﻿ 无标度网络中遗忘率变化的谣言传播模型研究
 文章快速检索 高级检索

1. 上海工程技术大学 管理学院, 上海 200444;
2. 上海交通大学 中美物流研究院, 上海 200030

Rumor spreading model with variable forgetting rate in scale-free network
WANG Xiao-li1, ZHAO Lai-jun2, XIE Wan-lin1
1. School of Management, Shanghai University of Engineering Science, Shanghai 200444, China;
2. Sino-US Global Logistics Institute, Shanghai Jiaotong University, Shanghai 200030, China
Abstract:This paper studies the rumor spreading model with a function of forgetting rate changing over time in scale-free networks. The corresponding mean-field equations are derived. Further, numerical simulations are conducted on Renren Network, an online social platform, to better understand the performance of the model. Results show that forgetting rate has a significant impact on the final size of rumor spreading: the larger the initial forgetting rate or the faster the forgetting speed, the smaller the final size of the rumor spreading; The final size of rumor spreading is much smaller under variable forgetting rate compared to that under a constant forgetting rate. Simulations also show that the spreading speed is faster and the final size is smaller in scale-free network than in homogeneous network.
Key words: rumor spreading model     scale-free network     homogeneous network     variable forgetting rate

0 引言

 图 1 遗忘率变化的谣言传播过程结构

 $S_{k1}(t-1)=I_k(t-2)-I_k(t-1)$ (2)

 $S_{k1}'(t-1)=S_{k1}(t-1)\cdot\bigg[k\eta \sum\limits_{k'}S_{k'}(t-1)P(k'/k)+k\gamma \sum\limits_{k'}R_{k'}(t-1)P(k'/k)\bigg]$ (3)

 $S_{k1}(t)=S_{k1}'(t-1)\cdot\delta(1)$ (4)

 $S_{k1}(t)=[I_k(t-2)-I_k(t-1)]\cdot\bigg[k\eta \sum\limits_{k'}S_{k'}(t-1)P(k'/k)+k\gamma \sum\limits_{k'}R_{k'}(t-1)P(k'/k)\bigg]\cdot\delta(1)$ (5)

2) $\tau=2$: 在$t-2$ 时刻转化而来的度为$k$ 的传播者密度为:
 $S_{k2}(t-2)=I_k(t-3)-I_k(t-2)$ (6)

 $S_{k2}'(t-2)=S_{k2}(t-2)\cdot\bigg[1-\bigg(k\eta \sum\limits_{k'}S_{k'}(t-2)P(k'/k)+k\gamma \sum\limits_{k'}R_{k'}(t-2)P(k'/k)\bigg)\bigg]$ (7)

 $S_{k2}(t-1)=S_{k2}'(t-2)\cdot(1-\delta(1))$ (8)

 $S_{k2}'(t-1)=S_{k2}(t-1)\cdot\bigg[1-\bigg(k\eta \sum\limits_{k'}S_{k'}(t-1)P(k'/k)+k\gamma \sum\limits_{k'}R_{k'}(t-1)P(k'/k)\bigg)\bigg]$ (9)

 $S_{k2}(t)=S_{k2}'(t-1)\cdot\delta(2)$ (10)

 \begin{aligned} S_{k2}(t)= &[I_k(t-3)-I_k(t-2)]\cdot\bigg[1- \bigg(k\eta \sum\limits_{k'}S_{k'}(t-2)P(k'/k)+k\gamma \sum\limits_{k'}R_{k'}(t-2)P(k'/k)\bigg)\bigg]\cdot(1-\delta(1)) \\ \\ &\cdot\bigg[1-\bigg(k\eta \sum\limits_{k'}S_{k'}(t-1)P(k'/k)+k\gamma \sum\limits_{k'}R_{k'}(t-1)P(k'/k)\bigg)\bigg]\cdot\delta(2)\\ \end{aligned} (11)

3) 同理可得$\tau=3,4,\cdots,t-1$ 时度为$k$ 的传播者密度变化.

