Simplicial Contagion Models Overview

• Simplicial Contagion Models are contagion processes defined on simplicial complexes that incorporate both pairwise and group-based infection channels.
• They yield nonlinear infection terms with critical-mass effects, leading to discontinuous transitions and bistability not seen in standard pairwise models.
• SCM methodologies—from homogeneous mean-field to pair-based approximations—offer insights into social spreading phenomena, influence maximization, and threshold dynamics.

Simplicial Contagion Models (SCM) are contagion processes defined on simplicial complexes and related higher-order network representations, in which infection can propagate not only along edges but also through simplices that encode genuinely group-based interactions. In the canonical SIS formulation, a susceptible node can be infected pairwise by one infected neighbor or, for example, through a 2-simplex when the other two nodes are infected, so SCM embed reinforcement directly in the transmission mechanism rather than in an exogenous threshold rule. This construction yields nonlinear infection terms, critical-mass effects, discontinuous transitions, hysteresis, and bistability that are absent from standard pairwise SIS mean-field dynamics, and it has subsequently been extended to pair-based closures, clique-level microscopic descriptions, signed and temporal higher-order networks, multigroup systems, coupled contagions, and arbitrary-order hypergraph interactions (Iacopini et al., 2018).

1. Formal definition and canonical ingredients

A simplicial complex is a collection of simplices satisfying downward closure: if a set of nodes forms a $d$-simplex, then all of its lower-dimensional faces are also present. Nodes are $0$-simplices, edges are $1$-simplices, filled triangles are $2$-simplices, tetrahedra are $3$-simplices, and so on. In SCM, this inclusion property is not a technicality; it is the structural assumption that distinguishes simplicial complexes from general hypergraphs and permits pairwise and higher-order contagion channels to coexist on nested interaction supports (Iacopini et al., 2018).

The standard SCM state space is binary. Each node is susceptible or infected, and recovery is SIS-like with rate $\mu$. For the order-$D$ mean-field SCM introduced in the foundational formulation, the infection parameters are $B=\{\beta_1,\beta_2,\dots,\beta_D\}$, where $\beta_\omega$ is the infection probability per unit time for a susceptible node participating in an $\omega$-simplex whose other $0$0 nodes are infectious. The resulting homogeneous mean-field dynamics for the prevalence $0$1 is

$0$2

Here $0$3 is the generalized degree of node $0$4, namely the number of incident $0$5-simplices, and $0$6 is its network average (Iacopini et al., 2018).

For the frequently studied $0$7 case, the two infection channels are pairwise transmission along edges and triadic reinforcement on 2-simplices. In the pair-based treatment of the SIS SCM, the microscopic reactions are

$0$8

with pairwise infection rate $0$9, triangle infection rate $1$0, and recovery rate $1$1. In that setting, structural parameters are made explicit through the average degree $1$2, the global clustering coefficient $1$3, the fraction $1$4 of triangles that are actual 2-simplices, and the average number of incident 2-simplices per node,

$1$5

This parameterization makes clear that SCM depend not only on interaction order but also on the density of genuine simplices relative to ordinary closed triplets (Malizia et al., 2023).

2. Mean-field equations and the canonical $1$6 SCM

Specializing the general SCM to triangles yields the mean-field equation

$1$7

With the rescaled parameters

$1$8

the nonzero steady states are

$1$9

The stationary equation is cubic, with the absorbing state $2$0 and up to two additional fixed points $2$1 and $2$2. When $2$3, the transition is continuous at $2$4. When $2$5, the saddle-node threshold

$2$6

satisfies $2$7, and the interval $2$8 becomes bistable: $2$9 and $3$0 are stable, while $3$1 is unstable and acts as a critical-mass separatrix (Iacopini et al., 2018).

Equivalent formulations appear in later SIS SCM work. In homogeneous notation,

$3$2

with $3$3 the normalized pairwise infectivity and $3$4 the normalized triangular infectivity. The same cubic structure implies that pairwise interactions dominate the earliest stage of invasion, because the term proportional to $3$5 is negligible when prevalence is small, whereas higher-order simplices dominate later once enough infected nodes are present to activate many group interactions. In that formulation, the bistable regime extends down to $3$6 whenever $3$7, so sufficiently strong higher-order transmission can maintain contagion even after pairwise transmission is removed, provided the infected fraction is already above the unstable branch (Li et al., 2021).

