Group interactions modulate critical mass dynamics in social convention

The theory of critical mass posits that stable social conventions can be overturned when a committed minority reaches a critical size. Prior work across contexts has reported a broad range of tipping fractions: qualitative analyses suggest 30–40% may be needed in gender-related corporate norms, controlled experiments on social coordination observed about 25%, while linguistic norm changes have been attributed to minorities as small as 0.3%. To understand these tipping points, models inspired by statistical physics have been developed, notably the naming game (NG), which shows how shared conventions can emerge without central coordination. In the NG with committed minorities, a critical mass of about 10% can flip norms, with subsequent generalizations yielding thresholds between 10% and 40%, and even vanishing critical masses in some extensions. Despite these results, less attention has been paid to how the initial minority forms and the role of group interactions. Real social systems frequently involve interactions in groups rather than pairs, and higher-order (non-pairwise) interactions can significantly alter dynamics, producing abrupt transitions, bi-stability, and critical mass phenomena. Motivated by this, the study investigates how group interactions shape critical mass dynamics.

Empirical and theoretical studies report varied critical mass sizes: 30% or higher in some organizational settings, ~25% in controlled coordination experiments, and ~0.3% for linguistic norm changes. The NG framework has been central to theory, with tipping points around 10% in basic models and 10–40% depending on commitment strength, and with modifications allowing near-zero thresholds. Mechanisms hypothesized for minority growth include recruitment by an initial core, mobilization via high-resource individuals, homophily and local coalition formation, and engagement of peripheral non-committed individuals. Beyond pairwise interactions, higher-order models (majority rule on hypergraphs, evolutionary and opinion dynamics on simplicial complexes and hypergraphs) reveal rich behavior such as abrupt transitions, bi-stability, and critical mass effects. These works suggest that capturing group interactions is essential for explaining the diversity of observed tipping points.

The study generalizes the naming game to include group (higher-order) interactions represented by hyperlinks in a hypergraph. At each time step, a hyperlink (group) e is selected at random and a random member acts as speaker; all others are hearers. The speaker utters a name (convention) A chosen at random from its vocabulary. Agreement rules for group interactions are defined: (i) Unanimity (intersection) rule: agreement is possible only if all hearers already have A in their vocabularies; (ii) Union rule: agreement is possible if at least one hearer has A. Intermediate thresholds could also be defined. Social influence is modulated by a parameter β in [0,1]. If agreement is possible, with probability β all nodes in the group adopt A exclusively (erasing other names); with probability 1−β no convergence occurs but any node lacking A adds it. If agreement is not possible, hearers add A to their vocabularies. A fraction p of agents are committed to A (Ac) and never change or update their vocabulary. The state variables include n_x(t) for the fraction supporting x at time t and stationary values n_x. Simulations are run on empirical hypergraphs constructed from six datasets (workplace InVS15; primary school LyonSchool; conference SFHH; high school Thiers13; Email-EU; Congress-bills), capturing group size distributions up to 24. Aggregation procedures include 15-minute windows and maximal cliques (for SocioPatterns datasets), retaining cliques appearing more than once; Email-EU and Congress-bills are already in simplicial form. Agent-based stochastic simulations: at each step, choose a group, select a speaker, update according to rules, iterating to absorbing or steady states; steady-state densities are averaged over samples from late steps; results reported as medians and percentiles over multiple runs with random committed seeds. Synthetic analyses use (k−1)-uniform hypergraphs (all groups of size k) to isolate group-size effects. A mean-field (MF) approach for 3-node interactions (triads) derives ODEs for densities n_A, n_B, n_AB under unanimity and union rules, integrating numerically to map stationary outcomes as functions of β and p; pairwise benchmark equations are provided in Supplementary Note 1.

