社交网络上的谣言传播

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Rumor is an important form of social communications, and the spread of rumors plays a significant role in a variety of human affairs. There are two approaches to investigate the rumor spreading process: the microscopic models and the macroscopic models. The macroscopic models propose a macro view about this process are mainly based on the widely used Daley-Kendall and Maki-Thompson models. Particularly, we can view rumor spread as a stochastic process in social networks. While the microscopic models are more interested more on the micro interactions between individuals.

谣言（Rumor）是社会交往（social communication）的一种重要形式，谣言的传播在人类生活的各种事物中都发挥着重要作用。研究谣言传播过程有两种方法: 微观模型和宏观模型。宏观模型提出的宏观视角主要是基于广泛使用的戴利-肯德尔（Daley-Kendall）模型和马奇-汤普森（Maki-Thompson）模型。特别是，我们可以把谣言看作是社交网络中的随机过程（stochastic process）。而微观模型对个体间的微观相互作用更感兴趣。

谣言传播模型Rumor Propagation Models

In the last few years, there has been a growing interest in rumor propagation in Online social networks problems where different approaches have been proposed to investigate it. By carefully scrutinizing the existing literature, we categorize the works into macroscopic and microscopic approaches.

在过去的几年里，人们对在线社交网络中的谣言传播问题越来越感兴趣，人们提出了不同的方法来研究它。通过仔细研究现有文献，我们将这些著作分为宏观和微观两类方法。

宏观模型Macroscopic models

The first category is mainly based on the Epidemic models where the pioneering research engaging rumor propagation under these models started during the 1960s.

第一类主要基于传染病模型。通过这些模型，从事谣言传播的开创性研究始于1960年代。

传染病模型Epidemic models

A standard model of rumor spreading was introduced by Daley and Kendall,. Assume there are N people in total. And those people in the network are categorized into three groups: ignorants, spreaders and stiflers, which are denoted as I, S, and R respectively hereinafter:

一个谣言传播的标准模型是由戴利（Daley）和肯德尔（Kendall）。假设总共有N个人，传播网络中的这些人可以分为三类:无知者(ignorants)、传播者(spreaders)和谣言抑制者(stiflers)，分别在下文中表示为I、S和R：

• I: people who are ignorant of the rumor;
• S: people who actively spread the rumor;
• R: people who have heard the rumor, but no longer are interested in spreading it.

• I: 无知者(ignorants)，对谣言一无所知的人;
• S: 传播者(spreaders)，积极散布谣言的人;
• R: 谣言抑制者(stiflers)，听说过谣言但不再有兴趣散布谣言的人。

The rumor is propagated through the population by pair-wise contacts between spreaders and others in the population. Any spreader involved in a pair-wise meeting attempts to “infect” the other individual with the rumor. In the case this other individual is an ignorant, he or she becomes a spreader. In the other two cases, either one or both of those involved in the meeting learn that the rumor is known and decided not to tell the rumor anymore, thereby turning into stiflers.

通过传播者与跟他一对一接触的人，谣言在人群中传播。任何两人一对一时，传播者都试图用谣言“感染”另一个人。一种情况，另一个人是无知者时，（已经知道谣言的）他或她，就会成为一个传播者。另外两种情况，两个人都知道谣言，两个人或其中一个人，知道这个谣言是假的，并决定不再传播谣言，他们就会成为谣言抑制者。

One variant is the Maki-Thompson model .In this model, rumor is spread by directed contacts of the spreaders with others in the population. Furthermore, when a spreader contacts another spreader only the initiating spreader becomes a stifler. Therefore, three types of interactions can happen with certain rates.

[math]\displaystyle{ \begin{matrix}{}\\ S+I\xrightarrow{\alpha}2S \\{}\end{matrix} }[/math]

(1)

which says when a spreader meet an ignorant, the ignorant will become a spreader.

[math]\displaystyle{ \begin{matrix}{}\\ S+S\xrightarrow{\beta}S+R \\{}\end{matrix} }[/math]

(2)

which says when two spreaders meet with each other, one of them will become a stifler.