The generalization to higher simplex order preserves the same logic but increases the polynomial degree. For $3$8, the mean-field SCM adds terms of the form $3$9, and in the special case where only the top order $\mu$0 is active, the critical condition for a discontinuous transition is

$\mu$1

with the two nontrivial fixed points coalescing at $\mu$2 at onset. This shows that discontinuity is not restricted to triangles; it is a generic feature of sufficiently strong higher-order reinforcement (Iacopini et al., 2018).

3. Correlations, closures, and microscopic approximations beyond individual mean field

The individual-based mean-field description is analytically convenient but neglects dynamical correlations among neighboring nodes and among subsets inside simplices. The pair-based mean-field approximation (PBMF) introduced for simplicial SIS addresses this limitation by tracking node fractions $\mu$3, $\mu$4, pair fractions $\mu$5, $\mu$6, $\mu$7, the 2-simplex variable $\mu$8, and additional four-node motifs needed to represent infection through triangles. Triplets are closed by a $\mu$9-weighted interpolation between open-wedge and closed-triangle approximations,

$D$0

with

$D$1

Before closure, the pair system is exact and preserves conservation laws such as $D$2; after closure, the Kirkwood-type terms need not preserve higher-order conservation identities. The paper therefore integrates the full PBMF system to avoid negative states and limit error propagation (Malizia et al., 2023).

PBMF changes the epidemic-onset prediction qualitatively. Introducing the fast variables

$D$3

the early-time growth rate can be written as

$D$4

The corresponding PBMF threshold is

$D$5

which decreases as $D$6 increases. This contrasts with the IBMF prediction $D$7, independent of higher-order strength. Numerically, PBMF predicts both epidemic thresholds decreasing with $D$8, gives a more accurate bistable interval, better captures the transition type, and more closely matches the temporal evolution and endemic prevalence observed in simulations on random simplicial complexes (Malizia et al., 2023).

A different route to correlations is the network clique cover approximation for discrete-time SIS on simplicial complexes. That framework introduces a microscopic model with correlated concurrent infection channels inside each clique and an edge-disjoint edge clique cover (EECC) to prevent over-counting when cliques overlap. Linearization near threshold yields a generalized eigenvalue problem,

$D$9

with epidemic onset defined by

$B=\{\beta_1,\beta_2,\dots,\beta_D\}$0

Because the higher-order couplings enter the coefficients of $B=\{\beta_1,\beta_2,\dots,\beta_D\}$1 and $B=\{\beta_1,\beta_2,\dots,\beta_D\}$2, the threshold depends explicitly on the higher-order infection probabilities $B=\{\beta_1,\beta_2,\dots,\beta_D\}$3. By contrast, any mean-field approximation that treats node states as independent collapses to $B=\{\beta_1,\beta_2,\dots,\beta_D\}$4, independent of higher-order couplings. The clique-based approach therefore extends the threshold dependence on higher-order interactions to structured populations rather than only homogeneous mean-field reductions (Burgio et al., 2021).

For fully connected simplicial complexes, exact stochastic analysis is available. In that setting, the Kolmogorov forward equations for the number of infected nodes admit a hydrodynamic limit under the scalings $B=\{\beta_1,\beta_2,\dots,\beta_D\}$5, $B=\{\beta_1,\beta_2,\dots,\beta_D\}$6, and $B=\{\beta_1,\beta_2,\dots,\beta_D\}$7. The resulting exact mean-field equation for prevalence $B=\{\beta_1,\beta_2,\dots,\beta_D\}$8 takes the canonical form

$B=\{\beta_1,\beta_2,\dots,\beta_D\}$9

with $\beta_\omega$0. This result shows that the familiar SCM polynomial right-hand side is not only heuristic: for fully connected simplicial complexes it is the exact macroscopic limit of the underlying Markov chain (Kiss et al., 2023).

4. Phase transitions, critical mass, bistability, and multistability

The canonical SCM phase diagram is organized by the coexistence of absorbing and endemic attractors. In the triangle model, the unstable branch $\beta_\omega$1 is the critical mass: trajectories with $\beta_\omega$2 decay to the disease-free state, while trajectories with $\beta_\omega$3 converge to the endemic branch $\beta_\omega$4. This is the mathematical origin of hysteresis and seed-size dependence in SCM, and it is the mechanism used to interpret tipping-point-like social adoption in the original model (Iacopini et al., 2018).

