This is a discrete-time Susceptible–Believer–Skeptic (S/B/K) contagion model — the same SIR structure epidemiologists use for disease, applied to a rumor. Each of 42 "students" starts Susceptible (hasn't heard it) except a handful of seeded Believers. Every step, each Believer attempts to convert every Susceptible neighbor with the transmission probability p; Believers also spontaneously lose interest and become Skeptic (immune, no longer spreads it) at a small fixed recovery rate.
P(S→B in one step) = 1 − (1 − p)^(believer neighbors)
P(B→K in one step) = r (fixed ≈ 0.06)
epidemic threshold ≈ p·⟨k⟩ / r
The two populations shown have the same number of students and are seeded identically — the only difference is topology. The "classroom" network is a Watts–Strogatz small-world graph: mostly local ties (like desk neighbors and friend groups) with a few long-range shortcuts, giving it a high clustering coefficient and a short average path length at once. The "loose network" has fewer, randomly-placed ties and no local clustering — more like scattered acquaintances than a tight-knit class. Real classrooms behave like the small-world graph: because a rumor's neighbors are also each other's neighbors, belief gets locally reinforced (several friends confirming it) even while the network's shortcuts let it jump across the room fast — which is why dense, clustered social groups saturate with a rumor far quicker than the same number of loosely-connected people, even at equal average connectivity.
- Transmission probability — how convincing a single retelling is; higher values make each Believer–Susceptible contact more likely to convert.
- Network density — average number of social ties per student in both networks (kept equal so the comparison isolates topology, not connection count).
- Initial believers — how many students start already convinced; more seeds shorten the early lag before the rumor takes hold.