• Even extremely small committed minorities can overturn norms when social influence is imperfect. In Thiers13 (N=327), a single committed agent (≈0.3%) triggers takeover under unanimity rule with β=0.336; with β=1 the minority fails to spread, highlighting β’s central role.
• Two dynamical regimes emerge across empirical structures depending on β: (i) Low β (e.g., β=0.28): stable coexistence with substantial prevalence of A and many agents holding both names (A,B); (ii) Higher β (e.g., β=0.41): abrupt transition where the committed minority converts the whole population to A after a transient.
• Stationary-state phase behavior versus β: Without committed agents (p=0), low β yields symmetric coexistence; increasing β favors the initial majority (B), leading to its dominance beyond a structure-dependent threshold, with an intermediate regime of coexistence but B-dominance. With a committed minority (p=0.03), three regimes appear as β increases: coexistence with A already outperforming B at low β; complete takeover by A within a finite β-interval Δβ; and, at large β, failure of the minority to overturn the majority (except committed agents and neighbors tending to have (A,B)).
• Group size distributions matter: Data sets with larger groups (Email-EU, Congress-bills; group sizes up to 24) exhibit wider Δβ ranges where the minority prevails. Reshuffling preserving group-size distributions but destroying correlations retains these differences, implicating group sizes as key determinants.
• Non-monotonic group-size effect in uniform hypergraphs (unanimity rule): As k increases from 2 to 5, the takeover range Δβ shrinks; for larger k (e.g., k=10, 40), Δβ grows again. Thus, both small and large groups favor minority success relative to intermediate sizes. Larger p broadens Δβ.
• Mechanistic interpretation: For unanimity, larger k reduces the chance of agreement on a minority name but increases conversion impact per agreement, yielding non-monotonic β_max. For union, β_min increases monotonically with k, while β_max remains non-monotonic.
• Even when takeover fails at large β, group interactions sustain a large fraction of (A,B) agents, which increases with k at fixed β (e.g., β=0.71), showing propagation of the minority norm’s knowledge beyond p.
• Mean-field (triads) phase diagrams confirm three regimes with unanimity: low-β coexistence with n_A >> p; intermediate β with full takeover by A; high β where the majority prevails. Under the union rule, for p=0.08 the third regime vanishes and Δβ extends up to β=1. MF predictions match stochastic simulations.

The findings address how minorities can overturn established social conventions by showing that higher-order group interactions and imperfect social influence fundamentally reshape critical mass dynamics. Imperfect influence (β<1) allows minorities to seed widespread knowledge of the alternative norm, sustaining coexistence or enabling abrupt takeovers, sometimes from extremely small seeds. Group size modulates these processes in a non-monotonic fashion: both very small and very large groups facilitate minority success, while intermediate sizes can hinder it. This reconciles disparate empirical critical mass estimates by revealing that tipping thresholds depend on both social influence strength and interaction group sizes. The results underscore the importance of modelling higher-order interactions to capture real social dynamics and suggest avenues for interventions, such as leveraging group structures or tuning influence conditions, with relevance to accelerating desired norm changes or countering undesired ones.

This work extends the naming game to higher-order group interactions with imperfect social influence and committed minorities, validated on empirical hypergraphs and synthetic structures. It shows that critical mass can be extremely small under limited susceptibility (low β), that takeover regimes depend sensitively and non-monotonically on group size, and that group interactions can sustain alternative norms even without full takeover. Analytical mean-field results corroborate the simulation findings and clarify the roles of β, p, and agreement rules. Future research directions include: testing other opinion/cooperation models; designing controlled experiments to vary p and emulate β; incorporating community structure and temporal higher-order interactions; allowing β to depend on agent or group properties; modeling co-evolution of interaction structure and norms; and studying competing committed minorities.

The results are derived within a single modeling framework (the naming game) and thus may not generalize to all opinion dynamics. While more realistic than pairwise models, the interaction patterns still simplify real-world complexity. Additional limitations include absence of explicit community structure and temporal dynamics in the main analyses, homogeneous β across agents and groups, fixed group selection probabilities, and single-issue commitment without competing committed groups. The authors suggest extending to other models, conducting controlled experiments to probe group effects and β, incorporating community and temporal higher-order networks, allowing β and group activity to vary with properties, enabling co-evolving structures, and considering multiple competing committed minorities.