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Theory of Rumour Spreading in Complex Social Networks
Maziar Nekovee, Y. Moreno, G. Bianconi, M. Marsili
TL;DR
The paper addresses how rumour dynamics should be modeled on complex social networks rather than simplified homogeneous populations. It introduces a stochastic network model and derives mean-field equations, finding thresholds and spreading rates that depend strongly on topology and degree correlations. The results indicate that scale-free networks are especially prone to rumour spreading, with implications for Internet-mediated dissemination processes.
Problem
Existing rumour models provide limited quantitative treatment of network topology, although large Internet-mediated social networks have highly heterogeneous connectivity.
Method
The paper introduces a stochastic rumour model on networks, formulates it with Interactive Markov Chains, and derives mean-field equations for several network classes and degree correlations.
Results
The model has a spreading-rate threshold in ER networks, a vanishing threshold for infinite scale-free networks, faster initial spreading on scale-free networks, and further acceleration from assortative correlations.
Takeaways & Limitations
Scale-free social networks are prone to rumour spreading, with findings relevant to chain emails, viral advertising, and large-scale Internet information dissemination.
Takeaways & Limitations
The analysis assumes a static, time-independent network topology, whereas many social and communication networks are dynamic.
Abstract
from arXiv · showhide
We introduce a general stochastic model for the spread of rumours, and derive mean-field equations that describe the dynamics of the model on complex social networks (in particular those mediated by the Internet). We use analytical and numerical solutions of these equations to examine the threshold behavior and dynamics of the model on several models of such networks: random graphs, uncorrelated scale-free networks and scale-free networks with assortative degree correlations. We show that in both homogeneous networks and random graphs the model exhibits a critical threshold in the rumour spreading rate below which a rumour cannot propagate in the system. In the case of scale-free networks, on the other hand, this threshold becomes vanishingly small in the limit of infinite system size. We find that the initial rate at which a rumour spreads is much higher in scale-free networks than in random graphs, and that the rate at which the spreading proceeds on scale-free networks is further increased when assortative degree correlations are introduced. The impact of degree correlations on the final fraction of nodes that ever hears a rumour, however, depends on the interplay between network topology and the rumour spreading rate. Our results show that scale-free social networks are prone to the spreading of rumours, just as they are to the spreading of infections. They are relevant to the spreading dynamics of chain emails, viral advertising and large-scale information dissemination algorithms on the Internet.
1 Introduction
Existing rumour models often simplify or omit the topology of large Internet-mediated social networks. This paper introduces a network-based model and examines how network structure, degree correlations, and spreading rate shape rumour dynamics.
- Motivation: Homogeneous-mixing and simplified-topology models inadequately represent large social networks, particularly Internet-mediated networks.Such networks commonly have highly right-skewed degree distributions with many highly connected nodes.
- Contribution: The paper derives mean-field equations for a general rumour model on homogeneous, random, uncorrelated scale-free, and assortatively correlated scale-free networks.The analysis combines approximate analytical and exact numerical solutions to study steady-state and time-dependent behavior.
- Results: The model shows a critical spreading-rate threshold on networks with bounded degree fluctuations, whereas this behavior is absent on scale-free networks with unbounded fluctuations.The threshold separates conditions under which a rumour can or cannot propagate in the bounded-fluctuation cases.
- Results: The initial spreading rate is much higher on scale-free networks than on random graphs and increases further with assortative degree correlations.The final fraction of nodes that hear the rumour depends on the interaction between model parameters and degree-degree correlations.
- Applications: The findings apply to chain emails, Internet hoaxes, viral advertising, and large-scale information dissemination through gossip protocols.The introduction also connects rumour-like mechanisms with public opinion, financial markets, panic, and marketing.
2 A general model for rumour dynamics on social networks
The paper models rumour spreading as a stochastic process on an explicit social network, using plausible behavioural rules for transmission and cessation. Its rules extend standard rumour dynamics by representing both stifling and forgetting.
- Model motivation: The model represents a complex socio-psychological process through a set of simple, plausible behavioural rules.The authors aim to formalize behavioural mechanisms while simplifying the underlying process.
- State model: Individuals are classified as ignorants, spreaders, or stiflers, and spreaders contact others only along links of an undirected social network G = (V, E).The network vertices and edges represent individuals and social connections, respectively.
- Transition rules: A spreader converts an ignorant into a spreader at rate λ, while contact with another spreader or stifler turns the initiating spreader into a stifler at rate α.These rules encode acceptance of a rumour and cessation after contacts with informed individuals.
- Cessation mechanisms: Unlike standard Daley-Kendall and Maki-Thompson variants, the model includes forgetting or disinclination to spread as a second cessation mechanism alongside stifling.The authors describe stifling and forgetting as distinct mechanisms that can end rumour spreading.
3 Interactive Markov chain mean-field equations
The model’s network dynamics are formulated as Interactive Markov Chains and then coarse-grained by node degree. This yields deterministic mean-field equations whose network dependence enters through degree correlations and neighbour-state probabilities.
