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Asymmetrically interacting spreading dynamics on complex layered networks

Wei Wang, Ming Tang, Hui Yang, Younghae Do, Ying-Cheng Lai, GyuWon Lee

arXiv:1405.1905v1physics.soc-phcs.SI

TL;DR

The paper studies asymmetrically coupled information and epidemic spreading across communication and physical-contact network layers using mean-field analysis and computational experiments. It finds that epidemic outbreaks can trigger information outbreaks, while rapid information spreading and inter-layer correlation can raise the epidemic threshold without changing the information threshold.

  • Problem

    The paper addresses how information diffusion and epidemic spreading interact asymmetrically across separate communication and physical-contact network layers.

  • Method

    The authors analyze a two-layer network model using mean-field theory and extensive numerical computations, considering both uncorrelated and inter-layer-correlated networks.

  • Results

    Epidemic outbreaks can trigger information outbreaks; high information-transmission rates increase the epidemic threshold, while stronger inter-layer correlation leaves the information threshold largely unchanged but suppresses epidemic spreading.

  • Takeaways & Limitations

    Information diffusion can mitigate epidemic spreading by promoting timely immunization, particularly among high-degree or high-centrality nodes.

  • Takeaways & Limitations

    The mean-field rate equations cannot be written for SF-ER networks with fixed correlation because the required conditional degree probabilities are unavailable, and the analysis strictly holds in the thermodynamic limit.

Abstract

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The spread of disease through a physical-contact network and the spread of information about the disease on a communication network are two intimately related dynamical processes. We investigate the asymmetrical interplay between the two types of spreading dynamics, each occurring on its own layer, by focusing on the two fundamental quantities underlying any spreading process: epidemic threshold and the final infection ratio. We find that an epidemic outbreak on the contact layer can induce an outbreak on the communication layer, and information spreading can effectively raise the epidemic threshold. When structural correlation exists between the two layers, the information threshold remains unchanged but the epidemic threshold can be enhanced, making the contact layer more resilient to epidemic outbreak. We develop a physical theory to understand the intricate interplay between the two types of spreading dynamics.

Results

The paper models disease spreading on a physical-contact layer and information spreading on a communication layer, allowing distinct structures, asymmetric coupling, and simultaneous dynamics. Analysis and simulations show that information can raise epidemic thresholds and reduce disease prevalence, while epidemics can trigger information outbreaks; inter-layer correlation mainly strengthens epidemic suppression.

  • Network structure: The analysis treats uncorrelated layers first, then constructs correlated layers by independently generating subnetworks and rematching corresponding nodes.Inter-layer dependence is characterized by how node degrees are matched across the two layers.
  • Model: The model uses an SIR process for information on communication layer A and an SIRV process for disease on contact layer B, with awareness enabling vaccination.The layers may have different structures and are coupled through corresponding nodes.
  • Thresholds: An epidemic outbreak can trigger information spreading on layer A even when information transmission within that layer is inefficient.For βB > βBu, epidemic-induced information can occur for βA ≤ βAu, while vaccination remains limited and βBc ≈ βBu.
  • Thresholds: When information spreads faster than disease, epidemic threshold enhancement is associated with communication-layer heterogeneity, higher βA, and higher vaccination probability p.This regime is identified by θ < 1; when θ > 1, disease spreads faster and vaccination can be neglected.
  • Uncorrelated networks: On SF-ER networks, βBc is unchanged by βA for βA ≤ βAu ≈ 0.06 but increases with βA above that regime.For p = 0.9 and βA ≥ 0.49, epidemic outbreak is ruled out; finite-size effects contribute to deviations from theory.
  • Final outbreak ratios: Increasing βA decreases the final disease recovery density RB, with RB → 0 at βB = 0.2 and βA ≥ 0.5.The final information recovery density RA increases rapidly with βA and βB in the low-rate regime.
  • Correlated networks: Positive inter-layer correlation leaves βAc largely unchanged but increases βBc by preferentially vaccinating high-degree contact-layer nodes.As q increases, RA and RB increase while the vaccinated density VB decreases.

Discussion

The model reveals asymmetric coupling between communication and epidemic spreading: disease can trigger information, while information can raise epidemic thresholds and reduce disease prevalence through vaccination. Inter-layer correlation leaves the information threshold largely unchanged but can suppress epidemics by immunizing high-degree nodes.

  • The study combines an asymmetrically interacting two-layer model, mean-field analysis, and computational investigation of epidemic and information thresholds and final infected fractions.
  • Information outbreak can be triggered by either its own spreading dynamics or an epidemic outbreak on the counter-layer.
  • Information spreading raises the epidemic threshold only when the information-transmission rate is sufficiently high.
  • Stronger inter-layer correlation leaves the information threshold unchanged but suppresses epidemics through timely immunization of large-degree nodes.