[math]\displaystyle{ \begin{matrix}{}\\ S+R\xrightarrow{\beta}2R \\{}\end{matrix} }[/math]

(3)

which says when a spreader meet a stifler, the spreader will lose the interest in spreading the rumor, so become a stifler.

一种变体是梅基-汤普森（Maki-Thompson）模型。在这个模型中，谣言是通过传播者与人群中其他人的直接接触来传播的。此外，当一个传播者遇到另一个传播者时，只有先前的传播者会成为谣言抑制者。这样，三种类型的相互作用可以以一定的速率发生。

1

({{{3}}})

就是说，当传播者遇到无知者时，无知者会成为一个传播者。

[math]\displaystyle{ \begin{matrix}{}\\ S+S\xrightarrow{\beta}S+R \\{}\end{matrix} }[/math]

(2)

就是说，当两个传播者相遇时，其中一个传播者会变成谣言抑制者。

[math]\displaystyle{ \begin{matrix}{}\\ S+R\xrightarrow{\beta}2R \\{}\end{matrix} }[/math]

(3)

就是说，当一个传播者遇到一个谣言抑制者时，他会失去传播谣言的兴趣，也成为一个谣言抑制者。

Of course we always have conservation of individuals:

[math]\displaystyle{ N=I+S+R }[/math]

当然，总人数是守恒的: [math]\displaystyle{ N=I+S+R }[/math]

The change in each class in a small time interval is:

[math]\displaystyle{ \Delta S \approx - \Delta t \alpha IS/N }[/math]

[math]\displaystyle{ \Delta I \approx \Delta t [{\alpha IS \over N} - {\beta I^2 \over N} - {\beta IR \over N}] }[/math]

[math]\displaystyle{ \Delta R \approx \Delta t [{\beta I^2 \over N}+{\beta IR \over N}] }[/math]

每类在很小时间间隔内的变化是:

[math]\displaystyle{ \Delta S \approx - \Delta t \alpha IS/N }[/math]

[math]\displaystyle{ \Delta I \approx \Delta t [{\alpha IS \over N} - {\beta I^2 \over N} - {\beta IR \over N}] }[/math]

[math]\displaystyle{ \Delta R \approx \Delta t [{\beta I^2 \over N}+{\beta IR \over N}] }[/math]

Since we know [math]\displaystyle{ S }[/math], [math]\displaystyle{ I }[/math] and [math]\displaystyle{ R }[/math] sum up to [math]\displaystyle{ N }[/math], we can reduce one equation from the above, which leads to a set of differential equations using relative variable [math]\displaystyle{ x=I/N }[/math] and [math]\displaystyle{ y=S/N }[/math] as follows

[math]\displaystyle{ {dx \over dt} = x \alpha y - \beta x^2 - \beta x(1-x-y) }[/math]

[math]\displaystyle{ {dy \over dt} = - \alpha xy }[/math]

which we can write

[math]\displaystyle{ {dx \over dt} = (\alpha + \beta)xy - \beta x }[/math]

[math]\displaystyle{ {dy \over dt} = - \alpha xy }[/math]

由于我们知道[math]\displaystyle{ S }[/math], [math]\displaystyle{ I }[/math] 的和 [math]\displaystyle{ R }[/math] 等于[math]\displaystyle{ N }[/math]，我们可以从上面去掉一个方程，用相对变量[math]\displaystyle{ x=I/N }[/math] 和 [math]\displaystyle{ y=S/N }[/math] 导出一组微分方程如下:

[math]\displaystyle{ {dx \over dt} = x \alpha y - \beta x^2 - \beta x(1-x-y) }[/math]

[math]\displaystyle{ {dy \over dt} = - \alpha xy }[/math]

可以写成：

[math]\displaystyle{ {dx \over dt} = (\alpha + \beta)xy - \beta x }[/math]

[math]\displaystyle{ {dy \over dt} = - \alpha xy }[/math]

