Cross-inhibition leads to group consensus despite the presence of strongly opinionated minorities and asocial behaviour

The paper investigates how groups reach consensus via local interactions without centralized control, focusing on robustness to asocial behavior (noise) and strongly opinionated minorities (zealots). The classical voter model, widely applied across biological and social systems, is known to be fragile: even minimal noise or the presence of inflexible individuals prevents stable agreement. The authors propose and analyze the cross-inhibition model, where pairwise interactions between agents with opposing opinions cause the listener to become undecided (rather than directly adopting the speaker’s opinion as in the voter model). Only undecided agents adopt others’ opinions. Asocial behavior is considered in two forms: (i) noise, where individuals sporadically change their opinion independently of peers; and (ii) zealots, who never change their opinion upon social signals. Though mathematically equivalent at the macroscopic level in some conditions, these mechanisms differ microscopically. Inhibitory signaling is widespread in natural systems (e.g., neuronal networks, social insects), and prior work showed cross-inhibition breaks symmetry in noise-free contexts. The research question is whether cross-inhibition enables stable consensus despite asocial behavior, and how it compares to the (weighted) voter model when options have equal or unequal qualities. The study is motivated both by biological explanations for inhibitory signaling and by applications to resilient, decentralized robot swarms.

The study combines analytical, semi-analytical, computational, and robotic experimental methods. - Models and states: Individuals can be in states X (opinion X), Y (opinion Y), or U (undecided). Two symmetric zealot groups Zx and Zy are considered when applicable. Asocial behavior is modeled as: (i) noise, with two variants in cross-inhibition—type 1 (transitions via U) and type 2 (direct X↔Y switching, as in the voter model); or (ii) zealots (inflexible individuals for each option). Option qualities qx and qy allow unequal-quality comparisons; define q = qx/qy. - Voter and weighted voter models: Individuals directly switch to the speaker’s opinion (with rates proportional to option qualities for the weighted variant). Asocial noise adds independent switching; zealots influence others but never change. - Cross-inhibition model: Upon disagreement in a pairwise interaction, the listener becomes undecided (U); undecided individuals can be recruited by committed peers (with rates proportional to quality). Two noise implementations are considered: type 1 (via U) and type 2 (direct switch). Zealots recruit but never change. - Chemical reaction formalism: Both models are specified as reaction schemes defining transition rates among states (including recruitment, cross-inhibition, and noise). This provides the basis for deriving ODE and master equation formulations. - Mean-field ODE analysis: For large populations, the authors derive ODEs for the proportions x, y, u, z and analyze fixed points and bifurcations. For the voter/weighted voter model with noise or zealots, closed-form stable fixed points are presented for q ≠ 1. For cross-inhibition, multiple fixed points exist, with stability changing via pitchfork and fold bifurcations as asocial behavior varies (exact expressions for asymmetric qualities are obtained symbolically but interpreted via phase portraits due to complexity). - Master equations and stationary probability distributions (SPDs): Finite-size stochastic dynamics are analyzed via birth–death processes (voter model) and 2D processes (cross-inhibition), deriving SPDs analytically using detailed balance when applicable (voter model with noise/zealots; cross-inhibition with zealots and noise type 1). For cross-inhibition with noise type 2, the SPD is computed numerically since detailed balance is not tractable. - Stochastic simulations: Gillespie’s stochastic simulation algorithm (10^4 runs per condition) estimates SPDs and transient behaviors, verifying analytical results and handling cases without closed-form SPDs. - Mean switching time (MST): The MST from a consensus for the inferior option to a stable consensus for the superior option is computed via 500 SSA runs (up to 10^6 time units), with a consensus threshold derived from long-run fluctuations. - Swarm robotics experiments: 100 Kilobot robots in a 1×1 m^2 arena interact locally via IR messages (range ~10 cm) and update opinions every 30 s. Robots display state via LED (red=X, blue=Y, green=U). Zealot robots broadcast but never update; zealots are split equally between options. Experiments start from a 50–50 split and last 60 minutes. Four conditions are reported: voter model with 20 zealots; cross-inhibition with 20 zealots; voter model with 4 zealots; cross-inhibition with 30 zealots. Tracking via overhead camera and ARK system; control code provided as open-source.