Later analyses clarify which polynomial structures are capable of generating which bifurcations. For a single homogeneous population with pairwise and three-body interactions, the reduced mean-field equation

$\beta_\omega$5

admits either a forward transcritical transition or a backward transcritical with bistability, depending on whether $\beta_\omega$6 or $\beta_\omega$7. Adding four-body interactions produces the quartic

$\beta_\omega$8

which supports a third regime: if $\beta_\omega$9 and $\omega$0, the branch develops two folds and the system can sustain two distinct stable endemic equilibria. The same work shows that a symmetric two-population model with pairwise and three-body interactions can generate multistability through symmetry breaking and pitchfork bifurcations, even though a single population with only three-body terms cannot (Kiss et al., 2024).

Exact higher-order SIS on fully connected simplicial complexes confirms that multistability is intrinsic to the infection mechanism rather than to structural heterogeneity. For pairwise plus triadic contagion, bistability occurs when the triadic strength is sufficiently large; in the exact parametrization this requires $\omega$1 for an interval of $\omega$2 with simultaneous stability of $\omega$3 and a positive endemic state. With tetrahedral infection added, the quartic fixed-point equation can undergo cusp bifurcations; the condition $\omega$4 marks the regime in which three positive endemic equilibria can exist, two of them stable. This establishes that two distinct stable endemic steady states can arise even on a fully connected simplicial complex (Kiss et al., 2023).

A recurrent misconception is that higher-order interactions merely renormalize pairwise spreading rates. Several results argue against that reading. In the canonical mean-field SCM, $\omega$5 changes the number and stability of fixed points rather than simply shifting a linear threshold. In pair-based and clique-based approximations, higher-order terms also modify the invasion threshold itself through local correlations. At the same time, temporal and linear-stability analyses show that pairwise channels still dominate the infinitesimal invasion regime in some formulations. The literature therefore supports a more precise statement: higher-order contagion generically reorganizes the nonlinear phase portrait, while its effect on the onset threshold depends on the approximation level and on whether the analysis resolves the relevant correlations (Malizia et al., 2023).

5. Signed, temporal, multigroup, coupled, and arbitrary-order generalizations

Several extensions modify the basic SCM by changing either the interaction substrate or the contagion channels. One major line introduces signed simplicial complexes. In the trust/distrust model, pairwise infection occurs along trusted edges at rate $\omega$6, pairwise distrust can induce recovery at rate $\omega$7, and triadic infection or recovery depends on the sign pattern of the triangle. The mean-field prevalence equation is

$\omega$8

with coefficients determined by trusted and distrusted edge counts and by the counts $\omega$9 of incident triangles with exactly $0$00 positive edges. Linearization gives the threshold

$0$01

so distrust raises the epidemic threshold. The transition is discontinuous when $0$02 and continuous when $0$03, while biased balanced triads can either promote or impede contagion depending on whether they are predominantly fully trusted or of the one-positive type (Kemmeter et al., 2023).

A related signed model with emotional group interactions restricts higher-order contagion to balanced triangles containing exactly two infected nodes and one susceptible, provided the susceptible has at least one positive edge to the infected pair. In mean field,

$0$04

where $0$05 and $0$06 with

$0$07

Because the pairwise term decreases linearly with the negative-edge fraction $0$08 but the group term decreases through the cubic polynomial $0$09, the influence of group interactions is more nonlinear and more sensitive to low levels of distrust. As $0$10 increases, the model crosses from discontinuous to continuous transitions by driving $0$11 below $0$12 (Ma et al., 2024).

Temporality changes the activation of simplices themselves. In the microscopic Markov-chain approach for temporal SCM,

$0$13

For homogeneous random simplicial complexes, uncorrelated temporality suppresses critical-mass effects in finite-size systems: the forward onset is then controlled entirely by $0$14, with $0$15 independent of $0$16, whereas temporal persistence restores higher-order effects and lowers the $0$17 needed for endemicity. In heterogeneous scale-free simplicial complexes, the suppressive effect of temporality is weaker, and seeding on hubs markedly lowers the critical higher-order infectivity (Chowdhary et al., 2021).

