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Epidemic thresholds of the Susceptible-Infected-Susceptible model on networks: A comparison of numerical and theoretical results
Silvio C. Ferreira, Claudio Castellano, Romualdo Pastor-Satorras
TL;DR
Different theories predict qualitatively different SIS epidemic thresholds on networks, especially for finite systems and heterogeneous degree distributions. The paper uses large-scale simulations with quasi-stationary sampling and susceptibility peaks to compare HMF and QMF predictions, finding that QMF is generally more accurate in scaling but often misses the prefactor.
Problem
Different theoretical approaches give qualitatively different estimates of SIS epidemic thresholds on networks, while finite-size threshold accuracy requires detailed quantitative investigation.
Method
The paper performs large-scale SIS simulations using the quasi-stationary method and susceptibility peaks to estimate effective finite-size thresholds across several network types.
Results
QMF generally improves threshold estimates over HMF and captures the vanishing threshold for power-law networks, but for γ = 2.75 its accuracy is about 30% because the prefactor differs from unity.
Takeaways & Limitations
QMF captures the network-size scaling of thresholds more reliably than HMF, but improved analytical methods are needed for accurate threshold predictions.
Abstract
from arXiv · showhide
Recent work has shown that different theoretical approaches to the dynamics of the Susceptible-Infected-Susceptible (SIS) model for epidemics lead to qualitatively different estimates for the position of the epidemic threshold in networks. Here we present large-scale numerical simulations of the SIS dynamics on various types of networks, allowing the precise determination of the effective threshold for systems of finite size N. We compare quantitatively the numerical thresholds with theoretical predictions of the heterogeneous mean-field theory and of the quenched mean-field theory. We show that the latter is in general more accurate, scaling with N with the correct exponent, but often failing to capture the correct prefactor.
I. INTRODUCTION
The SIS model on networks has competing theoretical threshold predictions, motivating quantitative finite-size tests. The paper compares heterogeneous and quenched mean-field approaches with large-scale simulations and finds QMF generally more accurate, though its prefactor can remain wrong.
- Motivation: SIS dynamics exhibit an epidemic transition between an absorbing healthy phase and an active endemic phase.
- Theoretical predictions: HMF predicts a threshold proportional to ⟨k⟩/⟨k^2⟩, vanishing when the degree distribution’s second moment diverges.
- Theoretical predictions: QMF uses the inverse largest adjacency-matrix eigenvalue and preserves the network’s quenched structure, unlike HMF’s annealed approximation.
- Theoretical predictions: QMF improves on HMF but both theories factorize neighboring-node activity probabilities and therefore neglect potentially important dynamical correlations.
- Paper objective: The paper tests these predictions through large-scale SIS simulations on heterogeneous annealed, random regular, star, and power-law networks.
- Paper objective: QMF captures the vanishing threshold for power-law networks across γ, but often predicts only the scaling with N rather than the correct prefactor.
II. NUMERICAL METHODS
The simulations implement continuous-time SIS dynamics by selecting recovery or transmission events probabilistically. Each event updates infection counts and advances time according to the current event rate.
- At each step, the algorithm counts infected nodes and links emanating from them before selecting a recovery or transmission event.
- Recovery occurs with probability Ni/(Ni + λNn), while transmission occurs with complementary probability λNn/(Ni + λNn).
- After each event, infection counts and related links are updated and time advances by Δt = 1/(Ni + λNn).
A. The quasi-stationary state method
The quasi-stationary method avoids the inefficiency of sampling only surviving runs by constraining SIS simulations to remain active. It stores and updates previously visited active configurations for this purpose.
- A. The quasi-stationary state method: The QS method replaces attempted visits to the absorbing state with active configurations randomly selected from a stored history.The history contains M active configurations and is continually updated during simulation.
- A. The quasi-stationary state method: The stored active-state list is updated by replacing a randomly selected configuration with the present active configuration at probability prΔt.The passage describes the update mechanism used to maintain the quasi-stationary ensemble.
- A. The quasi-stationary state method: The same simulation procedure has previously determined critical points and exponents for the contact process on annealed and quenched networks.This establishes prior use of the procedure across both network types.
B. Numerical determination of the epidemic threshold
Because the SIS threshold can depend on finite network size and vanish as N grows, conventional finite-size scaling is unsuitable. The paper instead uses susceptibility peaks to estimate the effective threshold.
- B. Numerical determination of the epidemic threshold: For SIS on power-law networks, conventional finite-size scaling fails because the effective threshold depends on N and tends to zero as system size increases.At any λ > 0, the activity density eventually reaches a finite limit once λc(N) becomes sufficiently small.
- B. Numerical determination of the epidemic threshold: At fixed N, the susceptibility maximum occurs at λp(N), which provides a finite-size estimate of the transition location.For a finite thermodynamic-limit threshold, the peak approaches λc(∞) with finite-size corrections.
- B. Numerical determination of the epidemic threshold: The paper assumes λp(N) − λc(N) ∼ N^-1/ν̄, so susceptibility peaks and size-dependent thresholds coincide asymptotically.The passage states that this assumption is explicitly checked and found correct where controllable.
- B. Numerical determination of the epidemic threshold: The susceptibility χ is computed from fluctuations of the infected density and differs from the standard definition χN.The chosen definition produces clearer numerical results while preserving the usual scaling properties.
III. NUMERICAL CHECK OF THE SUSCEPTIBILITY METHOD: ANNEALED SCALE-FREE NETWORKS
Annealed networks provide a benchmark because rewiring preserves degree information while destroying dynamical correlations, making HMF predictions exact. The susceptibility-peak estimate agrees with the corresponding threshold calculations.
