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An SICR rumor spreading model in heterogeneous networks

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  • This article discusses the spread of rumors in heterogeneous networks. Using the probability generating function method and the approximation theory, we establish an SICR rumor model and calculate the threshold conditions for the outbreak of the rumor. We also compare the speed of the rumors spreading with different initial conditions. The numerical simulations of the SICR model in this paper fit well with the stochastic simulations, which means that the model is reliable. Moreover the effects of the parameters in the model on the transmission of rumors are studied numerically.

    Mathematics Subject Classification: Primary: 05C80, 37N25; Secondary: 92D25.

    Citation:

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  • Figure 1.  Structure of the rumor spreading process

    Figure 2.  Simulation with $ \alpha = 0.7 $, $ \beta = 0.1 $, $ \delta = 0.2 $, $ \gamma = 0.5 $, $ \eta = 0.8 $. The blue dotted lines correspond to 100 random simulations for an SICR rumor model in a network with the Poisson degree distribution. The red solid lines show the numerical simulation based on the model (1). The green solid lines show the numerical simulation based on the mean field system in Zan [23]. (a) the trajectories of the densities of the infective. (b) the trajectories of the densities of the final size

    Figure 3.  Degree distribution of generated Power-law networks

    Figure 4.  Simulations with $ \alpha = 0.7 $, $ \beta = 0.1 $, $ \delta = 0.2 $, $ \gamma = 0.5 $, $ \eta = 0.8 $. The blue dotted lines correspond to 100 random simulations for an SICR rumor model in a network with the power law degree destribution. The red solid lines show the numerical simulation based on the model (1). The green solid lines show the numerical simulation based on the mean field system in Zan [23]. (a) the trajectories of the densities of the infective. (b) the trajectories of the densities of the final size

    Figure 5.  Numerical simulations of the model (1) with $ \alpha = 0.7 $, $ \beta = 0.1 $, $ \delta = 0.2 $, $ \gamma = 0.5 $, $ \eta = 0.8 $ in the networks with different degree distribution but with the same averaged degree. The red dotted lines and the blue solid lines correspond to numerical simulations trajectories based on the model (1) in a network with the refined power-law degree distribution and the Poisson degree distribution, respectively. (a) the trajectories of the densities of the infective. (b) the trajectories of the densities of the final size

    Figure 6.  (a) Time evolutions of the speeds of the rumor spreading $ v $ in the complex network with poisson degree distribution and Power-law distribution, respectively. (b) Degree distriutions

    Figure 7.  Time evolutions of the speeds of the rumor spreading in the nodes with degree $ k = 5, 10, 15. $, i.e. $ v_k $. (a) In network with Poisson ditribution. (b) In network with Power Law distribution

    Figure 8.  Time evolutions of the relative propagation velocity in the nodes with degree $ k = 5, 10, 15. $, i.e. $ \hat{v}_k $. (a) In network with Poisson ditribution. (b) In network with Power Law distribution

    Figure 9.  numerical simulations with $ \alpha = 0.7 $, $ \beta = 0.1 $, $ \delta = 0.2 $, $ \gamma = 0.2 $. (a) the trajectories of the densities of the infective over time under different persuading rate $ \eta $. (b) the trajectories of the densities of the final size over time under different persuading rate $ \eta $

    Figure 10.  Numerical simulations with $ \beta = 0.2 $, $ \gamma = 0.5 $, $ \eta = 0.8 $. (a) the trajectories of the densities of the susceptible over time under different spreading rate $ \alpha $ and refuting rate $ \delta $. (b) the trajectories of the densities of the infective over time under different spreading rate $ \alpha $ and refuting rate $ \delta $. (c) the trajectories of the densities of the final size over time under different spreading rate $ \alpha $ and refuting rate $ \delta $

    Figure 11.  Numerical simulation of SIR model that compare the infective nodes recover by themselves and recover by the structure of the network. (a) the trajectories of the densities of the susceptible. (b) the trajectories of the densities of the infective. (c) the trajectories of the densities of the final size

    Table 1.  Key variables and parameters

    Series Symbol Series Description
    $ \alpha $ Spreading rate. The constant rate at which a susceptible node
    becomes an infective node when it contacts an infective node
    $ \beta $ Ignoring rate. The constant rate at which a susceptible node becomes
    a refractory node when it contacts an infective node
    $ \delta $ Refuting rate. The constant rate at which a susceptible node becomes
    a counterattack node when it contacts an infective node
    $ \gamma $ Stifling rate. The constant rate at which an infective node becomes
    a refractory node when it contacts another infective or
    refractory node
    $ \eta $ Persuading rate. The constant rate at which an infective node
    becomes a refractory node when it contacts a counterattack node
    $ p_k $ The probability that a node will have degree $ k $
    $ g(x) $ The probability generating function for the degree distribution $ \{p_k\} $
    $ P_X^Y $ The probability that an arc with an ego in set X has an alter in Y
    $ \mathcal{A}_X $ Set of arcs (ego, alter) such that node ego is in set $ X $
    $ M_X $ Fraction of arcs in set $ \mathcal{A}_X $
    $ \mathcal{A}_{XY} $ Set of arcs (ego, alter) s.t ego$ \in X $ and alter$ \in Y $
    $ M_{XY} $ Fraction of arcs in set $ \mathcal{A}_{XY} $
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