Compared with the ordinary SIR model, we see that the only difference to the ordinary SIR model is that we have a factor [math]\displaystyle{ \alpha + \beta }[/math] in the first equation instead of just [math]\displaystyle{ \alpha }[/math]. We immediately see that the ignorants can only decrease since [math]\displaystyle{ x,y\ge 0 }[/math] and [math]\displaystyle{ {dy \over dt}\le 0 }[/math]. Also, if

[math]\displaystyle{ R_0={\alpha +\beta \over \beta} \gt 1 }[/math]

which means

[math]\displaystyle{ {\alpha \over \beta}\gt 0 }[/math]

the rumour model exhibits an “epidemic” even for arbitrarily small rate parameters.

与一般的SIR模型相比，我们发现与一般SIR模型的唯一区别在于，我们在第一个方程中有一个因子[math]\displaystyle{ \alpha + \beta }[/math]，而不仅仅是[math]\displaystyle{ \alpha }[/math]。我们立即看到，当[math]\displaystyle{ x,y\ge 0 }[/math] 和 [math]\displaystyle{ {dy \over dt}\le 0 }[/math]时，无知者开始减少。此外，如果:

[math]\displaystyle{ R_0={\alpha +\beta \over \beta} \gt 1 }[/math]

这意味着

[math]\displaystyle{ {\alpha \over \beta}\gt 0 }[/math]

即使在任意小的速率参数下，谣言模型也表现出“流行性”。

社会网络中的传染病模型Epidemic models in social network

We model the process introduced above on a network in discrete time, that is, we can model it as a DTMC. Say we have a network with N nodes, then we can define [math]\displaystyle{ X_i(t) }[/math] to be the state of node i at time t. Then [math]\displaystyle{ X(t) }[/math] is a stochastic process on [math]\displaystyle{ S=\{S,I,R\}^N }[/math]. At a single moment, some node i and node j interact with each other, and then one of them will change its state.

我们对上面介绍的过程，在离散时间的网络上建模，也就是说，我们可以将它建模为一个离散时间马尔可夫链（DTMC）。假设我们有一个N个节点的网络，那么我们可以定义[math]\displaystyle{ X_i(t) }[/math]为t时刻的节点 i 的状态。那么[math]\displaystyle{ X(t) }[/math]是[math]\displaystyle{ S=\{S,I,R\}^N }[/math]上的随机过程。在某一时刻，某个节点 i 和节点 j 相互作用，然后其中一个节点将改变其状态。

Thus we define the function [math]\displaystyle{ f }[/math] so that for [math]\displaystyle{ x }[/math] in [math]\displaystyle{ S }[/math],[math]\displaystyle{ f(x,i,j) }[/math] is when the state of network is [math]\displaystyle{ x }[/math], node i and node j interact with each other, and one of them will change its state.

这样，我们定义一个函数[math]\displaystyle{ f }[/math] ，使得对于[math]\displaystyle{ S }[/math]中的[math]\displaystyle{ x }[/math] ，[math]\displaystyle{ f(x,i,j) }[/math] 是，当网络状态为[math]\displaystyle{ x }[/math]时，节点 i 和节点 j 相互作用，其中一个将改变其状态。

The transition matrix depends on the number of ties of node i and node j, as well as the state of node i and node j. For any [math]\displaystyle{ y=f(x,i,j) }[/math], we try to find [math]\displaystyle{ P(x,y) }[/math]. If node i is in state I and node j is in state S, then [math]\displaystyle{ P(x,y)=\alpha A_{ji}/k_i }[/math]; if node i is in state I and node j is in state I, then [math]\displaystyle{ P(x,y)=\beta A_{ji}/k_i }[/math]; if node i is in state I and node j is in state R, then [math]\displaystyle{ P(x,y)=\beta A_{ji}/k_i }[/math]. For all other [math]\displaystyle{ y }[/math], [math]\displaystyle{ P(x,y)=0 }[/math].