Methods

The model represents information diffusion with SIR dynamics on a communication layer and disease transmission with SIRV dynamics on a contact layer. Mean-field equations account for within-layer transmission, counterpart-layer coupling, vaccination, and inter-layer degree correlation.

  • The mean-field theory derives rate equations from infection probabilities for counterpart nodes during an infinitesimal time interval.
  • Information reaches a susceptible layer-A node through informed neighbors or an infected counterpart node in layer B.
  • For uncorrelated layers, counterpart degree conditionals simplify to P(kB|kA) = PB(kB) and P(kA|kB) = PA(kA).
  • A susceptible layer-B node can become infected by a same-layer neighbor or vaccinated when its informed counterpart triggers vaccination with probability p.
  • Inter-layer correlation is quantified by the Spearman rank coefficient ms, with random matching giving ms ≈0 and equal degree ranks giving ms ≈1.
  • Random rematching converts maximally correlated layer pairs toward uncorrelated structure according to the rematching probability q.

Figure legends

The figures examine final densities, spreading thresholds, and rematching effects in layered networks, comparing SF-ER and ER-ER structures and varying transmission, vaccination, and correlation parameters.

  • Figure 1 illustrates asymmetrically coupled spreading processes on a simulated communication-contact double-layer network.
  • Figure 2 measures information susceptibility χ against βA and βAc against βB under different vaccination rates on SF-ER networks.Analytical predictions from Eq. (9) are shown for the threshold plot.
  • Figure 3 presents the epidemic threshold βBc as functions of information-transmission rate βA and vaccination rate p on SF-ER double-layer networks.The figure compares analytical predictions for p = 0.5 and p = 0.9 and includes the θ = 1 case.
  • Figure 4 plots final recovered densities RA and RB and vaccination density VB against βA and βB for SF-ER networks.The plotted curves are numerical solutions of Eqs. (1)-(8) in the limit t →∞, with p = 0.5.
  • Figures 5 and 6 examine how cross-layer spreading and rematching probability q affect outbreak thresholds across SF-ER and ER-ER networks.Figure 5 varies βB against βAc, while Figure 6 varies q for both βAc and βBc.
  • Figure 7 shows how rematching probability q changes final recovered densities RA and RB and vaccination density VB in SF-ER and ER-ER networks.

Additional information

The additional information defines the layer-specific state variables and mean-field equations used to describe information and epidemic spreading and to obtain final densities.

  • Layer A uses degree-specific densities for susceptible, informed, and recovered nodes, while layer B uses susceptible, infected, recovered, and vaccinated densities.
  • The mean-field rate equations describe information spreading in layer A and epidemic spreading in layer B.
  • ΘA(t) and ΘB(t) denote the probabilities that neighboring nodes in layers A and B are in the relevant informed or infected states.
  • State densities are obtained for each degree and state, and whole-system final densities follow by taking t →∞.

B. Linear analysis for the information threshold

Linear analysis represents early information growth with a block matrix whose maximum eigenvalue determines whether an information outbreak occurs.

  • At t →0, the layered system is treated as two coupled SI-epidemic subsystems for analyzing information spreading.
  • The linearized dynamics are written using a vector of infected densities and a block matrix C.
  • The elements of C are specified through the layer-specific coupling terms governing early spreading.
  • An information outbreak occurs when the maximum eigenvalue ΛC of C exceeds 1.Epidemic infection in layer B can instantaneously inform the counterpart node in layer A, facilitating information spreading.
  • The information threshold is determined from the larger of the maximum eigenvalues associated with the coupled and isolated layer matrices.

C. Competing percolation theory for epidemic threshold

The epidemic-threshold analysis compares the growth rates of disease and vaccination, then uses competing percolation on the residual contact network when vaccination spreads faster.

  • The early epidemic and information populations grow exponentially, allowing their growth-rate ratio θ to determine which process is faster.
  • When θ > 1, disease spreads faster than vaccination, so vaccination spread is treated as insignificant and neglected.
  • For θ < 1, vaccination and epidemic spreading are treated successively, and the epidemic threshold is derived using bond percolation.
  • After information spreading ends, the informed population determines the fraction of vaccinated or removed nodes in the contact layer.The informed density is obtained from the generating-function equations, and vaccination density is pSA.
  • A disease outbreak requires a giant residual cluster in the contact layer after vaccination removes nodes.
  • The epidemic threshold is obtained from the residual-network reproductive condition, with the threshold corresponding to eRi = 1 and βBc = 1/H′.
  • For correlated layers, conditional degree distributions and layer-specific mean-field equations are used to calculate state densities and thresholds.