- Interactive Markov-chain formulation: Interactive Markov Chains represent each node with an internal Markov chain whose transition probabilities depend on connected nodes’ states.The full system is a global Markov chain, but its exponentially large state space makes direct numerical solution difficult for large networks.
- Node transitions: For an ignorant node, the framework tracks probabilities of remaining ignorant or transitioning to the spreader state during a time interval.The number g denotes the node’s spreader neighbours at time t.
- Degree-class approximation: The microdynamics are coarse-grained by degree, averaging transition probabilities over nodes with the same connectivity.For a degree-k node, g is treated as a stochastic variable with a binomial distribution.
- Network dependence: The probability that an edge from a degree-k ignorant points to a spreader is expressed using degree correlations and spreader densities across connectivity classes.The approximation ignores dynamic correlations between neighbouring node states.
- Mean-field equations: Transition probabilities generate coupled Chapman-Kolmogorov equations and, after neglecting fluctuations around expected values, deterministic rate equations for ignorants, spreaders, and stiflers.The degree-class densities satisfy ρ_i(k, t) + ρ_s(k, t) + ρ_r(k, t) = 1.
- Implementation: The equations require only an analytical or numerical degree-degree correlation function, so the reported studies do not need to generate an actual network.The underlying network enters the equations solely through the degree-degree correlation function.
4 Analysis
The analysis derives mean-field rumour equations for homogeneous and inhomogeneous networks, then examines thresholds and final rumour size analytically. Forgetting creates a finite threshold in homogeneous networks, while in uncorrelated inhomogeneous networks the threshold follows the degree distribution and matches the SIR threshold to leading order.
- Homogeneous networks: The homogeneous-network equations use a constant degree or average degree when connectivity fluctuations decay exponentially.The analysis assumes negligible degree correlations and small degree fluctuations in this case.
- Homogeneous networks: Analytical integration yields a transcendental equation for R, the final fraction of nodes that ever hear the rumour.The final state is defined by ρ_s(∞)=0.
- Homogeneous networks: A non-zero final rumour size exists only when ϵ > 1; with forgetting, this produces a finite spreading threshold.When δ = 0, the equations instead always admit a non-zero solution because ϵ = 1 + λ/α > 1.
- Homogeneous networks: The homogeneous-network threshold is independent of the stifling rate α and matches the SIR threshold because forgetting contributes first-order terms while stifling contributes second-order terms initially.Near the initial state, the model therefore reduces dynamically to SIR spreading.
5 Numerical results
Numerical analyses compare rumour spreading on Erdős-Rényi, uncorrelated scale-free, and assortatively correlated scale-free networks. Thresholds, spreading speed, and final rumour size differ substantially across topologies and depend on spreading and cessation parameters.
- Network models: The study numerically solves mean-field equations for Erdős-Rényi and scale-free networks, including assortative degree correlations.Networks use prescribed degree distributions and correlation functions; simulations examine final size and time-dependent spreading.
- Uncorrelated networks: The ER network has a critical threshold λc = 0.12507, independent of the stifling parameter α, below which rumours cannot spread.The threshold agrees with the analytical result.
- Uncorrelated networks: In finite-size scale-free networks, the threshold is much smaller, is approached with zero slope, and vanishes as system size becomes infinite.The threshold is therefore not intrinsic to rumour spreading on infinite scale-free networks.
- Dynamics: Scale-free networks spread rumours initially faster than ER networks; with stifling alone, hubs accelerate early spread but later impede it after becoming stiflers.Under stifling-only cessation, the final rumour size is higher on ER networks despite slower initial spreading on ER.
- Assortative correlations: Assortative correlations increase the initial spreading rate, while their effect on final rumour size reverses with the rumour spreading rate.Below λ ≈ 0.5, correlated networks reach fewer nodes; at larger λ, they reach more than uncorrelated networks.
6 Conclusions
The paper presents a general rumour model for complex networks and derives mean-field equations using an Interactive Markov Chain formulation. Its analyses identify topology-dependent thresholds and spreading dynamics, while highlighting static-network assumptions as a scope boundary.
- Conclusions: The model incorporates both stifling and forgetting as distinct mechanisms that can end rumour spreading.This extends earlier rumour models within a general complex-network framework.
- Conclusions: Interactive Markov Chains yield deterministic mean-field equations used to analyze rumour dynamics on ER and scale-free networks with exponent γ = 3.The paper combines analytical and numerical investigations of critical behavior, dynamics, and stationary states.
- Conclusions: Assortative degree correlations speed initial spreading on scale-free networks, but their effect on the final fraction hearing a rumour depends on the spreading rate.The correlation analysis uses a local ansatz for the degree-degree correlation function.
- Limitations: The analysis assumes a static, time-independent network topology, whereas many real social and communication networks are dynamic.The authors identify evolving networks as a subject for future work.