4) $\tau=t$: 初始时刻的度为$k$ 的传播者密度,即$S_k(0)$. 其由于遗忘机制在$t$ 时刻转化为免疫者的密度变化可由以下公式获得:
 $S_{kt}(0)=S_k(0)$ (12)
 $S_{kt}'(0)=S_{kt}(0)\cdot\bigg[1-\bigg(k\eta \sum\limits_{k'}S_{k'}(0)P(k'/k)+k\gamma \sum\limits_{k'}R_{k'}(0)P(k'/k)\bigg)\bigg]$ (13)
 $S_{kt}(1)=S_{kt}'(0)\cdot(1-\delta(1))$ (14)
 $S_{kt}'(1)=S_{kt}(1)\cdot\bigg[1-\bigg(k\eta \sum\limits_{k'}S_{k'}(1)P(k'/k)+k\gamma \sum\limits_{k'}R_{k'}(1)P(k'/k)\bigg)\bigg]$ (15)
 $S_{kt}(2)=S_{kt}'(1)\cdot(1-\delta(2))$ (16)
 $S_{kt}'(2)=S_{kt}(2)\cdot\bigg[1-\bigg(k\eta \sum\limits_{k'}S_{k'}(2)P(k'/k)+k\gamma \sum\limits_{k'}R_{k'}(2)P(k'/k)\bigg)\bigg]$ (17)
 $$\cdot\cdot\cdot$$
 $S_{kt}(t-1)=S_{kt}'(t-2)\cdot(1-\delta(t-1))$ (18)
 $S_{kt}'(t-1)=S_{kt}(t-1)\cdot\bigg[1-\bigg(k\eta \sum\limits_{k'}S_{k'}(t-1)P(k'/k)+k\gamma \sum\limits_{k'}R_{k'}(t-1)P(k'/k)\bigg)\bigg]$ (19)
 $S_{kt}(t)=S_{kt}'(t-1)\cdot\delta(t)$ (20)

 \begin{aligned} S_{kt}(t)=&S_k(0)\cdot\bigg[1-\bigg(k\eta\sum\limits_{k'}S_{k'}(0)P(k'/k)+k\gamma\sum\limits_{k'}R_{k'}(0)P(k'/k) \bigg)\bigg](1-\delta(1))\\ &\cdot\bigg[1-\bigg(k\eta \sum\limits_{k'}S_{k'}(1)P(k'/k)+k\gamma \sum\limits_{k'}R_{k'}(1)P(k'/k)\bigg)\bigg]\\ &\cdot(1-\delta(2))\cdot\bigg[1-\bigg(k\eta \sum\limits_{k'}S_{k'}(2)P(k'/k)+k\gamma \sum\limits_{k'}R_{k'}(2)P(k'/k) \bigg)\bigg]\cdot\cdot\cdot(1-\delta(t-1))\\ &\cdot\bigg[1-\bigg(k\eta \sum\limits_{k'}S_{k'}(t-1)P(k'/k)+k\gamma \sum\limits_{k'}R_{k'}(t-1)P(k'/k)\bigg)\bigg]\cdot\delta(t) \end{aligned} (21)

 \begin{aligned} S_{k\delta}(t)=&\sum\limits_{\tau=2}^{t-1}\bigg\{[I_k(t-\tau-1)-I_k(t-\tau)]\prod\limits_{n=1}^{\tau}\bigg[1-\bigg(k\eta\sum\limits_{k'}S_{k'}(t-n)P(k'/k)\\ &+k\gamma \sum\limits_{k'}R_{k'}(t-n)P(k'/k)\bigg)\bigg]\cdot\prod\limits_{n=1}^{\tau-1}[1-\delta(n)]\cdot\delta(\tau)\bigg\}+[I_k(t-2)-I_k(t-1)]\\ &\cdot\bigg[1-\bigg(k\eta \sum\limits_{k'}S_{k'}(t-1)P(k'/k)+k\gamma \sum\limits_{k'}R_{k'}(t-1)P(k'/k)\bigg)\bigg]\cdot\delta(1)\\ &+S_k(0)\cdot\prod\limits_{n=1}^{t}\bigg[1-\bigg(k\eta \sum\limits_{k'}S_{k'}(t-n)P(k'/k)+k\gamma \sum\limits_{k'}R_{k'}(t-n)P(k'/k)\bigg)\bigg]\cdot\prod\limits_{n=1}^{t-1}[1-\delta(n)]\cdot\delta(t) \end{aligned} (22)

