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Coevolution spreading in complex networks

Wei Wang, Quan-Hui Liu, Junhao Liang, Yanqing Hu, Tao Zhou

arXiv:1901.02125v3physics.soc-phcs.SI

TL;DR

Coevolving diseases, behaviors, and information interact in complex networks, but their mechanisms and critical phenomena require systematic understanding. This review synthesizes theoretical and network-science progress across four coevolution spreading types, highlighting phase transitions, interactions, topology, and resource or awareness effects.

  • Problem

    Diseases, behaviors, and information rarely spread independently, creating a need to understand their interacting mechanisms, outbreaks, infection timing, and containment.

  • Method

    The review organizes theoretical and network-science progress across biological contagions, social contagions, awareness–epidemic spreading, and resources–epidemic spreading.

  • Results

    Coevolution spreading exhibits interaction- and topology-dependent phase transitions, including discontinuous social-contagion cascades, awareness-dependent epidemic thresholds, and abrupt outbreaks under limited resources.

  • Takeaways & Limitations

    The review provides a state-of-the-art framework for studying coevolution spreading dynamics and identifies theoretical challenges and open issues for future research.

  • Takeaways & Limitations

    Most known interaction mechanisms between spreading contagions, including synergy, competition, and asymmetry, have not been verified with empirical data.

Abstract

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The propagations of diseases, behaviors and information in real systems are rarely independent of each other, but they are coevolving with strong interactions. To uncover the dynamical mechanisms, the evolving spatiotemporal patterns and critical phenomena of networked coevolution spreading are extremely important, which provide theoretical foundations for us to control epidemic spreading, predict collective behaviors in social systems, and so on. The coevolution spreading dynamics in complex networks has thus attracted much attention in many disciplines. In this review, we introduce recent progress in the study of coevolution spreading dynamics, emphasizing the contributions from the perspectives of statistical mechanics and network science. The theoretical methods, critical phenomena, phase transitions, interacting mechanisms, and effects of network topology for four representative types of coevolution spreading mechanisms, including the coevolution of biological contagions, social contagions, epidemic-awareness, and epidemic-resources, are presented in detail, and the challenges in this field as well as open issues for future studies are also discussed.

1. Introduction

Real-world spreading processes are networked, heterogeneous, and often strongly coupled rather than independent. This review organizes recent coevolution-spreading research, standardizes its concepts, and examines four representative mechanisms.

  • Motivation: Spreading models address outbreak occurrence, infected population size, infection timing, and containment, with applications including epidemic management and e-commerce.These questions motivate the study of spreading dynamics in real systems.
  • Networked spreading: Real contact patterns differ from well-mixed populations because individuals interact with limited sets of contacts represented by network G(V, E).Network structure provides a framework for describing constrained interactions among individuals.
  • Networked spreading: Real networks exhibit heterogeneous degrees, small-world structure, communities, multiplexity, spatiality, and temporality that oversimplified models cannot fully capture.These structural features contribute to differences between analytical predictions and empirical observations.
  • Coevolution spreading: Diseases, behaviors, and information can coevolve through strong interactions, such as disease-induced susceptibility and awareness-mediated epidemic suppression.The SARS example illustrates how information-driven protective actions reduced infections.
  • Review scope: The review responds to fragmented literature by organizing results, unifying terminology and notation, and integrating application possibilities across fields.It is intended to support existing researchers, newcomers, and potential users of coevolution-spreading findings.
  • Review scope: Four representative mechanisms are biological contagions, social contagions, awareness–epidemic spreading, and resources–epidemic spreading.The review covers theoretical methods, critical phenomena, phase transitions, interaction mechanisms, and network-topology effects.

2. Coevolution of biological contagions

Biological contagions model processes in which a single activated source can transmit infection, with SIS and SIR providing the principal network examples.

  • Biological contagions: Biological contagions include epidemic, virus, and information diffusion processes in which a single activated source can be sufficient for transmission.This distinguishes them from social contagions requiring multiple reinforcing contacts.
  • Representative models: SIS and SIR are representative biological-contagion models on networks.SIS is reversible, whereas SIR is irreversible because recovered nodes do not return to susceptibility.
  • Representative models: In SIS dynamics, infected nodes transmit to susceptible neighbors at rate β and return to the susceptible state at rate γ.The model therefore permits repeated infection and recovery cycles.

