# American Institute of Mathematical Sciences

## An SICR rumor spreading model in heterogeneous networks

 1 School of Mathematical Sciences, Shanxi University, Taiyuan 030006, China 2 Shanxi Key Laboratory of Mathematical Techniques and Big Data Analysis on Disease Control and Prevention, Taiyuan 030006, China 3 Complex System Research Center, Shanxi University, Taiyuan 030006, China

* Corresponding author: lijinxian@sxu.edu.cn

Received  December 2018 Revised  July 2019 Published  November 2019

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.

Citation: Jinxian Li, Ning Ren, Zhen Jin. An SICR rumor spreading model in heterogeneous networks. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2019237
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Structure of the rumor spreading process
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
Degree distribution of generated Power-law networks
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
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
(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
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
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
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 $\eta$
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 $\delta$
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
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 $\mathcal{A}_{XY}$
 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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