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Simplicial models of social contagion
Iacopo Iacopini, Giovanni Petri, Alain Barrat, Vito Latora
TL;DR
Social contagion involving opinion, norms, or novelties is not always adequately described by pairwise interactions because peer influence and reinforcement can operate through groups. The paper models such processes on simplicial complexes, combining link- and group-based contagion, and finds analytically and numerically that higher-order interactions induce discontinuous transitions with coexisting healthy and endemic states. These results identify a critical-mass requirement for reaching the endemic state and leave data-driven higher-order dynamical effects as an area for further study.
Problem
Pairwise interactions are often insufficient to characterize social contagion involving complex influence and reinforcement mechanisms.
Method
The paper represents social systems as simplicial complexes, models contagion through groups of different sizes, and combines simulations with mean-field analysis.
Results
Higher-order interactions induce a discontinuous transition and a bistable region where healthy and endemic states coexist, with mean-field analysis correctly predicting the steady-state dynamics.
Takeaways & Limitations
The bistable region requires a critical mass to reach the endemic state, helping explain minimal committed minorities associated with initiating social changes.
Takeaways & Limitations
The framework may need carefully chosen hyperedges to better capture higher-order dynamical effects in real data-driven models.
Abstract
from arXiv · showhide
Complex networks have been successfully used to describe the spread of diseases in populations of interacting individuals. Conversely, pairwise interactions are often not enough to characterize social contagion processes such as opinion formation or the adoption of novelties, where complex mechanisms of influence and reinforcement are at work. Here we introduce a higher-order model of social contagion in which a social system is represented by a simplicial complex and contagion can occur through interactions in groups of different sizes. Numerical simulations of the model on both empirical and synthetic simplicial complexes highlight the emergence of novel phenomena such as a discontinuous transition induced by higher-order interactions. We show analytically that the transition is discontinuous and that a bistable region appears where healthy and endemic states co-exist. Our results help explain why critical masses are required to initiate social changes and contribute to the understanding of higher-order interactions in complex systems.
Introduction
Pairwise network models do not fully capture social contagion driven by peer influence, reinforcement, and group interactions. The paper proposes simplicial contagion, combining pairwise and higher-order transmission, and shows that higher-order interactions can produce discontinuous transitions, bistability, and critical-mass effects.
- Motivation: Pairwise interactions remain the structural basis of existing threshold and epidemic-like models, even when contagion requires multiple exposures.These models are therefore limited in representing contagion occurring directly through groups.
- Modeling framework: Simplicial complexes represent social systems with both pairwise links and higher-order group interactions.A k-simplex contains k + 1 vertices, while a simplicial complex also includes all sub-simplices of each simplex.
- Modeling framework: The proposed simplicial contagion model assigns different rates to contagion through links and through group interactions.It combines stochastic simple contagion with complex contagion in which an individual is simultaneously exposed to multiple sources.
- Analysis and simulations: Numerical simulations use both empirical and synthetic simplicial complexes, while analysis derives mean-field equations for infected-node density.The analytical approach describes the evolution of infection density and predicts steady-state behavior.
- Results: Higher-order interactions change the epidemic-threshold transition from continuous to discontinuous and create a bistable region.In this region, healthy and endemic asymptotic states coexist, and the mean-field analysis predicts the transition and bistable-region location.
- Results: A critical mass is needed to reach the endemic state within the bistable region.The paper connects this result to the observed minimal size of committed minorities required to initiate social changes.
Results
The simplicial contagion model produces continuous transitions under weaker higher-order effects but discontinuous transitions, bistability, and critical-mass dependence when higher-order contagion is sufficiently strong.
- Model: The model assigns binary susceptible or infectious states to vertices of a simplicial complex and simulates contagion from an initial infectious density.The simplicial complex represents the social structure, while infectious and susceptible states correspond to binary vertex values.
- Empirical structures: For λ∆=0.8, infectious density changes with λ similarly to simple contagion, producing a continuous transition.This behavior resembles the standard SIS case without higher-order effects.
- Empirical structures: For λ∆=2, an endemic state appears below the standard epidemic threshold, accompanied by a discontinuous transition and hysteresis.Healthy and endemic states coexist in a bistable region, and the final state depends on the initial infectious density.
- Validation: The mean-field predictions correctly reproduce the steady-state dynamics, transition position and nature, and bistable-region location.The reported phenomenology remains similar across empirical structures and the synthetic random simplicial complex.
- Synthetic complex: For λ∆=2.5, the synthetic random simplicial complex likewise exhibits bistability for λc < λ < 1, with outcomes determined by initial density.For initial densities 0.01 and 0.4, the system respectively converges to zero or a positive stationary density in this region.
- Mean-field analysis: In the bistable regime, the unstable fixed point separates attraction basins and defines the critical initial density required to reach the endemic state.The mean-field approximation captures the threshold locations, discontinuous transition, and bistable-region structure for λ∆>1.
- Scope and extensions: The analytical results were derived in a mean-field approximation, while broader hypergraph classes and data-driven higher-order dynamical models remain future directions.The discussion identifies more general hypergraphs and real-data-driven models as extensions.
Methods
The study constructs simplicial complexes from face-to-face interaction data and synthetic random models, then uses them as substrates for higher-order contagion simulations.
- Four face-to-face interaction datasets represent a workplace, conference, hospital, and high school social context.