B. Linear analysis for the information threshold

The early dynamics are treated as two coupled SI subsystems, yielding an information-outbreak threshold with two distinct outbreak mechanisms. Disease spreading can induce information spreading when the disease transmission exceeds its threshold.

  • B. Linear analysis for the information threshold: The initial dynamics are modeled as two coupled SI-epidemic subsystems evolving simultaneously across the two layers.The time evolution is described by the paper’s coupled equations.
  • B. Linear analysis for the information threshold: The information-outbreak threshold is obtained from the linearized dynamics and matches the corresponding main-text threshold.
  • B. Linear analysis for the information threshold: For βB ≤ βBu, only a small number of contact-layer nodes are infected, so disease impact on the information threshold is negligible.
  • B. Linear analysis for the information threshold: For βB > βBu, epidemic spreading can trigger an outbreak of information on the communication layer.

C. Competing percolation theory for epidemic threshold

The epidemic threshold is analyzed after information spreads on the communication layer and removes corresponding contact-layer edges through vaccination. A giant residual contact cluster then determines the epidemic threshold.

  • C. Competing percolation theory for epidemic threshold: When βA ≤ βAu, information has little effect on epidemic spreading, leaving βBc approximately equal to βBu.
  • C. Competing percolation theory for epidemic threshold: For βA > βAu, the epidemic-threshold effect of information spreading is assessed through the informed fraction SA and recovered-node densities on layer A.
  • C. Competing percolation theory for epidemic threshold: Vaccinating counterpart nodes randomly removes their connecting edges from the contact layer, producing a residual network with a new degree distribution.
  • C. Competing percolation theory for epidemic threshold: The epidemic threshold follows from requiring a giant cluster in the residual contact network and using the first two degree-distribution moments.

S3. Simulation results

Simulations initialize infection on the contact layer together with information on its counterpart communication node, then update both layers with asymmetric parallel dynamics. Final recovered and infected densities are recorded after spreading terminates.

  • S3. Simulation results: Each simulation starts by randomly infecting a contact-layer node and informing its counterpart communication-layer node.
  • S3. Simulation results: Parallel updates calculate informed and infected transmission probabilities from the numbers of informed and infected neighbors.
  • S3. Simulation results: When both counterpart nodes are susceptible, information and disease transmissions compete, with information transmission selected according to πA/(πA + πB).
  • S3. Simulation results: If the communication counterpart is informed or refractory while the contact node remains susceptible, only disease transmission can occur.
  • S3. Simulation results: The process ends after all informed and infected nodes recover, and the final densities RA, RB, and VB are recorded across 30 network realizations.The simulations use NA = NB = 2 × 10^4 and average degrees ⟨kA⟩ = ⟨kB⟩ = 8.

B. Uncorrelated double-layer networks

For uncorrelated double-layer networks, simulations broadly validate the theoretical thresholds while revealing finite-size deviations. Network heterogeneity lowers the information threshold and can slightly raise the epidemic threshold under low information transmission.

  • B. Uncorrelated double-layer networks: For βB ≤ βBu, the information threshold is approximately βAc ≈ βAu = ⟨kA⟩/(⟨kA^2⟩ − ⟨kA⟩), while βAc = 0.0 for βB > βBu.
  • B. Uncorrelated double-layer networks: As network size grows, the information threshold decreases as the second degree moment of layer A increases.
  • B. Uncorrelated double-layer networks: Theoretical and simulated thresholds generally agree, but finite-size deviations grow at high vaccination rates because competing percolation is not strictly valid there.
  • B. Uncorrelated double-layer networks: Smaller γA, representing stronger heterogeneity in layer A, makes information outbreaks easier and can slightly raise βBc at low information-transmission rates.
  • B. Uncorrelated double-layer networks: For SF-SF networks, theoretical–simulation threshold gaps narrow with increasing network size, supporting the thermodynamic-limit assumption.
  • B. Uncorrelated double-layer networks: Figure S8 compares susceptibility and information or epidemic thresholds across network sizes on SF-ER networks.

C. Correlated double-layer networks

Positive inter-layer correlation in SF-SF networks leaves the information threshold unchanged while enhancing the epidemic threshold, making the contact layer more robust to outbreak.

  • The information threshold remains unchanged as positive inter-layer correlation increases on SF-SF networks.Correlation is increased by reducing the rematching probability q.
  • The epidemic threshold increases with positive inter-layer correlation on SF-SF networks.The reported trend is consistent with corresponding results for ER-ER correlated networks.
  • Positive inter-layer correlation makes the contact layer more robust to epidemic outbreak.
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