Multistability, intermittency, and hybrid transitions in social contagion models on hypergraphs

The study investigates how group interactions, modeled via critical-mass dynamics, shape social contagion when many groups coexist and intersect. Motivated by empirical evidence that tipping points in social conventions vary widely (about 10–40% in several settings, as low as 0.3% in linguistic norm changes), the authors ask: (1) How does a collection of groups behave? (2) How do intersections among groups affect global dynamics? (3) Can smaller groups have higher critical-mass thresholds than the entire population? Building on prior hypergraph-based social contagion models that exhibited discontinuous transitions, bistability, and hysteresis, the paper explores richer dynamical phenomena—multistability, intermittency, and the nature (hybrid vs. continuous) of transitions—using analytical approximations, simulations on real and synthetic hypergraphs, and exact analysis on a symmetric structure.

The paper situates itself within research on critical-mass and threshold dynamics in social systems, including theoretical, observational, and experimental works showing tipping points for social convention change and the impact of committed minorities. Prior studies report thresholds commonly between 10–40%, with much lower values observed in linguistic norm changes. In higher-order network modeling, simplicial and hypergraph contagion models have exhibited discontinuous transitions, bistability, and hysteresis, contrasting with standard graph-based SIS dynamics. Recent advances include spectral thresholds for extinction on hypergraphs, mean-field analyses, and evidence for hybrid transitions in related higher-order models. The work extends these strands by connecting higher-order contagion with community structure and by characterizing hybrid transitions via susceptibility scaling and finite-size analyses.

Model definition: Society is modeled as a hypergraph H = (V, E), with nodes as individuals and hyperedges as group interactions of arbitrary size. Each node i has a binary state Y_i ∈ {0,1}. Active nodes deactivate via a Poisson process with rate δ_i. For each hyperedge e_j, T_j = ∑_{k∈e_j} Y_k counts active nodes. When T_j ≥ Θ_j, a Poisson process with rate λ_j triggers simultaneous activation of all inactive nodes in e_j after an exponentially distributed time. For pairwise hyperedges (|e_j|=2), Poisson processes are directed to recover standard SIS as a limiting case. Thresholds are set as Θ_j = ⎡Θ'|e_j|⎤ with global Θ' ∈ [0,1]. Spreading rates depend on hyperedge size, λ_j = λ f(|e_j|); simulations use f(ℓ) = log2(ℓ), ensuring pairwise rate equals 1. The exact dynamics form a continuous-time Markov chain on {0,1}^N. First-order (individual-based) approximation: Assuming independence among node states, the authors derive ODEs for y_i(t) = E[Y_i], where activation terms use the Poisson-binomial distribution of active neighbors in each hyperedge (stabilized via discrete Fourier transform). The ODE system is integrated with an adaptive Runge-Kutta-Fehlberg (4,5) method (GSL), with specified absolute/relative tolerances. Simulation methods: Continuous-time Monte Carlo using the Gillespie algorithm simulates activation and deactivation processes, dynamically creating/removing hyperedge activation processes based on threshold attainment. To characterize steady behavior in finite systems with an absorbing state, a quasi-stationary (QS) method maintains a reservoir of previously visited active states; upon absorption, the process is reset to a stored active state, allowing estimation of the state distribution P(n), order parameter ρ = ⟨n⟩/N, and susceptibility χ = (⟨n^2⟩ − ⟨n⟩^2)/⟨n⟩. An adaptive QS sampling procedure reduces computational cost by monitoring convergence of χ. Real-data experiment: The blues reviews hypergraph (Amazon reviewers grouped by genre-month; N=1106 nodes, 694 hyperedges, max |e|=83) is analyzed. Pairwise interactions alone yield a giant component of 24 nodes, while including hyperedges produces a giant component of all 1106 nodes. A hypergraph configuration-model rewiring (vertex-labeled; 10^7 rewirings) generates randomized counterparts. Synthetic hypergraphs with communities: A generative model builds two-community hypergraphs with specified within-community hyperedges (densities can differ), hyperedge sizes from an exponential distribution (bounded), and m_out bridging hyperedges across communities, controlling modularity. This isolates the effect of bridges on dynamics. Exact analysis on the hyperblob: On a symmetric structure consisting of a random regular graph (degree k) plus one global hyperedge including all nodes, exact master equations for the number of active nodes n are derived. Under QS constraints (forbidding direct transition to the absorbing state), the stationary distribution π_n is obtained via linear recursions with O(N) complexity. Finite-size scaling of order parameter and susceptibility identifies second-order and hybrid transition features. Parameter conventions: Unless otherwise noted, δ=1, Θ'=0.5, and f(ℓ)=log2(ℓ). Example λ values are varied to explore transitions and intermittency.

