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The effect of heterogeneity on hypergraph contagion models
Nicholas Landry, Juan G. Restrepo
TL;DR
Higher-order interactions complicate contagion dynamics, yet the effects of heterogeneity and correlations in their structure lack a general explanation. The paper develops a hyperdegree-based mean-field SIS model for hypergraphs with links and triangles, finding that structural organization changes bistability and explosive transitions. These predictions are compared with microscopic simulations and analytic results, while the study remains limited by its simple Markovian model and mean-field accuracy.
Problem
The paper addresses how heterogeneity and correlations in higher-order interactions affect bistability and explosive contagion transitions.
Method
The authors develop a hyperdegree-based mean-field description of SIS contagion on hypergraphs with links and triangles, considering alternative interaction structures and contagion mechanisms.
Results
Heterogeneity in pairwise degree structure postpones or suppresses explosive transitions, while higher-order degree heterogeneity promotes them; simulations and analytic predictions support these findings.
Takeaways & Limitations
The organization of higher-order interactions can substantially affect epidemic onset, bistability, hysteresis, and explosive transitions in hypergraph contagion.
Takeaways & Limitations
The study uses a simple Markovian SIS model, excludes non-Markovian dynamics, and relies on a mean-field description that is not quantitatively accurate for moderate infected populations.
Abstract
from arXiv · showhide
The dynamics of network social contagion processes such as opinion formation and epidemic spreading are often mediated by interactions between multiple nodes. Previous results have shown that these higher-order interactions can profoundly modify the dynamics of contagion processes, resulting in bistability, hysteresis, and explosive transitions. In this paper, we present and analyze a hyperdegree-based mean-field description of the dynamics of the SIS model on hypergraphs, i.e. networks with higher-order interactions, and illustrate its applicability with the example of a hypergraph where contagion is mediated by both links (pairwise interactions) and triangles (three-way interactions). We consider various models for the organization of link and triangle structure, and different mechanisms of higher-order contagion and healing. We find that explosive transitions can be suppressed by heterogeneity in the link degree distribution, when links and triangles are chosen independently, or when link and triangle connections are positively correlated when compared to the uncorrelated case. We verify these results with microscopic simulations of the contagion process and with analytic predictions derived from the mean-field model. Our results show that the structure of higher-order interactions can have important effects on contagion processes on hypergraphs.
I. INTRODUCTION
Higher-order interactions can produce bistability, hysteresis, and explosive epidemic transitions, but the role of structural heterogeneity and correlations remains insufficiently understood. The paper develops a mean-field framework to analyze these effects in hypergraph contagion models.
- Higher-order interactions can profoundly alter contagion dynamics, producing bistability, hysteresis, and explosive transitions.
- The paper addresses the lack of a general theory explaining how heterogeneity and correlations in higher-order interaction structure affect bistability.
- It presents a degree-based mean-field description of SIS dynamics on hypergraphs with heterogeneous interactions and correlations across edge sizes.
- Using links and triangles as an illustrative case, the paper derives conditions for bistability and hysteresis and studies how model parameters affect their onset.
II. MODEL
The model studies SIS contagion on hypergraphs, where pairwise and higher-order interactions represent distinct channels for spreading in epidemic and social settings.
- The model describes SIS contagion spreading through pairwise and higher-order interactions on a hypergraph.
- Pairwise interactions can represent face-to-face exposure, whereas higher-order interactions can represent contagion through shared group spaces.
A. Hypergraph model
The hypergraph framework represents nodes connected by hyperedges of multiple sizes and characterizes network structure through hyperdegrees, degree distributions, and cross-order connection probabilities. It supports heterogeneous higher-order interactions and correlations between interaction types.
- Nodes are coupled by undirected hyperedges of sizes 2 through M, with each hyperedge containing a set of nodes.
- A node’s hyperdegree records its participation in hyperedges of different sizes; second-order degree counts links, while higher-order degrees count larger hyperedges.
- Figure 1 illustrates infection transmission through size-2 and size-3 hyperedges, with rates β2 and β3, respectively.
- The mean-field framework assumes nodes with the same hyperdegree have identical statistical properties and uses P(k) and f_m to describe network structure.
- For the links-and-triangles case, links and triangles may be organized with heterogeneous degrees and correlations, without requiring triangle members to be linked pairwise.