2.1. Single biological contagions

Single-contagion behavior depends on both spreading dynamics and network topology. The review compares analytical approaches and shows that degree heterogeneity, clustering, communities, multiplexity, and temporal structure alter thresholds and epidemic evolution.

  • Theoretical approaches: The review emphasizes mainstream theoretical approaches and network-topology effects in single-contagion spreading.These approaches include mean-field-like, quenched mean-field, and dynamic message-passing methods.
  • Theoretical approaches: Pastor-Satorras and Vespignani’s SIS study on Barabási–Albert networks used heterogeneous mean-field theory to analyze epidemic spreading with power-law degree distribution P(k) ∼ k^-3.The work addressed the contrast between long virus lifetimes and low infected-population fractions.
  • Critical phenomena: When λ ≤ λc there is no global epidemic, whereas λ > λc permits a global epidemic; larger-degree nodes have higher infection probability.The threshold follows from linearizing around the infection-free state.
  • Theoretical approaches: Heterogeneous mean-field theory predicts annealed-network spreading well but cannot accurately predict quenched-network thresholds when local structures exceed degree-distribution information.Neighbor-state independence also neglects dynamical correlations.
  • Critical phenomena: The epidemic threshold and prevalence depend on network structure, with λc = ⟨k⟩/(⟨k2⟩−⟨k⟩) in the bond-percolation analysis and R ∼ (λ − λc)^αe near criticality.The exponent is αe = 1 for ER and SF networks with γD ≥ 4, and αe = 1/(γD − 3) when 3 < γD < 4.
  • Effects of network topology: In scale-free networks, degree heterogeneity can eliminate a finite threshold, while localized eigenvectors can confine infection mainly to hubs and their neighbors.Such localization makes epidemic growth slow and vulnerable to fluctuations.
  • Theoretical approaches: Dynamic message passing incorporates full network topology and predicts SIR spreading well on uncorrelated locally tree-like networks, but requires 2E + N differential equations.It has applications in containment, source localization, and network dismantling.
  • Theoretical approaches: For 56 real-world networks, dynamic message passing produces more accurate epidemic thresholds than mean-field-like predictions because it incorporates full topology information.For uncorrelated configuration networks, mean-field-like and dynamic message-passing predictions are identical for SIR thresholds.

2.2. Coevolution of two biological contagions

Biological contagions can spread successively, competitively, or cooperatively, with interactions and network structure determining thresholds, dominance, coexistence, and transition types.

  • Successive contagions: Successive epidemics use residual-network structure to define a coexistence threshold for the second contagion after the first spreads.The first epidemic is analyzed through bond percolation, while the second spreads on the remaining network.
  • Successive contagions: Overlapping links can suppress the second epidemic in successive spreading, whereas partial immunity can make its invasion easier.These effects depend on how the first contagion modifies susceptibility and network overlap.
  • Competing or cross-immunity contagions: Competing epidemics produce epidemic-free, absolute-dominance, or coexistence regimes, with the faster epidemic spreading first and the slower one using the residual network.The growth-rate boundary depends on transmissibilities, spreading rates, and the network threshold.
  • Cooperative contagions: The classical epidemic threshold is λc = ⟨k⟩/(⟨k2⟩−⟨k⟩), and topology determines whether cooperative transitions are continuous or discontinuous.Loops are crucial for discontinuous transitions, which are absent on some locally connected lattices and BA networks but appear on ER networks and other topologies.
  • Cooperative contagions: For degree exponent 2 < γD < 3, cooperative spreading has no threshold and a continuous transition, while larger γD permits discontinuous transitions above Hc.For 3 < γD < 4, Hc exceeds 2; for γD > 4, Hc = 2.
  • Cooperative contagions: Cooperative contagions can spread faster on clustered networks and exhibit tricritical behavior as cooperation increases.The transition changes from continuous to hybrid with increasing cooperation, and clustered structure accelerates spreading relative to equivalent random networks.

