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为了对网络中的反应扩散系统进行理论上的分析描述，我们允许节点之间具有异质的连接模式。对这些系统的第一种分析方法是考虑将基于节点度的平均场理论方法Degree-based mean-field approach（简称为DBMF方法）扩展到具有任意度分布的反应扩散系统网络中。为简单起见，我们首先考虑在一个简化的系统上应用DBMF方法，这样具有简单网络结构的系统可以描述为：系统的粒子之间没有相互作用或反应，粒子只会在网络上扩散，并且网络可以具有任意的拓扑形式。以这样的描述构建这个系统，我们如果以度<math> k </math>作为量化指标定义分类的话，可以将系统简单表示为：
\begin{equation}
\begin{equation}
\mathcal{N}_k=\frac{1}{N_k}\sum_{i\in\mathcal{V}(k)}\mathcal{N}(i)
\mathcal{N}_k=\frac{1}{N_k}\sum_{i\in\mathcal{V}(k)}\mathcal{N}(i)
\end{equation}
\end{equation}
其中<math>N_k=NP(k)</math>表示度为k的节点数量，<math>\mathcal{V}(k)</math>表示度为$k$的节点集合，<math>\mathcal{N}_k</math>表示度为<math>k</math>的节点的平均粒子数。基于度的平均场理论假设度为<math>k</math>的节点，同理节点中的粒子，在统计上是等效的。在这种近似的假设下，粒子随机扩散的动力学可以由如下平均场理论方程表示：
其中<math>N_k=NP(k)</math>表示网络中度为k的节点数量，<math>\mathcal{V}(k)</math>表示度为<math>k</math>的节点集合，<math>\mathcal{N}_k</math>表示度为<math>k</math>的节点中包含的平均粒子数。基于度的平均场理论方法（DBMF）近似地假设度为<math>k</math>的节点是同质的，同理可近似认为节点中的粒子在统计上也是等效的。在这种近似的假设下，粒子随机扩散的动力学可以由如下平均场理论方程表示：
\begin{equation}
\begin{equation}
\frac{d\mathcal{N}_k}{dt}=-d_k\mathcal{N}_k(t)+k\sum_{k'}P(k'|k)d_{k'k}\mathcal{N}_{k'}(t)
\frac{d\mathcal{N}_k}{dt}=-d_k\mathcal{N}_k(t)+k\sum_{k'}P(k'|k)d_{k'k}\mathcal{N}_{k'}(t)
\end{equation}
\end{equation}
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在每个单位时间内有比例为<math> d_k </math>的粒子从度为<math> k </math>的节点中离开，第二项表示外部粒子从邻居节点扩散到度为<math> k </math>的节点中，该项与节点的连边数<math> k </math>成正比，也就是说一个节点与其邻居节点的连接越多，外部粒子扩散进入该节点的可能性越大。条件概率<math>P(k'|k)</math>表示的是一条边的一端连接度为<math>k</math>的节点时，其另一端指向度为<math>k'</math>的概率。<math>d_{k'k}</math>表示的是两端连接了度为<math>k</math>和<math>k'</math>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<math>r</math>扩散，因此沿着每条连边的扩散率为<math>d_{k'k}=r/k'</math>。在无关联的网络上，<math>P(k'|k)=k'P(k')/\left<k\right></math>，因此在稳态<math>d\mathcal{N}/dt=0</math>时，很容易得到解（Colizza等，2007b； Noh和Rieger，2004）：
\begin{equation}
<nowiki>\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} </nowiki>
上面的公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，如果一个节点的度越大，则其被扩散中的粒子访问到的可能性就越大。
==亚种群模型==
==亚种群模型==
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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