B. Contagion model
The contagion model is an SIS process in which infected nodes heal at rate γ and susceptible nodes are infected through hyperedges according to alternative threshold rules.
- Each node is either susceptible or infected, and infected nodes return to the susceptible state at healing rate γ.
- Under collective contagion, a susceptible node is infected at rate β_m when all other members of its size-m hyperedge are infected.
- Under individual contagion, infection occurs at rate β_m when at least one other member of the hyperedge is infected.
- The framework could also treat quorum contagion, where at least j other hyperedge members must be infected.
III. MEAN-FIELD ANALYSIS
The paper develops a hyperdegree-based mean-field analysis of SIS contagion on hypergraphs with links and triangles, allowing heterogeneous structures and degree correlations. It shows how contagion organization and link-degree heterogeneity alter epidemic thresholds, bistability, hysteresis, and explosive transitions.
- Mean-field framework: The mean-field framework tracks infection among nodes sharing hyperdegree and specifies hyperedge formation through degree distributions and connection probabilities.It accommodates heterogeneous higher-order interactions and correlations between nodal degrees of different orders.
- Network structure: For links and triangles, degree-correlated structure concentrates triangles around high-link-degree nodes, whereas uncorrelated structure assigns triangle membership independently of link degree.The corresponding triangle probabilities are f3(k,k1,k2) = 2kk1k2/(N⟨k⟩)2 and f3(k,k1,k2) = 2⟨k⟩/N2, respectively.
- Epidemic threshold: The no-infection state becomes linearly unstable for β2 > γ⟨k⟩/⟨k2⟩ in the uncorrelated case, matching the threshold obtained for the correlated case.This threshold follows from linearizing the mean-field equations around the infection-free solution.
- Collective contagion: For small β3 the transition to infection is continuous, whereas larger β3 produces discontinuous transitions with hysteresis, bistability, and explosive growth.Mean-field equations reproduce the microscopic simulations qualitatively, although quantitative disagreement is attributed to mean-field assumptions.
- Heterogeneity and bistability: As link-degree heterogeneity increases, the bistability onset β3 increases in the uncorrelated case but remains almost unchanged in the degree-correlated case.In the uncorrelated case, contagion becomes dominated by pairwise-network hubs, suppressing bistability; correlated triangles and links increase their effectiveness together.
C. Higher-order healing: hipster effect
Higher-order healing has opposite effects depending on contagion mode: it suppresses explosive transitions under collective contagion but can create bistability under individual contagion.
- Interpretation: The higher-order healing mechanism represents reduced adoption when an idea or trend becomes popular within a group.Pairwise connections can still persuade the individual to adopt the trend.
- Collective contagion: Collective contagion with nonnegative higher-order healing rates does not produce explosive transitions.The higher-order healing mechanism does not change the epidemic threshold because triangle healing is absent from the linearization at zero.
- Individual contagion: Individual contagion permits explosive transitions across ranges of nonnegative pairwise and higher-order healing rates.The degree-correlated phase diagram contains a narrow bistable band between no infection and monostable infection at sufficiently large higher-order healing.
- Individual contagion: Sufficiently strong higher-order healing eliminates infection in the individual-contagion model.A narrow bistable region separates the no-infection and monostable-infection regimes.
D. Unfortunate series of events
The all-hyperedge extension examines individual contagion across social events of every size and identifies a moment-based propagation threshold.
- All hyperedge sizes: For hyperedges of all sizes, the analysis focuses on individual contagion and linearizes the disease-free solution to determine its instability threshold.The framework allows hyperedges representing parties, conferences, concerts, and sports events.
- Propagation condition: Social contagion propagates when the weighted sum over hyperedge sizes exceeds γ⟨k⟩/⟨k2⟩.The threshold compares higher-order transmission contributions with the healing rate and moments of the underlying degree distribution.
- Propagation condition: Restricting large social events or reducing βm through physical separation can lower the sum enough to prevent propagation.These interventions truncate or weaken higher-order contagion contributions.
IV. THE EFFECT OF DEGREE DISTRIBUTION ON β c
The paper studies how degree-distribution heterogeneity changes the onset of bistability and explosive transitions, combining numerical mean-field solutions with closed-form approximations.
- Numerical analysis: The critical β3 for bistability is computed numerically from mean-field equations across correlated and uncorrelated hypergraph structures.The analysis uses βc3 normalized by βc2 as a function of power-law exponent r and maximum degree kmax.
