuasion sits uneasily between the two: a single viral post can plant a claim, but durable belief change usually needs repetition and corroboration.

The social propagation playground borrows the reproduction number and threshold from epidemiology for its running simulation, and exposes the linear-threshold rule as a contrast object in its logic and calibration.

The model

The Viewer runs a discrete-time mean-field branching process at one-minute resolution. Rather than tracking individual accounts, it carries aggregate state: the number of currently active shares, cumulative reach, and a cumulative manipulation index. The core quantities are as follows.

Basic reproduction number. A fresh share reaches avgDegree followers, each of whom reshares with probability shareProb, optionally amplified by an algorithmic boost. The product

R0 = avgDegree x shareProb x amplification

is the branching analogue of R0 = contacts times transmission probability. With the default fan-out of 150, reshare probability 0.02, and amplification 1.0, R0 equals 3.0.

Effective reproduction number. Policy levers act as multipliers. Forward caps halve the effective fan-out (degree factor 0.5); question gating scales down the reshare probability (share factor 0.75); delayed visibility (coolup) damps ignition (damping 0.55). The effective reproduction number is

R_eff = R0 x degreeFactor x shareProbFactor x coolupIgnitionDamp

Epidemic threshold. The cascade is supercritical, meaning self-sustaining and growing, exactly when R_eff exceeds 1. Stacking enough friction to drive R_eff below 1 is the whole game of the policy scenario: it converts a growing cascade into one that fades.

Saturation and final size. Growth cannot continue forever in a finite audience. With susceptible fraction s = 1 minus reach over N, the per-step multiplier is R_eff times s, so growth halts at the fixed point where R_eff s = 1. The eventual attack-rate fraction is 1 minus 1 over R_eff, the social-contagion analogue of the SIR final-size relation. At R_eff = 2 this is one half of the audience.

Attention decay and manipulation. Persuasive potency fades with content age by a half-life: freshness equals 2 to the power of minus age over half-life. The effective per-exposure conversion probability is the baseline conversion times a gating-skepticism factor times freshness. The Political Manipulation Impact (PMI) index accumulates, at each minute, the new exposures times this effective conversion. Delayed-visibility policies reduce PMI not only by cutting reach but by ageing content before it is seen, so it lands with diminished potency.

Calibration

Because the live simulation is stochastic and tuned for pedagogical visibility rather than quantitative forecasting, only its deterministic skeleton is calibrated. Each calibration case checks a logic function against a closed-form target:

• R0 product form. Fan-out 150, reshare 0.02, amplification 1.0 gives R0 = 3.0 exactly.
• Threshold, supercritical side. Forward caps and gating leave R_eff = 1.125, still above 1, so the cascade spreads (boolean check).
• Threshold, subcritical side. Lower fan-out plus full friction drives R_eff below 1, so the cascade dies out.
• Final-size relation. At R_eff = 2.0 the eventual reach is 1 minus one half, that is half the audience.
• Freshness decay. Content one half-life old retains exactly half its potency.
• Complex contagion. A node with threshold 0.25 and 40 percent active neighbours adopts, as the linear-threshold rule requires.
• Cooldown clamp. A ten-hour posting lock caps any account at 24 over 10 = 2.4 posts per day.

All seven reproduce their analytic targets with zero error.

Limitations

The model is a deliberately transparent caricature and several of its simplifications matter.

The mean-field collapse assumes well-mixed contact. Real social graphs are clustered and heavy-tailed; hubs and communities mean a single R_eff is not enough, and on strongly heterogeneous networks the epidemic threshold can vanish entirely. The running simulation uses simple contagion, so where adoption is genuinely complex and threshold-gated it overstates spread. Policy levers are static multipliers with no adaptive adversary, yet real spammers buy aged accounts, shift timing, and evade caps, so measured effectiveness erodes over time. The PMI index is a proxy for expected conversions, not a model of belief change, persuasion heterogeneity, or downstream behaviour. Finally, finite stochastic cascades can fade out early even when R_eff exceeds 1, so the deterministic final-size figure is only an expectation over surviving cascades.

The playground is therefore a tool for reasoning about mechanism and comparative policy effect, not a calibrated forecaster of any specific information operation.

References

• Kermack, W. O. and McKendrick, A. G. (1927). A contribution to the mathematical theory of epidemics.
• Anderson, R. M. and May, R. M. (1991). Infectious Diseases of Humans.
• Diekmann, O., Heesterbeek, J. A. P. and Metz, J. A. J. (1990). On the definition and computation of the basic reproduction ratio R0.
• Pastor-Satorras, R. and Vespignani, A. (2001). Epidemic spreading in scale-free networks.
• Granovetter, M. (1978). Threshold models of collective behavior.
• Watts, D. J. (2002). A simple model of global cascades on random networks.
• Centola, D. and Macy, M. (2007). Complex contagions and the weakness of long ties.
• Wu, F. and Huberman, B. A. (2007). Novelty and collective attention.