 $\frac{\mathrm{d}I_k(t)}{\mathrm{d}t}=-\lambda kI_k(t)\sum\limits_{k'}S_{k'}(t)P(k'/k)$ (23)
 \begin{aligned} &\frac{\mathrm{d}S_k(t)}{\mathrm{d}t}\\ =&\lambda kI_k(t)\sum\limits_{k'}S_{k'}(t)P(k'/k)-k\eta \sum\limits_{k'}S_{k'}(t)P(k'/k)-k\gamma \sum\limits_{k'}R_{k'}(t)P(k'/k)\\ &-\sum\limits_{\tau=2}^{t-1}\bigg\{[I_k(t-\tau-1)-I_k(t-\tau)] \prod\limits_{n=1}^{\tau}\bigg[1-\bigg(k\eta \sum\limits_{k'}S_{k'}(t-1)P(k'/k)+k\gamma \sum\limits_{k'}R_{k'}(t-1) P(k'/k)\bigg)\bigg]\\ &\cdot\prod\limits_{n=1}^{\tau-1}[1-\delta(n)]\cdot\delta(\tau)\bigg\}-[I_k(t-2)-I_k(t-1)] \bigg[1-\bigg(\eta \sum\limits_{k'}S_{k'}(t-1)+\gamma \sum\limits_{k'}R_{k'}(t-1)\bigg)kP(k'/k)\bigg]\cdot\delta(1)\\ &-S_k(0)\cdot\prod\limits_{n=1}^{t}\bigg[1-\bigg(\eta \sum\limits_{k'}S_{k'}(t-n)+\gamma \sum\limits_{k'}R_{k'}(t-n)\bigg)kP(k'/k)\bigg]\cdot\prod\limits_{n=1}^{t-1}[1-\delta(n)]\cdot\delta(t) \end{aligned} (24)
 \begin{aligned} \frac{\mathrm{d}R(t)}{\mathrm{d}t}=&k\eta \sum\limits_{k'}S_{k'}(t)P(k'/k)+k\gamma\sum\limits_{k'}R_{k'}(t)P(k'/k) +\sum\limits_{\tau=2}^{t-1}\bigg\{[I_k(t-\tau-1)-I_k(t-\tau)]\\ &\cdot\prod\limits_{n=1}^{\tau}\bigg[1-k\eta \sum\limits_{k'}S_{k'}(t-n)P(k'/k)-k\gamma \sum\limits_{k'}R_{k'}(t-n)P(k'/k) \bigg]\\ &\cdot\prod\limits_{n=1}^{\tau-1}[1-\delta(n)]\cdot\delta(\tau)\bigg\}+[I_k(t-2)-I_k(t-1)]\\ &\cdot\bigg[1-k\eta \sum\limits_{k'}S_{k'}(t-1)P(k'/k)-k\gamma \sum\limits_{k'}R_{k'}(t-1)P(k'/k)\bigg]\cdot\delta(1)\\ &+S_k(0)\cdot\prod\limits_{n=1}^{t}\bigg[1-k\eta\sum\limits_{k'}S_{k'}(t-n)P(k'/k)+k\gamma \sum\limits_{k'}R_{k'}(t-n)P(k'/k)\bigg]\cdot\prod\limits_{n=1}^{t-1}[1-\delta(n)]\cdot\delta(t) \end{aligned} (25)

 图 2 当$\langle k\rangle=18.74$,$\lambda=0.5$, $\eta=\gamma=0.5$,(a) $v=1.1$ 和(b) $\beta=0.2$ 时, 传播者密度$S(t)$ 随时间$t$ 的变化图

 图 3 当$\langle k\rangle=18.74$,$\lambda=0.5$, $\eta=\gamma=0.5$,(a) $v=1.1$ 和(b) $\beta=0.2$ 时,$Rs$ 随$v$ 的变化图

 图 4 当$\langle k\rangle=18.74$,$\lambda=0.5$, $\eta=\gamma=0.5$ 时,遗忘率为常量和变量两种情况下的$S(t)$ 和$R(t)$ 变化图

 图 5 当$\langle k\rangle=18.74$,$\lambda=0.5$, $v=1.1$,$\beta=0.2$,(a) $\gamma=0.6$ 和(b) $\eta=0.2$ 时, 免疫者密度$R(t)$ 随时间\\ $t$ 的变化图

 图 6 当$\lambda=0.5$,$\eta=\gamma=0.5$,$v=1.1$, $\beta=0.2$ 时,无标度网络和均匀网络中的$S(t)$ 和$R(t)$ 变化图
3 结论

1) 在考虑传播者转化为免疫者不同途径差异和遗忘率变化的基础上, 建立了无标度网络中遗忘率变化的谣言传播模型, 并通过对此谣言传播过程的分析得到了对应的平均场方程.

2) 用有限差分法在社交网络人人网上对谣言传播过程进行了数值模拟. 结果表明: 遗忘率函数中的参数变化对谣言传播的规模有显著影响, 初始遗忘率的增大会导致谣言传播的最终规模减小, 遗忘速度的增大也会导致谣言的影响范围变小; 相同条件下, 当遗忘率为变量时,社交网络中谣言传播的最终规模更小.

3) 数值结果还表明: 传播者遇到免疫者转变为免疫者的比率$\gamma$ 对谣言传播的最终规模影响更大,即增大参数$\gamma$ 的值可以有效地降低谣言传播的最终规模; 网络结构对谣言的传播也有重要影响,与均匀网络相比, 谣言在无标度网络中传播速度更快,最终规模更小.

0

#### 文章信息

WANG Xiao-li, ZHAO Lai-jun, XIE Wan-lin

Rumor spreading model with variable forgetting rate in scale-free network

Systems Engineering - Theory & practice, 2015, 35(2): 458-465.