- Bistability criterion: The onset of bistability corresponds to two stable steady states and is identified through multiple nonzero mean-field solutions.The correlated case uses Eq. (10), whereas the uncorrelated case uses the coupled system of Eqs. (15)–(16).
- Numerical analysis: For a k-regular network, βc3/βc2 equals 1, while heterogeneous networks increase βc3 relative to βc2.The comparison is shown for both correlated and uncorrelated cases.
- Numerical analysis: Greater heterogeneity in the pairwise degree distribution generally suppresses explosive transitions, especially in the uncorrelated case.Increasing r or kmax raises heterogeneity under the stated parameterization, with exceptions for small r and large kmax in the degree-correlated case.
- Analytic approximations: For the uncorrelated case, saddle-node bifurcations can occur at positive V, requiring higher-order expansions and physical-root constraints.The relevant roots must lie within V ∈ [0,1].
- Analytic approximations: Higher-order expansions can improve accuracy for highly heterogeneous distributions, but their conditions become extremely complicated.The paper therefore notes limited utility in continuing the expansion order.
V. DISCUSSION
The discussion concludes that hyperedge organization and heterogeneity shape epidemic onset and discontinuous transitions, while identifying modeling limits and broader applications of the formalism.
- Main findings: Heterogeneous pairwise degree distributions suppress discontinuous transitions, whereas heterogeneity in higher-order hyperedge degrees promotes them.Pairwise and group infection act as competing contagion mechanisms.
- Main findings: The study treats independent and positively degree-correlated organizations of hyperedges as null models for comparing real-world hypergraph structure.Real networks may have substantially more complicated organization.
- Model scope: The methodology covers collective contagion, individual contagion, and higher-order healing within the SIS framework.Other higher-order contagion mechanisms are left for future research.
- Limitations: The study is limited by its simplest SIS model, exclusion of non-Markovian dynamics, mean-field inaccuracies at moderate infected-population values, and degree-based edge probabilities.The authors state that these assumptions may inadequately represent some real-world networks.
- Broader applications: The hyperdegree-based mean-field formalism could also be applied to synchronization, opinion formation, and other epidemic models on hypergraphs.The authors propose it as a tool for studying heterogeneity in these processes.
1. Microscopic simulation of the hypergraph SIS model
The microscopic SIS simulation models infection and healing as a synchronous stochastic process on hypergraph nodes, then probes bistability by sweeping infection parameters upward and downward.
- 1. Microscopic simulation of the hypergraph SIS model: Each node is represented by a binary state, with infection transmitted through pairwise links or triangles and healing occurring at a constant rate.Pairwise and hyperedge infection events are treated as independent, as are infections from neighboring nodes.
- 1. Microscopic simulation of the hypergraph SIS model: Nodes are synchronously updated at intervals of ∆t = 0.1 using conditional infection and healing rules.A uniform random variable is independently drawn for each modality and node at each time step.
- 1. Microscopic simulation of the hypergraph SIS model: The network starts with p = 0.001 infected nodes, and the population infection average is recorded over the simulation.A single node is reinfected if the population reaches the absorbing all-healthy state.
- 1. Microscopic simulation of the hypergraph SIS model: Bistability is detected by comparing equilibrium responses while incrementally increasing and decreasing β2 at fixed β3.Distinct equilibrium values at the same β2 indicate bistability, and repeated curves identify the β3 onset.
2. Network models
The network models isolate degree-distribution effects while varying triangle construction between degree-correlated and independently distributed structures, with bistability analyzed from mean-field solutions.
- 2. Network models: The simulations use configuration-model networks of size N = 10^4 to isolate the effect of the degree distribution.The generated graphs were not averaged over an ensemble because the realization was relatively large.
- 2. Network models: Triangles are generated either with a degree-correlated configuration-model construction or by independently sampling triples with replacement.Both constructions can produce self-loops and multi-edges, but these events are expected to be rare.
- 2. Network models: The bistability index is obtained from the largest positive mean-field solution at the pairwise epidemic threshold β2 = β2^c.This procedure applies the relevant mean-field equations to correlated and uncorrelated cases.
- 2. Network models: The critical triangle infection rate β3^c is found by bisection until the interval tolerance reaches 10^-4.The lower endpoint of the final interval is assigned as β3^c.