转移矩阵依赖于节点 i 和节点 j 的联系数，以及节点 i 和节点 j 的状态。对于任意[math]\displaystyle{ y=f(x,i,j) }[/math]，我们设法求[math]\displaystyle{ P(x,y) }[/math]。如果节点 i 处于状态I，节点 j 处于状态S，则[math]\displaystyle{ P(x,y)=\alpha A_{ji}/k_i }[/math]; 如果节点 i 处于状态I，节点 j 处于状态S，则 p (x，y) = beta a _ { ji }/k _ i; 如果节点 i 处于状态I，节点 j 处于状态R，则[math]\displaystyle{ P(x,y)=\beta A_{ji}/k_i }[/math]。对于所有其他[math]\displaystyle{ y }[/math]，[math]\displaystyle{ P(x,y)=0 }[/math]。

The procedure on a network is as follows :

ordered list

1= We initial rumor to a single node [math]\displaystyle{ i }[/math];

2= We pick one of its neighbors as given by the adjacency matrix, so the probability we will pick node [math]\displaystyle{ j }[/math] is
[math]\displaystyle{ p_j={A_{ji} \over k_i} }[/math]
where [math]\displaystyle{ A_{ji} }[/math] is from the adjacency matrix and [math]\displaystyle{ A_{ji}=1 }[/math] if there is a tie from [math]\displaystyle{ i }[/math] to [math]\displaystyle{ j }[/math], and [math]\displaystyle{ k_i= \textstyle \sum_{j=1}^N A_{ij} }[/math] is the degree for node [math]\displaystyle{ i }[/math];

3= Then have the choice:

ordered list

3.1= If node [math]\displaystyle{ j }[/math] is an ignorant, it becomes a spreader at a rate [math]\displaystyle{ \alpha }[/math];

3.2= If node [math]\displaystyle{ j }[/math] is a spreader or stifler, then node [math]\displaystyle{ i }[/math] becomes a stifler at a rate [math]\displaystyle{ \beta }[/math].

4= We pick another node who is a spreader at random, and repeat the process.

网络环境下的操作步骤如下 :

操作表

1= 我们把谣言初始化赋予给节点 [math]\displaystyle{ i }[/math];

2= 从临接矩阵（adjacency matrix）中，我们选择一个它的临近节点[math]\displaystyle{ j }[/math], 它成为谣言传播者的概率是

[math]\displaystyle{ p_j={A_{ji} \over k_i} }[/math]

其中[math]\displaystyle{ A_{ji} }[/math] 来自于临接矩阵，且如果有从节点[math]\displaystyle{ i }[/math]到节点[math]\displaystyle{ j }[/math]的连接，那么[math]\displaystyle{ A_{ji}=1 }[/math] ，且[math]\displaystyle{ k_i= \textstyle \sum_{j=1}^N A_{ij} }[/math] 是节点[math]\displaystyle{ i }[/math]的度（Degree）（图论);

3= 然后选择:

操作表

3.1= 如果节点 [math]\displaystyle{ j }[/math] 是无知者，它成为一个传播者的速率是 [math]\displaystyle{ \alpha }[/math]；

3.2= 如果节点 [math]\displaystyle{ j }[/math] 是一个传播者或谣言抑制者, 那么节点[math]\displaystyle{ i }[/math] 成为一个谣言抑制者的速率是 [math]\displaystyle{ \beta }[/math]。

4= 我们随机选择一个是传播者的节点, 并重复这一过程。

小世界中的传染病模型Epidemic Models in Small-World Network

We would expect that this process spreads the rumor throughout a considerable fraction of the network. Note however that if we have a strong local clustering around a node, what can happen is that many nodes become spreaders and have neighbors who are spreaders. Then, every time we pick one of those, they will recover and can extinguish the rumor spread. On the other hand, if we have a network that is Small World, that is, a network in which the shortest path between two randomly chosen nodes is much smaller than that one would expect, we can expect the rumor spread far away.