2.3. Coevolution of multiple biological contagions

Multiple biological epidemics can interact through cross-immunity, cooperative transmission, shared hosts, and recovery dynamics, producing varied prevalence patterns and temporal behavior. The reviewed models examine these interactions analytically and computationally across different epidemic structures and network settings.

  • Cross-immunity: Cross-immunity is modeled through prior exposure and current coinfection, with total prevalence depending on η and φ.The model assumes σj,l = ηjφl, with η describing prior exposure cross-immunity and φ describing current coinfection cross-immunity.
  • Cross-immunity: Cross-immunity against coinfection has a more prominent influence on multiple epidemics than prior exposure cross-immunity.
  • Spatiotemporal patterns: Short infectious periods produce one dominant cluster, whereas relatively long infectious periods allow many clusters to coexist and alternate.
  • Temporal dynamics: Interactions among epidemics can transform damped oscillations into sustained oscillations in an SICR-based model with host demography.
  • Cooperative transmission: A cooperative SIR model defines infection by the (ℓ + 1)-th epidemic with probability βℓ+1 and analyzes the resulting critical condition.The cooperative strength is defined as Hℓ = βℓ/λ1, and the model tracks the fraction of nodes infected by ℓ different epidemics.
  • Cooperative transmission: At the minimum cooperative strength H, the system exhibits a discontinuity of n/(n − 1) in the well-mixed setting.

2.4. Summary

For single biological contagions on complex networks, phase transitions remain continuous, while thresholds and critical behavior depend on network topology. The section also emphasizes trade-offs among theoretical approaches.

  • 2.4. Summary: Single biological contagions on complex networks always exhibit a continuous phase transition.
  • 2.4. Summary: Thresholds and critical behavior for single biological contagions are associated with network topologies.
  • 2.4. Summary: Widely used theoretical approaches differ in their limitations and advantages for quantitatively describing spreading dynamics.

3. Coevolution of social contagions

Social contagions describe the diffusion of behaviors, news, innovations, fads, health practices, and protests under both peer interaction and psychological or cognitive influences. The section examines how multiple social contagions interact, beginning with empirical evidence and foundational models.

  • Scope and foundations: Social contagions include news, innovations, cultural fads, health behaviors, and political protests.
  • Scope and foundations: Unlike biological contagions, social behavior adoption involves multiple confirmation and social reinforcement rather than a sufficient single contact.
  • Scope and foundations: Psychological, cognitive, and social-affirmation factors influence individual states alongside peer interactions.
  • Coevolution focus: The section focuses on how different forms of interplay affect the coevolution of two or more social contagions, including cooperation between behaviors.
  • Coevolution focus: The social-contagion discussion begins with empirical studies and fundamentals of mathematical models before addressing generalized threshold models.

3.1. Single social contagions

Single social contagions require reinforcement from multiple exposures, so their spread depends on adoption thresholds, memory, individual interactions, and network topology. Studies show that these mechanisms can alter cascade likelihood, adoption speed, and the continuity of phase transitions.

  • Social reinforcement: Social contagions differ from simple contagions because adoption typically requires multiple sources of activation through social reinforcement.Preventative behaviors are described as uncertain, risky, and costly, unlike infections that one infected individual can reproduce.
  • Threshold model: Threshold models make adoption depend on whether active-neighbor counts or fractions reach an individual’s threshold.The Watts model incorporates heterogeneity in individuals’ numbers of contacts and assigns thresholds from a distribution.
  • Threshold model: A giant vulnerable cluster allows a random seed to trigger a global cascade, while degree heterogeneity makes global cascades easier.For uniform thresholds, final adoption first increases continuously and then decreases discontinuously as average degree rises.
  • Network topology: Network structure changes social-contagion outcomes: positive degree correlations expand global-cascade regions, while clustering can reduce cascade size.Community structure separates activation peaks, and optimal intercommunity organization can maximize diffusion extent.
  • Generalized social contagions: Synergy and waiting-time heterogeneity modify contagion speed, while constructive synergy lowers epidemic thresholds and can produce explosive growth with hysteresis.Interfering synergy slows adoption; heterogeneously distributed wait times can either accelerate or decelerate it.
  • Memory effects: Memory and exposure-history effects can produce three equilibrium classes and change adoption transitions from discontinuous to continuous.With fixed memory length, the equilibrium class is determined by the probabilities of infection after one and two exposures.