The multigroup simplicial SIS model replaces scalar prevalence by a vector $0$18, with pairwise contact matrix $0$19, triadic incidence matrices $0$20, recovery matrix $0$21, and dynamics

$0$22

This formulation yields rigorous sufficient conditions for three domains. The disease-free equilibrium is globally exponentially stable if

$0$23

If $0$24 but higher-order incidence is large enough, the system is bistable; if $0$25, an endemic equilibrium exists, and for sufficiently small $0$26 it is unique and globally exponentially stable off the origin (Cisneros-Velarde et al., 2020).

Coupled contagions reveal another route by which SCM phenomenology can appear. In the simplicially driven simple contagion model, process $0$27 is simplicial and process $0$28 is pairwise, with uni-directional coupling $0$29. The driver obeys

$0$30

while the effective infectivity of the driven simple contagion is

$0$31

Because $0$32 can jump discontinuously at the simplicial saddle-node, $0$33 can cross its threshold abruptly, allowing a purely pairwise disease process to exhibit discontinuous transitions and bistability through hidden higher-order social driving (Lucas et al., 2022).

Finally, arbitrary-order formulations now span both exact and approximate settings. Exact hydrodynamic limits on fully connected complexes yield $0$34 (Kiss et al., 2023). Discrete-time tensor models on weighted, directed hypergraphs generalize this to

$0$35

with explicit healthy-state, endemic-state, bistability, multistability, and domain-of-attraction results (Liang et al., 2024). A separate agent-based framework shows that steady-state dynamics generated by higher-order hyperedges can be replicated by pairwise simulations through activity-based scaling, and that time-varying pairwise parameters can also reproduce transient trajectories. This suggests a practical modeling duality between explicit higher-order topology and suitably normalized lower-order dynamics at the population level (Tan et al., 28 Aug 2025).

6. Identification, optimization, and methodological limits

SCM have also generated algorithmic questions about how higher-order contagion should be inferred, controlled, or exploited. One inference strategy uses a single observed cascade and the order in which nodes become infected. On aggregated networks and simplicial complexes, the Spearman correlations

$0$36, $0$37, $0$38, and $0$39

separate simple contagion, threshold contagion, and higher-order contagion because infection order correlates differently with degree and triangle participation under different mechanisms. On the workplace dataset, $0$40 yields an AUC of approximately $0$41 for distinguishing threshold processes from the others, and a random-forest classifier using $0$42 attains high multiclass accuracy across several empirical datasets (Cencetti et al., 2023).

Influence maximization under SCM has been formulated on hypergraphs with SIR dynamics and pairwise plus 2-simplex transmission. Message passing around the disease-free state leads to a weighted non-backtracking operator

$0$43

whose spectral radius controls the linear threshold. On that basis, the paper defines a collective influence score

$0$44

and proposes the Collective Influence Adaptive (CIA) algorithm, which selects seeds with high collective influence while explicitly avoiding seed clustering. In synthetic and real hypergraphs, CIA outperforms degree-based, hyperdegree-based, pruning, and random baselines, especially in sparser regimes (Zhang et al., 2023).

Parameter learning is also available in discrete-time higher-order SIS. For node $0$45, the regression

$0$46

yields a least-squares estimator for $0$47 once trajectories and hyperedge weights are known. This provides a direct route from time-series data to node-wise pairwise and higher-order transmission parameters in the tensor-based discrete-time framework (Liang et al., 2024).

Across these developments, several limitations recur. Mean-field models neglect heterogeneity and correlations; PBMF and MECLE improve accuracy but depend on heuristic closures that can violate higher-order conservation relations or deteriorate when many group interactions share edges (Malizia et al., 2023). Temporal and signed extensions are mostly validated on synthetic topologies, and explicitly note that real signed higher-order data would be needed for parameter calibration and empirical validation (Kemmeter et al., 2023). Exact results are strongest on fully connected or highly symmetric structures, whereas arbitrary sparse simplicial complexes still require approximations, numerical continuation, or simulation (Kiss et al., 2023).

Taken together, the literature defines SCM as a family of higher-order contagion processes whose essential mathematical feature is the simplex-order-dependent nonlinear infection term. The field has moved from homogeneous mean-field equations to correlation-aware closures, exact fully connected limits, tensor-based discrete-time dynamics, and application-level tasks such as detection, learning, and influence maximization. A consistent conclusion across these strands is that higher-order interactions alter not only quantitative outbreak sizes but the qualitative geometry of contagion itself: they can create critical masses, reshape thresholds, sustain hysteresis, and generate bistable or multistable epidemic landscapes even when the underlying population structure is otherwise simple (Kiss et al., 2024).