- Interactions are aggregated in five-minute windows, maximal cliques are computed, and 2- and 3-cliques are weighted by frequency.
- For the D = 2 analysis, the final structure is a clique complex formed by 1- and 2-simplices.
- Thresholded simplicial complexes are augmented through a simplicial configuration model using duplicated facet-size and pure-degree lists, producing larger complexes with matching statistical properties.
- The synthetic random simplicial-complex model first adds links with probability p1 and then 2-simplices with probability p∆, controlling average degrees ⟨k⟩ and ⟨k∆⟩.
1 Generalized degree distributions of empirical and synthetic simplicial complexes
The analysis compares generalized degree distributions in empirical complexes with those generated by random simplicial-complex models across four social contexts.
- The four panels correspond to a workplace, conference, hospital, and high school, allowing distributions to be compared across social contexts.
- Generalized degrees k1 and k2 = k∆ count incident 1-simplices and 2-simplices, respectively.
- Vertical dashed lines indicate the corresponding average generalized degrees.
- Model realizations are compared with approximated average values calculated for the synthetic complexes.
2 Hysteresis and system size
Finite-size simulations examine hysteresis in synthetic D = 2 simplicial complexes, showing bistability and hysteresis when higher-order contagion is sufficiently strong.
- The simulations use complexes with ⟨k⟩∼20 and ⟨k∆⟩∼6 across system sizes N = 500, 1000, 2000, and 4000.
- The stationary infected fraction is plotted against λ = β⟨k⟩/µ with λ∆ = 2.5, the parameter regime displaying a discontinuous transition.
- At λ∆ = 2.5, a bistable region forms where healthy and endemic states coexist, accompanied by hysteresis.
- Different initial infected densities are used to probe dependence on initial conditions in the hysteretic regime.
- For reference, λ∆ = 0.8 exhibits a continuous transition without hysteresis.
3 Cases of higher dimensions
The higher-dimensional analysis extends the contagion dynamics to D = 3 and shows that higher-order interactions can generate multiple stationary states and discontinuous transitions.
- For D = 3, the model includes spreading through simple contagion, 2-simplices, and 3-simplices, with three spreading parameters β1, β2 = β∆, and β3.
- The higher-dimensional treatment restricts the analysis because the full phase diagram depends on three parameters and is cumbersome to represent.
- In the graphical analysis, f1(ρ) = λ + λ3ρ^2 and f2(ρ) = 1/(1 −ρ), and the sign of f1 − f2 determines temporal evolution.
- The steady-state condition yields a polynomial of degree 3 and therefore has at most three real roots.
- Three crossings produce two stable fixed points separated by an unstable state, so the long-run state can depend on the initial infected density.
- For λ < 1, increasing λ3 can create a positive endemic state above a critical value λ3 = λc3, producing a discontinuous transition.
General D, with β1 = · · · = βD−1 = 0
When contagion occurs only through groups of size D + 1, the model exhibits a discontinuous transition between extinction and a positive stationary infection density. For general D, the stationary-density equation is not analytically solvable, but the phase structure follows from the sign of F_D(ρ).
- For general D, no analytical solution exists for the stationary values of infectious density.
- When β1 = · · · = βD−1 = 0, the critical point satisfies ρ− = ρ+ = 1 − 1/D, and the transition is discontinuous for any D.The transition separates the absorbing state ρ = 0 from a stationary state with nonzero density ρ+ > 0.
- The discontinuity separates vanishing spreading at low βD from finite long-time infection density at large βD.
- The sign of d_tρ(t) is determined oppositely by F_D(ρ) = 1 − λ_Dρ^(D−1)(1 − ρ) over ρ ∈ [0, 1].
- If F_D(1 − 1/D) > 0, infectious density continually decreases and can only approach extinction.
- If F_D(1 − 1/D) < 0, two roots ρ− < ρ+ arise, creating distinct outcomes based on the initial density.Initial density below ρ− decreases, whereas density above ρ− converges to the positive state ρ+.
4 Hypergraphs and simplicial complexes
The paper distinguishes general hypergraphs from simplicial complexes by requiring simplicial complexes to contain every nonempty subset of each hyperedge. This structure matches the modeled social interactions and preserves analytical tractability, while the model can be extended to hypergraphs.
- A hypergraph consists of vertices V and hyperedges E, with each hyperedge able to join any number of vertices.
- Simplicial complexes are special hypergraphs that contain every nonempty subset of each hyperedge.
- This subset-closure requirement is appropriate for the paper’s social-interaction model and helps keep it simple and analytically solvable.
- The SCM can be straightforwardly extended to model complex contagion processes on more general hypergraphs.
5 Results on empirical simplicial complexes without data augmentation
The model is evaluated on simplicial complexes built from face-to-face contact data in workplaces, conferences, and high schools. Simulations plot stationary infection density against rescaled infectivity and reveal bistability under sufficiently strong higher-order effects.
- Simplicial complexes are constructed from high-resolution face-to-face contact data recorded in workplaces, conferences, and high schools.
- Stationary infected-node fraction is plotted against rescaled infectivity λ = β⟨k⟩/µ for higher-order interaction settings.
- The λΔ = 0 curve represents the standard SIS model and excludes higher-order effects.
- For λΔ = 2.5, a bistable region appears in which healthy and endemic states coexist.