- III. NUMERICAL CHECK OF THE SUSCEPTIBILITY METHOD: ANNEALED SCALE-FREE NETWORKS: Annealed networks rewire edges after each vertex-state change while preserving degrees and degree correlations.This procedure destroys dynamical correlations and renders HMF predictions exact.
- III. NUMERICAL CHECK OF THE SUSCEPTIBILITY METHOD: ANNEALED SCALE-FREE NETWORKS: The susceptibility peak is used to estimate thresholds in annealed networks because the resulting values agree with numerically evaluated HMF thresholds.The agreement holds both when the threshold vanishes with N and when it approaches a finite value.
IV. HOMOGENEOUS NETWORKS: THE RANDOM REGULAR NETWORK
Random regular networks test threshold predictions in a homogeneous setting where HMF and QMF both give 1/k. Simulations instead place the susceptibility peak near the pair-approximation value 1/(k −1), indicating that dynamical correlations matter.
- IV. HOMOGENEOUS NETWORKS: THE RANDOM REGULAR NETWORK: For random regular networks of degree k, HMF predicts the constant threshold λc^HMF = 1/k.All nodes have the same degree, while links are randomly distributed without self-connections or multiple connections.
- IV. HOMOGENEOUS NETWORKS: THE RANDOM REGULAR NETWORK: QMF gives the same 1/k prediction for random regular networks because the adjacency matrix has largest eigenvalue k.The all-ones vector is an eigenvector with eigenvalue k by Perron-Frobenius.
- IV. HOMOGENEOUS NETWORK: THE RANDOM REGULAR NETWORK: For annealed scale-free networks, the susceptibility is more efficient than χN for determining the effective size-dependent threshold.This comparison is shown using γ = 2.25 networks of different sizes.
- IV. HOMOGENEOUS NETWORKS: THE RANDOM REGULAR NETWORK: The susceptibility peak for degree-k = 10 random regular networks approaches λc^pair = 1/(k −1), rather than the HMF or QMF value 1/k.The peak becomes increasingly close to the pair-approximation prediction as N increases.
- IV. HOMOGENEOUS NETWORKS: THE RANDOM REGULAR NETWORK: HMF and QMF provide reasonable but non-exact threshold approximations because they neglect dynamical correlations among neighboring node states.Pair-approximation approaches account for these correlations more effectively.
V. HETEROGENEOUS NETWORKS: THE STAR GRAPH
The star graph provides a simple heterogeneous network where the epidemic threshold scales as kmax^-1/2, but HMF predicts the wrong scaling and QMF misses the prefactor.
- The star graph consists of one hub of degree kmax connected to kmax leaves of degree 1.
- The largest eigenvalue of the star graph's adjacency matrix scales as kmax^1/2.
- The susceptibility curves collapse when plotted against λ kmax^1/2, showing that λc scales as kmax^-1/2.
- The threshold prefactor is approximately 1.5 rather than 1, consistent with the rigorous bound λc ≥ kmax^-1/2.
- HMF fails for the star graph, whereas QMF gives the correct threshold scaling but not the exact prefactor.
POWER-LAW DEGREE DISTRIBUTED GRAPHS
Large-scale SIS simulations on power-law networks compare effective finite-size thresholds with HMF and QMF predictions across degree-exponent regimes. QMF generally captures the network-size scaling more accurately, while discrepancies in prefactors and competing transition mechanisms remain important.
- γ = 2.25: For γ = 2.25, susceptibility-peak thresholds agree with both HMF and QMF predictions, which nearly coincide.The numerical threshold follows the theoretical scaling, and the agreement indicates similar prefactors in this regime.
- γ = 2.25: χ is the best threshold indicator because its maximum remains clearly defined and diverges with network size.The fitted exponents give β/ν̄ = 0.65, γ′/ν̄ = −0.28, and (γ′ + β)/ν̄ = 0.37.
- γ = 2.25: At γ = 2.25, the effective order-parameter exponent lies between HMF's β = 4/3 and QMF's β = 1, excluding neither prediction.The estimate comes from networks with N = 10^7.
- γ = 2.75: For γ = 2.75, numerical thresholds follow the inverse-largest-eigenvalue scaling rather than HMF behavior, with QMF accuracy around 30%.QMF captures the scaling but uses a prefactor different from unity relative to the numerical threshold.
- γ = 3.5: For γ = 3.5, susceptibility develops competing peaks: a low-λ peak associated with the largest hub and a higher-λ peak associated with the maximum k-core.The hub-related transition follows QMF scaling, whereas the higher-λ transition fluctuates strongly across network realizations.
- γ = 3.5: At asymptotically large N, the hub-driven mechanism is expected to dominate, but the crossover is too slow to test fully at accessible sizes.The authors therefore cannot determine the accuracy of the QMF expression in this regime in detail.
VII. CONCLUSIONS
The paper develops accurate finite-size SIS threshold estimates using QS simulations and susceptibility peaks, then compares HMF and QMF across network structures. It finds that both theories can be inaccurate on random regular graphs, while QMF is sufficient to yield the correct result for the star graph.
- VII. CONCLUSIONS: The QS method and susceptibility-peak analysis provide accurate numerical threshold estimates for finite SIS networks.The approach is validated on annealed networks, where HMF is exact.
- VII. CONCLUSIONS: On random regular graphs, both HMF and QMF may provide inaccurate epidemic thresholds despite the networks’ homogeneous structure.The theories disregard dynamical correlations that influence the threshold.
- VII. CONCLUSIONS: For the strongly heterogeneous star graph, QMF theory is sufficient to yield the correct result.