3.2. Coevolution of two social contagions

Two social contagions may spread successively or simultaneously while cooperating, inhibiting, or competing. Models and empirical analyses show that these interactions can change cascade extent, transmission thresholds, and phase-transition types.

  • Successive contagions: Successively interacting social contagions model two behaviors spreading in sequence, with behavior 1 altering behavior 2’s adoption probability.The model uses modified SAR threshold dynamics for both behaviors.
  • Successive contagions: Inhibition can change behavior 2’s continuous transition into a discontinuous one, whereas synergy can reverse that change.The transition type is analyzed using bifurcation conditions involving the coupled spreading equations.
  • Simultaneous contagions: Empirical cascades show cooperation when Google Play Music and YouTube usage rhythms synchronize, but competition when one URL shortener gains attention as another loses it.A correlated-cascades model was proposed to predict product-adoption behavior and reportedly improved prediction accuracy.
  • Simultaneous contagions: Synergistic contagions on two layers can greatly enhance spreading and alter phase-transition types across behaviors with different adoption thresholds.A small transmission rate for a low-threshold behavior can induce a discontinuous transition in a high-threshold behavior, while a large rate can produce a continuous transition.
  • Simultaneous contagions: Inclusive adoption permits individuals to acquire both behaviors, while dormancy discounts dormant neighbors’ influence during adoption decisions.Lower synergy increases global-cascade susceptibility, especially on lattices, and faster diffusion of one dormant contagion may block the other.
  • Simultaneous contagions: Interdependent networks couple a simple SIS layer with a complex Watts-threshold layer, affecting both transition points and transition natures.The two layers represent distinct adoption processes linked by interconnections.

3.3. Coevolution of multiple social contagions

Multiple social contagions can compete or cooperate under finite user attention, producing critical spreading behavior and heavy-tailed popularity patterns.

  • Coevolution of multiple social contagions: Finite screen and memory capacities make memes compete for users’ limited attention during diffusion.Shorter retention windows increase competition, whereas longer windows decrease it.
  • Coevolution of multiple social contagions: Meme popularity growth follows a critical branching process, yielding very heavy-tailed popularity distributions when innovation vanishes.The asymptotic result applies in the zero-innovation limit, µ = 0.
  • Coevolution of multiple social contagions: Generalized models with memory times and heterogeneous activity rates remain analytically solvable and reproduce time-dependent heavy-tailed hashtag popularity.
  • Coevolution of multiple social contagions: Competing contagions decrease one another’s transmission rates, whereas cooperating contagions increase them.Evaluation on 18,000 simultaneously spreading Twitter contagions found a 71% average relative change in spreading probability from contagion interactions.

3.4. Summary

Social contagions commonly exhibit reinforcement, in which repeated contacts are needed to trigger adoption and affect cascade-size transitions.

  • 3.4. Summary: Social contagions across online and offline settings exhibit social reinforcement effects.Accumulated multiple contacts between adopted and susceptible individuals can be necessary to trigger infection.

4. Coevolution of awareness diffusion and epidemic spreading

Awareness and epidemics coevolve asymmetrically: awareness suppresses epidemics, while epidemics promote awareness diffusion. Network topology, multiplex coupling, delays, and behavioral costs shape thresholds and outbreak outcomes.

