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| | ==网络内容传播== | | ==网络内容传播== |
| − | Content in a [[complex network]] can spread via two major methods: conserved spread and non-conserved spread.<ref>Newman, M., Barabási, A.-L., Watts, D.J. [eds.] (2006) The Structure and Dynamics of Networks. Princeton, N.J.: Princeton University Press.</ref> In conserved spread, the total amount of content that enters a complex network remains constant as it passes through. The model of conserved spread can best be represented by a pitcher containing a fixed amount of water being poured into a series of funnels connected by tubes. Here, the pitcher represents the original source and the water is the content being spread. The funnels and connecting tubing represent the nodes and the connections between nodes, respectively. As the water passes from one funnel into another, the water disappears instantly from the funnel that was previously exposed to the water. In non-conserved spread, the amount of content changes as it enters and passes through a complex network. The model of non-conserved spread can best be represented by a continuously running faucet running through a series of funnels connected by tubes. Here, the amount of water from the original source is infinite. Also, any funnels that have been exposed to the water continue to experience the water even as it passes into successive funnels. The non-conserved model is the most suitable for explaining the transmission of most [[infectious diseases]].
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| | [[复杂网络]]中的内容主要通过两种方式传播：保守传播和非保守传播。<ref>Newman, M., Barabási, A.-L., Watts, D.J. [eds.] (2006) The Structure and Dynamics of Networks. Princeton, N.J.: Princeton University Press.</ref> 在保守传播中，进入复杂网络的内容总量在传播时保持不变。这个保守传播的模型可以用一个水罐来描述，这个水罐中有一定量的水被注入一系列由管子连接的漏斗中。在这里，水罐代表原始资源，而水则表示被传播的内容。漏斗和连接管分别表示节点和节点之间的连接。当水从一个漏斗流到另一个漏斗时，水立即从先前的漏斗中消失。在非保守传播中，内容的数量在进入和通过复杂网络时发生变化。非保守传播模型可以用一个持续流水的水龙头流过一系列由管子连接的漏斗来表示。在这里，来自原始水源的水量是无限的。而且，即使水已经进入下一个漏斗，任何之前已经接触过水的漏斗也会继续接触水。非保守模型最适合解释大多数[[传染病]]的传播。 | | [[复杂网络]]中的内容主要通过两种方式传播：保守传播和非保守传播。<ref>Newman, M., Barabási, A.-L., Watts, D.J. [eds.] (2006) The Structure and Dynamics of Networks. Princeton, N.J.: Princeton University Press.</ref> 在保守传播中，进入复杂网络的内容总量在传播时保持不变。这个保守传播的模型可以用一个水罐来描述，这个水罐中有一定量的水被注入一系列由管子连接的漏斗中。在这里，水罐代表原始资源，而水则表示被传播的内容。漏斗和连接管分别表示节点和节点之间的连接。当水从一个漏斗流到另一个漏斗时，水立即从先前的漏斗中消失。在非保守传播中，内容的数量在进入和通过复杂网络时发生变化。非保守传播模型可以用一个持续流水的水龙头流过一系列由管子连接的漏斗来表示。在这里，来自原始水源的水量是无限的。而且，即使水已经进入下一个漏斗，任何之前已经接触过水的漏斗也会继续接触水。非保守模型最适合解释大多数[[传染病]]的传播。 |
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| | ===SIR 模型=== | | ===SIR 模型=== |
| − | In 1927, W. O. Kermack and A. G. McKendrick created a model in which they considered a fixed population with only three compartments, susceptible: <math>S(t)</math>, infected, <math>I(t)</math>, and recovered, <math>R(t)</math>. The compartments used for this model consist of three classes:
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| − | * <math>S(t)</math> is used to represent the number of individuals not yet infected with the disease at time t, or those susceptible to the disease
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| − | * <math>I(t)</math> denotes the number of individuals who have been infected with the disease and are capable of spreading the disease to those in the susceptible category
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| − | * <math>R(t)</math> is the compartment used for those individuals who have been infected and then recovered from the disease. Those in this category are not able to be infected again or to transmit the infection to others.
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| − | The flow of this model may be considered as follows:
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| − | : <math>\mathcal{S} \rightarrow \mathcal{I} \rightarrow \mathcal{R} </math>
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| − | Using a fixed population, <math>N = S(t) + I(t) + R(t)</math>, Kermack and McKendrick derived the following equations:
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| − | : <math>
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| − | \begin{align}
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| − | \frac{dS}{dt} & = - \beta S I \\[8pt]
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| − | \frac{dI}{dt} & = \beta S I - \gamma I \\[8pt]
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| − | \frac{dR}{dt} & = \gamma I
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| − | \end{align}
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| − | </math>
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| − | Several assumptions were made in the formulation of these equations: First, an individual in the population must be considered as having an equal probability as every other individual of contracting the disease with a rate of <math>\beta</math>, which is considered the contact or infection rate of the disease. Therefore, an infected individual makes contact and is able to transmit the disease with <math>\beta N</math> others per unit time and the fraction of contacts by an infected with a susceptible is <math>S/N</math>. The number of new infections in unit time per infective then is <math>\beta N (S/N)</math>, giving the rate of new infections (or those leaving the susceptible category) as <math>\beta N (S/N)I = \beta SI</math> (Brauer & Castillo-Chavez, 2001). For the second and third equations, consider the population leaving the susceptible class as equal to the number entering the infected class. However, infectives are leaving this class per unit time to enter the recovered/removed class at a rate <math>\gamma</math> per unit time (where <math>\gamma</math> represents the mean recovery rate, or <math>1/\gamma</math> the mean infective period). These processes which occur simultaneously are referred to as the [[Law of mass action|Law of Mass Action]], a widely accepted idea that the rate of contact between two groups in a population is proportional to the size of each of the groups concerned (Daley & Gani, 2005). Finally, it is assumed that the rate of infection and recovery is much faster than the time scale of births and deaths and therefore, these factors are ignored in this model.
