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

===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>.

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 />

<math>p_j={A_{ji} \over k_i}</math> <br />

<math>p_j={A_{ji} \over k_i}</math> <br />

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

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