  • 4. Coevolution of awareness diffusion and epidemic spreading: Disease information can prompt protective actions such as masking or staying home, which suppress epidemic outbreaks.
  • 4.1. Empirical analyses: ILI visits and Google Flu Trends showed similar macroscopic trends and matching relative-growth patterns over nearly 200 weeks in the United States.Cross-correlations were positive for Wl = 3 and negative for Wl = 20.
  • 4.3. Coevolution on multiplex networks: The epidemic threshold depends on physical-network topology and awareness outbreak size; without an awareness outbreak, it reduces to the baseline epidemic threshold.
  • 4.3. Coevolution of awareness and epidemics on multiplex networks: The UAU+SIS model has a topology- and awareness-dependent metacritical point, with MMCA averaging approximately 98% accuracy.The metacritical point is bounded by [0, 1/Λmax(A)] × [0, 1/Λmax(B)].
  • 4.3. Coevolution of awareness and epidemics on multiplex networks: Heterogeneous awareness–epidemic interactions can produce hybrid phase transitions combining continuous transitions and bistable states.
  • 4.3. Coevolution of awareness and epidemics on multiplex networks: Overlap edges promote epidemic outbreaks when the social reinforcement strength α is positive by decreasing the epidemic threshold.For α = 0, the threshold equals the classical epidemic threshold.
  • 4.4. Summary: Immunization decisions can depend on both communication-layer awareness and epidemic conditions, rather than current perceptive awareness alone.A non-Markovian extension instead uses cumulative awareness to determine adoption probability.
  • 4.2. Coevolution on single networks: Awareness diffusion increases the epidemic outbreak threshold, while epidemic spreading promotes awareness diffusion.

5. Coevolution of resource diffusion and epidemic spreading

Resource diffusion and epidemic spreading coevolve through global inputs, local production, individual behavior, and network-layer structure. Limited or uneven resources can generate abrupt epidemic transitions, while allocation and behavioral rules alter vaccination, infection, and control outcomes.

  • Summary: Resource input can reduce transmission or improve recovery, but limited resources may trigger abrupt phase transitions and catastrophic outbreaks after slight shortages.Inadequate resource production can similarly cause abrupt outbreaks in coevolving systems.
  • Global resources: Multiple stable epidemic states and a critical resource value Rc arise, with widespread epidemics occurring only when R < Rc.The transition regime depends on the infection rate β, which separates discontinuous, hybrid, and continuous phase transitions.
  • Multilayer resources: Time delays between network layers can induce discontinuous, continuous, or hybrid transitions, and sufficiently long delays can control spreading even with little resource.The outcome depends on resource amount, interlayer infectious strength, and internal network structure.
  • Behavioral resources: Behavioral incentives create policy-dependent vaccination responses: partial subsidies vary monotonically with rationality, whereas free subsidies produce nonmonotonic dependence.Preferential imitation benefits targeted subsidies only when individuals favor subsidized neighbors; continuous vaccination strategies can lower both vaccination and epidemic levels when vaccination is cheap.
  • Individual resources: Equal resource distribution can maximize the outbreak threshold and minimize infected fractions, whereas degree-biased allocation can produce discontinuous transitions in homogeneous networks.Heterogeneous networks instead retain discontinuous transitions under the studied resource distributions.

6. Conclusions and outlooks

Coevolution spreading dynamics cannot generally be understood through single-spreading analysis because interacting processes produce distinctive collective phenomena. The review organizes the field’s current progress and identifies empirical, topological, theoretical, and control-related priorities for future work.

  • Single-spreading analysis cannot explain phenomena such as epidemic containment by induced awareness, discontinuous viral growth, or bistable outbreak sizes under limited resources.
  • The review organizes coevolution spreading into biological and social contagions, plus epidemic–awareness and epidemic–resource processes.
  • Empirical data are needed to determine whether currently modeled interactions, including synergy, competition, and asymmetry, capture real coevolution dynamics.
  • Network topology substantially affects coevolution spreading, while common analytical assumptions include large, sparse, local-tree-like, or static networks.
  • Human memory, burstiness, and mobility heterogeneity can alter interaction patterns and transmissibility, motivating temporal and spatial network models.
  • Influential-node identification must be redefined for coevolution because nodes that promote one process may inhibit spreading under asymmetrical interactions.
  • Existing methods designed for single spreading dynamics cannot generally handle the stronger correlations of coevolution, motivating approaches beyond message passing.
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