我们可以预期这个过程会在网络的相当一部分中传播谣言。但是请注意，如果我们在一个节点周围有一个强大的本地集群（local clustering） ，那么可能发生的情况是，许多节点成为传播者，并且有作为传播者的邻居。然后，每次我们选择其中之一（辟谣），他们将恢复，并可以消除谣言传播。另一方面，如果我们有一个小世界（small world）的网络，也就是说，在一个网络中，两个随机选择的节点之间的最短路径比预期的要小得多，我们可以预期，谣言会传播得很广。

Also we can compute the final number of people who once spread the news, this is given by
[math]\displaystyle{ r_\infty=1-e^{-({\alpha +\beta \over \beta})r_\infty} }[/math]
In networks the process that does not have a threshold in a well mixed population, exhibits a clear cut phase-transition in small worlds. The following graph illustrates the asymptotic value of [math]\displaystyle{ r_\infty }[/math] as a function of the rewiring probability [math]\displaystyle{ p }[/math].

我们还可以计算最终传播消息的人数，他由下面的公式给出：
[math]\displaystyle{ r_\infty=1-e^{-({\alpha +\beta \over \beta})r_\infty} }[/math]
完全混合的人群网络中，一个没有阈值的传播过程，在小世界中表现出明显的相变。下图说明了[math]\displaystyle{ r_\infty }[/math]的渐近值作为重布线概率[math]\displaystyle{ p }[/math]的函数。

微观模型Microscopic models

The microscopic approaches attracted more attention in the individual's interaction: "who influenced whom." The known models in this category are the information cascade and the linear threshold models, the energy model, HISB model and Galam's Model.

微观方法在个体的相互作用中吸引了更多的注意力： “谁影响了谁”。这一类中，知名的模型有：信息级联模型（information cascade）和线性阈值模型，能量模型, HISB模型 ，以及格莱姆（Galam）模型。

HISB模型 HISB model

The HISB model is a rumor propagation model that can reproduce a trend of this phenomenon and provide indicators to assess the impact of the rumor to effectively understand the diffusion process and reduce its influence. The variety that exists in human nature makes their decision-making ability pertaining to spreading information unpredictable, which is the primary challenge to model such a complex phenomenon.Hence, this model considers the impact of human individual and social behaviors in the spreading process of the rumors.

HISB模型是一个谣言传播模型，它可以再现这种现象的趋势，并提供指标评估谣言的影响，从而有效地了解其扩散过程并减少其影响。人性中存在的多样性，使得人们传播信息的决策能力不可预测，这是对这样一个复杂现象建模的主要挑战。因此，该模型考虑了人类个体行为和社会行为对谣言传播过程的影响。

The HISB model proposes an approach that is parallel to other models in the literature and concerned more with how individuals spread rumors. Therefore, it try to understand the behavior of individuals, as well as their social interactions in OSNs, and highlight their impact on the dissemination of rumors. Thus, the model, attempts to answer the following question: "When does an individual spread a rumor? When does an individual accept rumors? In which OSN does this individual spread the rumors?.

HISB模型提出了一种与文献中其他模型平行的方法，更关注个人如何传播谣言。因此，它试图了解个人的行为，以及他们在在线社交网络（OSN）的社会互动，并突显其对谣言传播的影响。该模型试图回答以下问题:“一个人什么时候散布谣言?一个人什么时候会接受谣言？这个人在哪个社交网站上散布谣言。”

First, it proposes a formulation of individual behavior towards a rumor analog to damped harmonic motion, which incorporates the opinions of individuals in the propagation process. Furthermore, it establishes rules of rumor transmission between individuals. As a result, it presents the HISB model propagation process, where new metrics are introduced to accurately assess the impact of a rumor spreading through OSNs.

首先，它提出了一种类似于阻尼谐波运动的谣言个体行为公式，该公式在传播过程中纳入了个体的意见。此外，它还建立了个体间谣言传播的规则。就这样，它呈现了HISB模型传播过程，其中引入了新的指标，以准确评估谣言通过在线社交网络（OSN）传播的影响。

参考文献References

Category:Social networks

分类: 社交网络

This page was moved from wikipedia:en:Rumor spread in social network. Its edit history can be viewed at 社交网络上的谣言传播/edithistory