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| − | More can be read on this model on the [[Epidemic model]] page.
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| | 1927年， W. O. Kermack 和 A. G. McKendrick 建立了一个仅包含三种人群的固定人口模型，即易感者： <math>S(t)</math>，被感染者， <math>I(t)</math>和康复者 <math>R(t)</math>。该模型中使用的分类可分为三种： | | 1927年， W. O. Kermack 和 A. G. McKendrick 建立了一个仅包含三种人群的固定人口模型，即易感者： <math>S(t)</math>，被感染者， <math>I(t)</math>和康复者 <math>R(t)</math>。该模型中使用的分类可分为三种： |
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| | 更多详细信息请查询[[流行病模型]]页面。 | | 更多详细信息请查询[[流行病模型]]页面。 |
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| | ===主方程法=== | | ===主方程法=== |
| − | A [[master equation]] can express the behaviour of an undirected growing network where, at each time step, a new node is added to the network, linked to an old node (randomly chosen and without preference). The initial network is formed by two nodes and two links between them at time <math>t = 2</math>, this configuration is necessary only to simplify further calculations, so at time <math>t = n</math> the network have <math>n</math> nodes and <math>n</math> links.
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| − | The master equation for this network is:
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| − | : <math>p(k,s,t+1) = \frac 1 t p(k-1,s,t) + \left(1 - \frac 1 t \right)p(k,s,t),</math>
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| − | where <math>p(k,s,t)</math> is the probability to have the node <math>s</math> with degree <math>k</math> at time <math>t+1</math>, and <math>s</math> is the time step when this node was added to the network. Note that there are only two ways for an old node <math>s</math> to have <math>k</math> links at time <math>t+1</math>:
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| − | * The node <math>s</math> have degree <math>k-1</math> at time <math>t</math> and will be linked by the new node with probability <math>1/t</math>
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| − | * Already has degree <math>k</math> at time <math>t</math> and will not be linked by the new node.
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| − | After simplifying this model, the degree distribution is <math>P(k) = 2^{-k}. </math><ref name="dorogovtsev-mendes">{{cite book|last1=Dorogovtsev|first1=S N|last2=Mendes|first2=J F F|title=Evolution of Networks: From Biological Nets to the Internet and WWW|date=2003|publisher=Oxford University Press, Inc.|location=New York, NY, USA|isbn=978-0198515906}}</ref>
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| − | Based on this growing network, an epidemic model is developed following a simple rule: Each time the new node is added and after choosing the old node to link, a decision is made: whether or not this new node will be infected. The master equation for this epidemic model is:
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| − | : <math>p_r(k,s,t) = r_t \frac 1 t p_r(k-1,s,t) + \left(1 - \frac 1 t \right) p_r(k,s,t),</math>
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| − | where <math>r_t</math> represents the decision to infect (<math>r_t = 1</math>) or not (<math>r_t = 0</math>). Solving this master equation, the following solution is obtained: <math>\tilde{P}_r(k) = \left(\frac r 2 \right)^k. </math><ref name="cotacallapa-hase">{{cite journal|last1=Cotacallapa|first1=M|last2=Hase|first2=M O|title=Epidemics in networks: a master equation approach|journal=Journal of Physics A|date=2016|volume=49|issue=6|page=065001|doi=10.1088/1751-8113/49/6/065001|bibcode=2016JPhA...49f5001C|arxiv=1604.01049}}</ref>
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| − | [[主方程]]可以描述一个无向生长网络的行为，其中，每一个时间步长添加一个新节点，将其与一个已有节点相连（无偏好地随机选择）。在<math>t = 2</math>时刻，网络初始化为两个节点以及它们之间的两条边，这样的初始化是为了简化之后的计算。所以在<math>t = n</math>时刻，网络有<math>n</math>个节点和<math>n</math>条边。
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| − | 这个网络的主方程是：
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| − | : <math>p(k,s,t+1) = \frac 1 t p(k-1,s,t) + \left(1 - \frac 1 t \right)p(k,s,t),</math>
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| | 其中 <math>p(k,s,t)</math> 是 <math>t</math>时刻[[用户:Jxzhou|Jxzhou]]（[[用户讨论:Jxzhou|讨论]]）（原文中是t+1，我觉得应该是t）[[用户:Jxzhou|Jxzhou]]（[[用户讨论:Jxzhou|讨论]]）节点<math>s</math>的度为<math>k</math>的概率，<math>s</math>是该节点添加到网络中的时间步长。使旧节点<math>s</math>在<math>t+1</math>时刻的度为<math>k</math>的方法只有两种： | | 其中 <math>p(k,s,t)</math> 是 <math>t</math>时刻[[用户:Jxzhou|Jxzhou]]（[[用户讨论:Jxzhou|讨论]]）（原文中是t+1，我觉得应该是t）[[用户:Jxzhou|Jxzhou]]（[[用户讨论:Jxzhou|讨论]]）节点<math>s</math>的度为<math>k</math>的概率，<math>s</math>是该节点添加到网络中的时间步长。使旧节点<math>s</math>在<math>t+1</math>时刻的度为<math>k</math>的方法只有两种： |