When Small Acts Scale: Ethical Thresholds in Network Diffusion

Much ethical evaluation treats actions dyadically: one agent acts on one recipient. In networked, platform-mediated environments, this lens misses how public acts diffuse. We introduce a minimal message-passing model in which an initiating act with baseline valence w spreads across a social graph with exposure b, per-hop salience $alpha$, compliance $q$, and depth (horizon) d. The model yields a closed-form \emph{network multiplier} relative to the dyadic baseline and identifies a threshold at r=b.alpha.q=1 separating subcritical (saturating), critical (linear), and supercritical (geometric) regimes. We show how common platform design levers – reach and fan-out (affecting b), ranking and context (affecting alpha), share mechanics and friction (affecting q), and time-bounds (affecting d) – systematically change expected downstream responsibility Applications include pandemic mitigation and vaccination externalities, as well as platform amplification of prosocial and harmful norms.

💡 Research Summary

The paper “When Small Acts Scale: Ethical Thresholds in Network Diffusion” develops a concise mathematical framework to capture how a single moral act propagates through a social network and how that propagation amplifies the initiator’s ethical responsibility. The authors model the act as a message that spreads from the origin across a branching process of depth d. Four parameters characterize the diffusion: b (the average number of immediate contacts or exposure per agent), α (the per‑hop salience or attenuation factor, 0 < α ≤ 1), q (the probability that an exposed agent adopts and retransmits the behavior), and d (the effective horizon of influence). The baseline moral weight of the act is w > 0.

At hop k the expected number of newly impacted agents is Cₖ = bᵏ q^{k‑1}. Each hop’s impact is further multiplied by α^{k‑1} to reflect diminishing influence with distance. Summing over all hops yields the total expected responsibility attributable to the initiator:

T(w; b, α, q, d) = w ∑{k=1}^{d} b^{k}(αq)^{k‑1} = w b ∑{j=0}^{d‑1}(bαq)^{j}.

The series is geometric with ratio r = bαq, which the authors term the “effective diffusion ratio.” The closed‑form solution is

T = ⎧ w b