Simplicial models of social contagion

Introduction

• The framework represents social groups as simplices, enabling pairwise and higher-order interactions while assuming higher-order interactions include their lower-order interactions.The authors describe this simplicial representation as more suited than networks to capturing coexisting pairwise and higher-order interactions.Source: Introduction · pp. 5–6
• Unlike models built on pairwise links, simplicial contagion allows contagion through both links and group interactions, with different rates for the two routes.The model combines stochastic simple contagion through pairwise interactions with complex contagion in which an individual is simultaneously exposed to multiple sources.Source: Introduction · pp. 4, 6
• The authors test the model through simulations on empirical and synthetic simplicial complexes and derive mean-field equations for infected-node density.They use the analytical approach to describe the density's evolution.Source: Introduction · p. 6
• Higher-order interactions change the epidemic-threshold transition from continuous to discontinuous and create a parameter region where healthy and endemic states coexist.The mean-field analysis predicts the steady-state dynamics, transition nature, and location of the bistable region.Source: Introduction · p. 7
• In the bistable region, a critical mass is needed to reach the endemic state, echoing the reported need for a minimum committed minority to initiate social change.The introduction relates this threshold to a recently observed minimal size of committed minorities.Source: Introduction · p. 7

The contagion model.

• The SCM assigns each vertex a binary susceptible or infectious state and permits infection through simplex subfaces when their other members are infectious.Infected nodes recover with a node-independent probability µ.Source: The contagion model · p. 7, Figure 1 · p. 8

Simplicial contagion on real-world simplicial complexes.

• The empirical complexes encode face-to-face contacts in workplace, conference, hospital, and high-school settings, using simplices through dimension 2.The study uses these structures to examine contagion in distinct social contexts.Source: The contagion model · pp. 9–10
• At λ∆=2, the empirical complexes show a lower-threshold discontinuous transition with hysteresis and coexisting healthy and endemic states whose outcome depends on initial infection density.This phenomenology appears across the tested social contexts.Source: Figure 2 · p. 11, The contagion model · p. 12
• At λ∆=0.8, prevalence changes continuously with λ, similarly to the standard SIS case at λ∆=0.The simulations compare these cases with λ∆=2, which exhibits a different transition pattern.Source: The contagion model · p. 12

Simplicial contagion on synthetic simplicial complexes.

• The proposed RSC model holds mean degree fixed while tuning full-triangle incidence, creating links and triangles independently rather than promoting graph cliques.For this study, the model is restricted to dimension D=2.Source: The contagion model · p. 13

The contagion model.

• For λ∆ = 0.8, contagion resembles the simple-contagion case (λ∆ = 0), with a continuous transition at λc = 1.The simulation results are plotted in Fig. 3a for λ∆ = 0, 0.8, and 2.5.Source: The contagion model · p. 14
• Despite differences in the underlying structure, model dynamics on the random simplicial complex are very similar to those on real-world simplicial complexes.Source: The contagion model · p. 14
• At λc = 1, epidemic-free and endemic states coexist for λc < λ < 1, producing a hysteresis loop; from initial densities 0.01 and 0.4, long-time density is respectively zero and positive.The hysteresis loop’s size dependence is very small.Source: The contagion model · pp. 14–15
• Single-run dynamics show a seed-density threshold: below it infection dies out, while above it infection reaches a non-trivial steady state.The temporal dynamics illustrate the dependence on initial conditions.Source: The contagion model · p. 15

Mean field approach.

• The mean-field description assumes homogeneous mixing, independent node states, and globally averaged local connectivity represented by average generalized degree.It describes the temporal evolution of the infected-node density.Source: Mean field approach · p. 15
• For λ∆≤1, the higher-order interactions leave the epidemic threshold unchanged, and the transition remains continuous at λ = 1.In this regime, higher-order interactions increase the infected density in the endemic state.Source: Mean field approach · pp. 18–19, Figure 4 · p. 20
• In the bistable range, the unstable steady state separates the basins: initial infection below it decays to the healthy state, while infection above it reaches an endemic state.The unstable branch defines the critical initial density needed to reach the endemic state.Source: Mean field approach · pp. 18, 19, 21
• For D = 2 and λ∆ > 1, the transition is discontinuous at λc = 2√λ∆−λ∆, with a bistable region for λc < λ < 1.The phase diagram shows the threshold shift and discontinuity when λ∆ > 1; for λ∆≤1 the transition remains continuous at λ = 1.Source: Mean field approach · p. 19, Figure 4 · p. 20
• The mean-field approximation captures the thresholds and discontinuous transition for the D = 2 simplicial contagion model, consistent with simulations.Figure 3 compares the analytical mean-field solution with numerical results.Source: Figure 3 · p. 16, Mean field approach · pp. 19, 21
• Although a general solution for arbitrary parameters and D remains out of reach, specific cases with D ≥3 also show discontinuous transitions as high-order contagion increases while simple contagion is below threshold.The reported cases are D = 3 with β2 = 0 and general D > 3 with β1 = · · · = βD−1 = 0.Source: Mean field approach · p. 21