• Real hypergraph (blues reviews) exhibits multistability and intermittency:
  • QS Monte Carlo on the real hypergraph shows multiple stable branches with bimodal state distributions across a range of λ; the rewired hypergraph exhibits a single discontinuous transition without multistability, indicating the role of correlations/community structure.
  • Bimodality types: (i) near-absorbing-state bimodality akin to SIS (e.g., λ=0.036 shows modes near absorbing state and around n≈250); (ii) intermittency-driven bimodality where the system oscillates between low and high activity regimes (e.g., λ≈0.0816, 0.0848), producing susceptibility peaks.
  • Quantitative context: N=1106 nodes, 694 hyperedges, max |e|=83; pairwise-only giant component size 24 vs. full-hypergraph giant component of 1106.
  • Susceptibility diverges or exhibits pronounced peaks at transitions between branches; multistability and intermittency depend on initial conditions and microstate localization.
• Localization and community structure:
  • Node-level activation probabilities reveal that different branches localize on distinct communities or subsets; e.g., an identified community C1 (~600 nodes) sustains activity in a lower (localized) branch, while other branches involve bridges and/or full-network activation.
  • Intermittency corresponds to activation/deactivation of a sparser periphery periodically seeded by a denser core.
• ODE (first-order) approximation reproduces multiple branches:
  • Numerical integration of the individual-based ODEs (Eq. 5) from microstates sampled in simulations recovers five steady-state branches, aligning with peaks of QS state distributions. The approximation neglects correlations/fluctuations but captures branch structure and supports that multistability is intrinsic, not a simulation artifact.
• Synthetic community hypergraphs clarify the role of bridges:
  • With few bridges (m_out=200), multistability occurs (different initial conditions lead to distinct active branches) without intermittency.
  • With more bridges (m_out=400 or 600), multistability diminishes and intermittency emerges, producing bimodal distributions and susceptibility peaks. As m_out increases, the susceptibility peak shifts to lower λ.
  • Varying average hyperedge size μ and threshold Θ' shows: larger μ favors intermittency; higher Θ' favors multistability.
• Hybrid phase transitions on the hyperblob:
  • Exact QS analysis reveals a second-order transition at low λ' followed by a hybrid transition between lower and upper branches, characterized by both a discontinuous jump in the order parameter and scaling of susceptibility.
  • Finite-size scaling shows variance measures ⟨(ρ^2)⟩ − ⟨ρ⟩^2 and ⟨(χ^2)⟩ − ⟨χ⟩^2 tend to zero with system size, indicating a discontinuity in the thermodynamic limit. Estimated scaling exponent μ ≈ 0.437 (<1) is consistent with hybrid transitions, and susceptibility peaks diverge with N.
• Initial condition sensitivity:
  • Steady states depend strongly on microstate properties of initial seeds (which community is seeded), not solely on the total prevalence, leading to different branches for identical macroscopic initial conditions. Overall, the study shows that higher-order interactions plus community structure can induce multistability, intermittency, multiple transitions, and hybrid critical phenomena in social contagion.

The findings address the central questions by showing that a collection of interacting groups (modeled as hyperedges) can produce multiple coexisting macrostates due to thresholded group activation and the structure of group overlaps. Intersections among groups (bridges) modulate whether the system exhibits localized multistable states (few bridges) or system-wide intermittency (many bridges). Smaller, denser communities can sustain activity locally, while sparser regions activate intermittently through cascades triggered by intersecting hyperedges. Transitions between activity levels are often hybrid, combining discontinuities with scaling, a hallmark of higher-order interactions not seen in standard SIS on graphs. These insights reconcile disparate empirical tipping-point estimates: high thresholds within single groups can coexist with much lower population-level tipping due to cascading across intersecting groups. The pronounced sensitivity to initial microstates implies that intervention strategies may require targeted seeding in specific communities rather than uniform increases in initial prevalence. The interplay between higher-order interactions and modular organization thus fundamentally shapes macroscopic contagion outcomes, including susceptibility peaks, localization, and temporal intermittency.

The paper demonstrates that social contagion on hypergraphs exhibits rich dynamics beyond discontinuous transitions and hysteresis, notably multistability, intermittency, and hybrid phase transitions. Empirical (blues reviews) and synthetic analyses link these effects to community structure and the presence of bridging hyperedges: scarce bridges support multiple stable localized branches; abundant bridges promote intermittency via bimodal state distributions. Exact results on a symmetric hypergraph (hyperblob) corroborate the hybrid nature of transitions. Future research directions include deriving necessary and sufficient structural conditions for multistability/intermittency, extending the model to incorporate mechanisms like backlash or cultural opposition, exploring impacts on other dynamical processes (e.g., synchronization, diffusion, opinion dynamics), and designing experiments to detect intermittency in small populations or online platforms. Broader applications include reinterpreting tipping phenomena where group-level thresholds and higher-order interactions are central.

• Finite-size analysis is generally infeasible on single real-world hypergraphs, limiting precise classification of transition types; QS-based proxies and combined ODE/simulation evidence are used instead.
• Measuring localization via standard graph tools (e.g., leading eigenvector of an adjacency matrix) is nontrivial for hypergraphs; a formal spectral framework capturing higher-order localization remains open.
• The model assumes that group activation increases the likelihood of further activation (no backlash or cultural opposition); such countervailing mechanisms are not included and could alter dynamics.
• Sensitivity to initial conditions complicates exhaustive branch sampling; discovering all branches may be computationally demanding and uncertain.
• Structural constraints determining stability of the absorbing state and critical points in general hypergraphs are not fully characterized; further spectral and numerical studies are needed.