- Mean-field (infinite population): Cross-inhibition exhibits two stable fixed points (majority consensus for either option) up to a substantial asocial behavior level. With symmetric zealots, two stable consensus attractors persist until a supercritical pitchfork bifurcation when zealots exceed approximately 40% of the population; beyond that, only an indecision attractor remains. In contrast, the (weighted) voter model loses stable consensus with even minimal asocial behavior, leading to persistent indecision. - Finite-size stochastic effects: For the voter model, small populations and low noise can show noise-induced bistability with transient consensus, but these consensuses are highly unstable and vanish as noise or system size increases (transition to indecision). Zealots more strongly lock the voter model into indecision even with relatively small zealot factions. Cross-inhibition robustly breaks symmetry and sustains stable majority consensus despite relatively high noise or zealotry, for both noise type 1 (via U) and type 2 (direct switching), across tested swarm sizes. - Robotics validation (100 Kilobots): With 20 zealots (10 per option), the voter model failed to reach agreement over one hour, whereas cross-inhibition rapidly reached and maintained stable consensus. With only 4 zealots, the voter model’s agreement was largely unstable and often undecided. With 30 zealots, cross-inhibition could still form large majorities but stability degraded, with consensus vacillating during the hour. These experiments corroborate theoretical predictions under local interactions. - Best-of-n (unequal qualities): The weighted voter model achieves higher accuracy (more often converges to the best option) but yields slim, unstable majorities when qualities are similar and can fall into indecision with asocial behavior. Cross-inhibition prioritizes stability, producing larger, more stable majorities and higher expected reward rates, but can lock onto the inferior option when qualities are close. As zealotry increases, the lower-branch (inferior-option consensus) disappears via a fold bifurcation; the threshold for this transition decreases as the quality difference grows (largest parameter region for the inferior-option branch at q=1, shrinking as q>1). Overall, except at extremely high zealot proportions, cross-inhibition yields larger majorities for the superior option than the (weighted) voter model. - Broader insight: Incorporating inhibitory signaling—a biologically pervasive mechanism—dramatically alters collective dynamics, enabling symmetry breaking and resilience to noise/zealots. Moderate asocial behavior can improve cross-inhibition decisions by removing inaccurate stable states while preserving consensus.

The findings demonstrate that replacing direct opinion adoption (voter model) with inhibition leading to an undecided state (cross-inhibition) fundamentally changes collective dynamics. Cross-inhibition enables stable majority consensus even in the presence of significant asocial behavior, addressing the well-known fragility of voter-like dynamics. This helps explain the prevalence of inhibitory signaling in biological systems (e.g., neurons, honeybees) that must coordinate under noise. In engineered systems, cross-inhibition offers a minimal, implementable mechanism for robust consensus in decentralized swarms subject to malfunction, independent actions, or adversarial agents. In best-of-n scenarios, the trade-off between accuracy and stability is clarified: weighted voter maximizes accuracy but is unstable, whereas cross-inhibition secures larger, more stable majorities and maximizes reward in value-based tasks, converging to the best option when quality differences are sufficiently large. Theoretical predictions from ODEs and master equations align with stochastic simulations and physical robot experiments, underscoring the practical relevance of the model.

This work shows that cross-inhibition provides a simple, biologically inspired mechanism that yields robust consensus under asocial behavior, in stark contrast to (weighted) voter models. Analytically and empirically, the model supports stable majority decisions up to large zealot proportions (~40%), with resilience confirmed by 100-robot swarm experiments. In best-of-n problems, cross-inhibition trades accuracy for stability when options are similar but increases expected reward and produces larger majorities, converging to the superior option as quality differences grow. The results help explain the ubiquity of inhibitory signaling in natural collective decision systems and offer a practical design pattern for resilient robot swarms and social networks. Future work may extend analysis to higher-dimensional choice sets, explore time-varying cross-inhibition, and integrate additional biologically plausible mechanisms while maintaining analytical tractability.

- Mean-field analysis assumes well-mixed interactions; real systems rely on local networks, although experiments partially validate applicability. - Stable states in cross-inhibition are not absorbing; finite-size fluctuations can induce switches between consensus states. - Analytical SPDs rely on detailed balance and are unavailable for cross-inhibition with noise type 2; those cases use numerical simulations. - In best-of-n with similar qualities, cross-inhibition may lock the group into a consensus for an inferior option (reduced accuracy). - Experimental validation is limited to 100 Kilobots in a specific environment and time horizon; generalization to other settings and scales remains to be tested.