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More in general models of social behavior and human mobility are often framed as reaction-diffusion processes where each node $i$ is allowed to host any nonnegative integer number of particles $\mathcal{N}(i)$, so that the total particle population of the system is $\mathcal{N}=\sum_i\mathcal{N}(i)$. This particle-network framework considers that each particle diffuses along the edges connecting nodes with a diffusion coefficient that depends on the node degree and/or other node attributes. Within each node particles may react according to different schemes characterizing the interaction dynamics of the system. A simple sketch of the particle-network framework is represented in the Figure.

More in general models of social behavior and human mobility are often framed as reaction-diffusion processes where each node $i$ is allowed to host any nonnegative integer number of particles $\mathcal{N}(i)$, so that the total particle population of the system is $\mathcal{N}=\sum_i\mathcal{N}(i)$. This particle-network framework considers that each particle diffuses along the edges connecting nodes with a diffusion coefficient that depends on the node degree and/or other node attributes. Within each node particles may react according to different schemes characterizing the interaction dynamics of the system. A simple sketch of the particle-network framework is represented in the Figure.

一般来说，社会行为和人类流动性模型经常被构建为反应扩散过程，其中每个节点$i$可以容纳任何非负整数$\mathcal{N}(i)$个粒子，因此系统的总粒子数为$\mathcal{N}=\sum_i\mathcal{N}(i)$。 该粒子——网络框架中的节点内的每个粒子只能沿着连接节点的连边扩散，扩散系数取决于节点度或其他节点属性等。不同系统里的节点内的粒子的反应规则不同。粒子——网络的框架简单示意图如图所示。

 

一般来说，社会行为和人类流动性模型经常被构建为反应扩散过程，其中每个节点$i$可以容纳任何非负整数$\mathcal{N}(i)$个粒子，因此系统的总粒子数为<math>\mathcal{N}=\sum_i\mathcal{N}(i)</math>。 该粒子——网络框架中的节点内的每个粒子只能沿着连接节点的连边扩散，扩散系数取决于节点度或其他节点属性等。不同系统里的节点内的粒子的反应规则不同。粒子——网络的框架简单示意图如图所示。

 

 

The first term of the equation considers that only a fraction $d_k$ of particles moves out of the node per unit time. The second term instead accounts for the particles diffusing from the neighbors into the node of degree k. This term is proportional to the number of links k times the average number of particles coming from each neighbor. This is equal to the average over all possible degrees $k'$ of the fraction of particles moving on that edge, $d_{k'k}$, according to the conditional probability $P(k'|k)$ that an edge belonging to a node of degree k is pointing to a node of degree $k'$. Here the term $d_{k'k}$ is the diffusion rate along the edges connecting nodes of degree k and $k'$. In the simplest case of homogeneous diffusion each particle diffuses with rate $r$ from the node in which it is and thus the diffusion per link $d_{k'k}=r/k'$. On uncorrelated networks $P(k'|k)=k'P(k')/\left<k\right>$ and hence one easily gets in the stationary state $d\mathcal{N}/dt=0$ the solution <ref>Noh J D, Rieger H. Random walks on complex networks[J]. Physical review letters, 2004, 92(11): 118701.</ref><ref>Colizza V, Pastor-Satorras R, Vespignani A. Reaction–diffusion processes and metapopulation models in heterogeneous networks[J]. Nature Physics, 2007, 3(4): 276-282.</ref>

The first term of the equation considers that only a fraction $d_k$ of particles moves out of the node per unit time. The second term instead accounts for the particles diffusing from the neighbors into the node of degree k. This term is proportional to the number of links k times the average number of particles coming from each neighbor. This is equal to the average over all possible degrees $k'$ of the fraction of particles moving on that edge, $d_{k'k}$, according to the conditional probability $P(k'|k)$ that an edge belonging to a node of degree k is pointing to a node of degree $k'$. Here the term $d_{k'k}$ is the diffusion rate along the edges connecting nodes of degree k and $k'$. In the simplest case of homogeneous diffusion each particle diffuses with rate $r$ from the node in which it is and thus the diffusion per link $d_{k'k}=r/k'$. On uncorrelated networks $P(k'|k)=k'P(k')/\left<k\right>$ and hence one easily gets in the stationary state $d\mathcal{N}/dt=0$ the solution <ref>Noh J D, Rieger H. Random walks on complex networks[J]. Physical review letters, 2004, 92(11): 118701.</ref><ref>Colizza V, Pastor-Satorras R, Vespignani A. Reaction–diffusion processes and metapopulation models in heterogeneous networks[J]. Nature Physics, 2007, 3(4): 276-282.</ref>