Discussion.

• The simplicial contagion model captures basic higher-order interaction mechanisms, with robust phenomenology on empirical datasets.The authors describe the framework as general and report that its phenomenology is robust in empirical-data results.Source: Discussion · p. 21
• Analytical results rely on a mean-field approximation and were quantitatively compared with nondescript random simplicial complexes akin to ER complexes.Source: Discussion · p. 21
• Future work could test SCM on complexes with heterogeneous generalized degree distributions, communities, hyperbolic geometry 58–60, or temporal evolution 52.Source: Discussion · pp. 21–22
• With carefully chosen hyperedge types, SCM maps to hypergraphs, motivating studies of complex contagion on broader hypergraph classes 61,62.Source: Discussion · p. 22
• The authors propose extending higher-order effects beyond spreading to other dynamics, such as Kuramoto-like models, to better represent real data-driven systems.Source: Discussion · p. 22

Data description and processing.

• Four face-to-face datasets cover distinct social contexts: a workplace (InVS15) 63, a conference (SFHH) 64, a hospital (LH10) 65, and a high school (Thiers13) 66, where interactions were measured.Source: Data description and processing · p. 22
• For D = 2, data were aggregated into 5-minute windows, all maximal cliques were computed, and all 2- and 3-cliques were frequency-weighted to form a complex of 1- and 2-simplices.Higher-dimensional cliques were not included in the final simplicial complex.Source: Data description and processing · p. 22
• The empirical complexes retained the 20% of simplices with the most appearances, yielding the thresholded complexes used in the study.These complexes are shown in Supplementary Figure 6, and their connectivity properties are summarized in Table 1.Source: Data description and processing · p. 22
• To reduce finite-size effects, each thresholded complex’s facet-size and node pure-degree lists were duplicated five times and input to the simplicial configuration model.The resulting complexes preserve the input’s statistical properties while having significantly larger size.Source: Data description and processing · pp. 22–23
• The random simplicial-complex model creates D = 2 complexes on N vertices by adding pairwise links with probability p1 and 2-simplices with probability p∆.The initial expected degree is (N − 1)p1, while the average number of incident 2-simplices is ⟨k∆⟩ = (N − 1)(N − 2)p∆/2.Source: Data description and processing · p. 23
• For p1, p∆ ≪ 1, the expected average degree combines link and simplex contributions as ⟨k⟩ ≈ (N − 1)p1 + 2⟨k∆⟩(1 − p1).A node’s degree rises by an average of 2(1 − p1) for each incident 2-simplex.Source: Data description and processing · pp. 23–24

• The SocioPatterns data sets were downloaded from sociopatterns.org/datasets.Source: Data availability · p. 24

• Empirical distributions are shown for workplace, conference, hospital, and high-school contexts, with k1 and k2 denoting incident 1-simplices and 2-simplices.Source: Figure 1 · p. 34
• For model-generated random simplicial complexes, the distributions show k1 and k2, and compare realization-based average degrees with approximated values.Source: Figure 2 · p. 35, §1 · p. 35

2 Hysteresis and system size

• Finite-size effects on hysteresis are explored for order-D = 2 synthetic random simplicial complexes with N = 500, 1000, 2000, and 4000.Figure 3 plots average stationary infected fraction against rescaled infectivity λ = β⟨k⟩/µ; Figure 4 compares initial infected densities ρ0 = 0.4 and ρ0 = 0.01.Source: Figure 3 · p. 36, §2 · p. 36, Figure 4 · p. 37
• At λ∆ = 2.5, the study observes a discontinuous transition that forms a bistable region, where healthy and endemic states co-exist and hysteresis appears; the cited initial infected density is ρ0 = 0.01.Source: §2 · p. 36
• At λ∆ = 0.8, the transition is continuous and shows no hysteresis.This case is shown for reference using black squares.Source: §2 · p. 36

