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第131行： − + + + + + + + + + + + − <br/> 第139行：
第148行： − + 第153行：
第162行： − − − 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> X_i(t)</math>为t时刻的节点i的状态。那么<math>X(t)</math>是<math>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。 − − − 这样，我们定义一个函数<math>f</math> ，使得对于<math>S</math>中的<math>x</math> ，<math>f(x,i,j)</math> 是，当网络状态为<math>x</math>时，节点i和节点j相互作用，其中一个将改变其状态。 − − − 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>y=f(x,i,j)</math>，我们设法求<math>P(x,y)</math>。如果节点i处于状态I，节点j处于状态S，则<math>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>P(x,y)=\beta A_{ji}/k_i</math>。对于所有其他<math>y</math>，<math>P(x,y)=0</math>。 − −
→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> X_i(t)</math> to be the state of node i at time t. Then <math>X(t)</math> is a stochastic process on <math>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. Thus we define the function <math>f</math> so that for <math>x</math> in <math>S</math>,<math>f(x,i,j)</math> is when the state of network is <math>x</math>, node i and node j interact with each other, and one of them will change its state. 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>y=f(x,i,j)</math>, we try to find <math>P(x,y)</math>. If node i is in state I and node j is in state S, then <math>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>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>P(x,y)=\beta A_{ji}/k_i</math>.
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> X_i(t)</math> to be the state of node i at time t. Then <math>X(t)</math> is a stochastic process on <math>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. Thus we define the function <math>f</math> so that for <math>x</math> in <math>S</math>,<math>f(x,i,j)</math> is when the state of network is <math>x</math>, node i and node j interact with each other, and one of them will change its state. 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>y=f(x,i,j)</math>, we try to find <math>P(x,y)</math>. If node i is in state I and node j is in state S, then <math>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>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>P(x,y)=\beta A_{ji}/k_i</math>.
For all other <math>y</math>, <math>P(x,y)=0</math>.
For all other <math>y</math>, <math>P(x,y)=0</math>.<br/>
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> X_i(t)</math>为t时刻的节点i的状态。那么<math>X(t)</math>是<math>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。
这样，我们定义一个函数<math>f</math> ，使得对于<math>S</math>中的<math>x</math> ，<math>f(x,i,j)</math> 是，当网络状态为<math>x</math>时，节点i和节点j相互作用，其中一个将改变其状态。
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>y=f(x,i,j)</math>，我们设法求<math>P(x,y)</math>。如果节点i处于状态I，节点j处于状态S，则<math>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>P(x,y)=\beta A_{ji}/k_i</math>。对于所有其他<math>y</math>，<math>P(x,y)=0</math>。
The procedure on a network is as follows<ref name=Brockmann> Brockmann, D. 2011 Complex Networks and Systems, Lecture Notes, Northwestern University</ref> :
The procedure on a network is as follows<ref name=Brockmann> Brockmann, D. 2011 Complex Networks and Systems, Lecture Notes, Northwestern University</ref> :
1= We initial rumor to a single node <math>i</math>;
1= We initial rumor to a single node <math>i</math>;
2= We pick one of its neighbors as given by the [[adjacency matrix]], so the probability we will pick node <math>j</math> is <br />
2= We pick one of its neighbors as given by the [[adjacency matrix]], so the probability we will pick node <math>j</math> is <br />
<math>p_j={A_{ji} \over k_i}</math> <br />
<math>p_j={A_{ji} \over k_i}</math> <br />
4= We pick another node who is a spreader at random, and repeat the process.
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<ref name=Brockmann> Brockmann, D. 2011 Complex Networks and Systems, Lecture Notes, Northwestern University</ref> :
网络环境下的操作步骤如下<ref name=Brockmann> Brockmann, D. 2011 Complex Networks and Systems, Lecture Notes, Northwestern University</ref> :
操作表
操作表
1= 我们把谣言初始化赋予给节点 <math>i</math>;
1= 我们把谣言初始化赋予给节点 <math>i</math>;
2= 从[[临接矩阵（adjacency matrix）]]中，我们选择一个它的临近节点<math>j</math>, 它成为谣言传播者的概率是 <br />
2= 从[[临接矩阵（adjacency matrix）]]中，我们选择一个它的临近节点<math>j</math>, 它成为谣言传播者的概率是 <br />