方程中的第一项表示每单位时间内有$d_k$部分的粒子从度为$k$的节点中离开，第二项表示粒子从邻居节点扩散到度为$k$的节点中，该项与连边数$k$成正比。条件概率$P(k'|k)$表示的是一条边的一端连接度为$k$的节点时，其另一端指向度为$k'$的概率。$d_{k'k}$表示的是两端连接了度为$k$和$k'$节点的连边的扩散率。在均匀扩散的最简单情况下，每个粒子从其所在的节点以速率$r$扩散，因此沿着每条连边的扩散率为$d_{k'k}=r/k'$。在无关联的网络上，$P(k'|k)=k'P(k')/\left<k\right>$，因此在稳态$d\mathcal{N}/dt=0$时，很容易得到解（Colizza等，2007b； Noh和Rieger，2004）：

<nowiki>方程中的第一项表示每单位时间内有$d_k$部分的粒子从度为$k$的节点中离开，第二项表示粒子从邻居节点扩散到度为$k$的节点中，该项与连边数$k$成正比。条件概率$P(k'|k)$表示的是一条边的一端连接度为$k$的节点时，其另一端指向度为$k'$的概率。$d_{k'k}$表示的是两端连接了度为$k$和$k'$节点的连边的扩散率。在均匀扩散的最简单情况下，每个粒子从其所在的节点以速率$r$扩散，因此沿着每条连边的扩散率为$d_{k'k}=r/k'$。在无关联的网络上，$P(k'|k)=k'P(k')/\left<k\right>$，因此在稳态$d\mathcal{N}/dt=0$时，很容易得到解（Colizza等，2007b； Noh和Rieger，2004）：

\begin{equation}

\begin{equation}

\mathcal{N}_k=\frac{k}{\left<k\right>}\frac{\mathcal{N}}{N}

\mathcal{N}_k=\frac{k}{\left<k\right>}\frac{\mathcal{N}}{N}

\end{equation}

\end{equation}

The above equation explicitly brings the diffusion of particles in the description of the system and points out the importance of network topology in reaction-diffusion processes. This expression indicates that the larger the degree of a node, the larger the probability to be visited by the diffusing particles.

The above equation explicitly brings the diffusion of particles in the description of the system and points out the importance of network topology in reaction-diffusion processes. This expression indicates that the larger the degree of a node, the larger the probability to be visited by the diffusing particles.</nowiki>

上式显式地给出了系统中粒子扩散的描述，并指出网络拓扑结构在反应扩散过程中的重要性。从式中可以看到，一个节点的度越大，则其被扩散中的粒子访问的可能性就越大。

上式显式地给出了系统中粒子扩散的描述，并指出网络拓扑结构在反应扩散过程中的重要性。从式中可以看到，一个节点的度越大，则其被扩散中的粒子访问的可能性就越大。

== 亚种群模型 ==

==亚种群模型==

The above approach can be generalized to reacting particles with different states by adding a reaction term to the above equations <ref>Colizza V, Pastor-Satorras R, Vespignani A. Reaction–diffusion processes and metapopulation models in heterogeneous networks[J]. Nature Physics, 2007, 3(4): 276-282.</ref>. We now describe a generalization to this setting of the standard SIS model in discrete time, with probability per unit time $\beta$ of infection and probability $\mu$ of recovery. We consider $\mathcal{N}$ individuals diffusing in a heterogeneous network with $N$ nodes and degree distribution $P(k)$.

The above approach can be generalized to reacting particles with different states by adding a reaction term to the above equations <ref>Colizza V, Pastor-Satorras R, Vespignani A. Reaction–diffusion processes and metapopulation models in heterogeneous networks[J]. Nature Physics, 2007, 3(4): 276-282.</ref>. We now describe a generalization to this setting of the standard SIS model in discrete time, with probability per unit time $\beta$ of infection and probability $\mu$ of recovery. We consider $\mathcal{N}$ individuals diffusing in a heterogeneous network with $N$ nodes and degree distribution $P(k)$.

其中$\bar{S}(t)=\sum_k P(k)S_k$。

其中$\bar{S}(t)=\sum_k P(k)S_k$。

== 参考文献 References ==   

==参考文献 References==   

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