Case D = 3

• For maximum simplex dimension D = 3, the model has spreading parameters β1, β2, and β3, with evolution terms proportional to ρ, ρ², and ρ³, respectively.The terms are weighted by the corresponding mean simplex counts and by the susceptible fraction (1 − ρ).Source: Case D = 3 · p. 38
• For tractability, the analysis sets βΔ = 0, restricting contagion to simple contagion or size-4 cliques with three infectious nodes.This simplification avoids explicit cubic-root solutions and allows a graphical analysis.Source: Case D = 3 · p. 38
• After rescaling time, the sign of the density’s evolution is determined by f1 − f2, where f1(ρ) = λ + λ3ρ² and f2(ρ) = 1/(1 − ρ); their equality has at most three real roots.The stationary states are identified graphically from intersections of the functions over 0 ≤ ρ < 1.Source: Case D = 3 · pp. 38–39
• For λ > 1, the system always reaches a positive-density stationary state; with three crossings, two stable states are separated by an unstable state, making the outcome initial-condition-dependent.With one crossing, every positive initial density converges to that stationary state.Source: Case D = 3 · pp. 39–40
• If the initial density ρ(0) is below ρ−, dρ/dt is negative and ρ decreases; if it is above ρ−, the system converges to ρ+, which is positive.Source: Case D = 3 · p. 40
• For λ < 1, the transition at λ3 = λc3 is discontinuous, with f1 and f2 tangent at a positive density before two intersections appear above threshold.Figure 5 illustrates the tangency and two-intersection cases for λ = 0.5 and λ2 = 0.Source: Case D = 3 · p. 40, Figure 5 · p. 41

General D, with β1 = · · · = βD−1 = 0

• For general D, stationary infection density has no analytical solution; the analysis instead uses the sign of F_D(ρ)=1−λ_Dρ^(D−1)(1−ρ) to determine dynamics on 0≤ρ≤1.Source: General D, with β1 = · · · = βD−1 = 0 · p. 42
• With β1 = · · · = βD−1 = 0, spreading through cliques of size D+1 yields a discontinuous transition for any D, from vanishing spread to positive stationary density.Source: General D, with β1 = · · · = βD−1 = 0 · pp. 42–43
• If F_D(1 − 1/D) > 0, dρ/dt is negative for every ρ > 0, so infection density can only decrease.Source: General D, with β1 = · · · = βD−1 = 0 · p. 43
• If F_D(1 − 1/D) < 0, roots ρ− and ρ+ divide initial conditions: ρ(0) < ρ− decays, while ρ(0) > ρ− converges to ρ+ > 0.Source: General D, with β1 = · · · = βD−1 = 0 · p. 43

4 Hypergraphs and simplicial complexes

• A hypergraph generalizes a graph by allowing hyperedges to join any number of vertices; formally, H = (V, E), with E a subset of the power set of V.Source: §4 · p. 44
• A simplicial complex is a hypergraph in which every nonempty subset of each hyperedge is also an edge.Source: §4 · p. 44
• The subset requirement is considered appropriate for the social interactions studied and keeps the model simple and analytically tractable.Source: §4 · p. 44
• The simplicial contagion model can also be extended to complex contagion processes on general hypergraphs.Source: §4 · p. 44

5 Results on empirical simplicial complexes without data augmentation

• At λ∆ = 2.5, simulations on workplace, conference, and high-school contact simplicial complexes show coexisting healthy and endemic states.The figure also plots λ∆ = 0.8 and the standard SIS model at λ∆ = 0.Source: Figure 6 · p. 45
• The figure plots stationary infected-node fractions against rescaled infectivity λ = β⟨k⟩/µ for order D = 2, using face-to-face contact data without augmentation.The data come from a workplace, conference, and high school; curves are shown for λ∆ = 0.8, λ∆ = 2.5, and standard SIS at λ∆ = 0.Source: Figure 6 · p. 45