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==反应扩散过程==

==反应扩散过程==

[[File:Snipaste_2020-10-23_22-40-33.png|thumb|right|(a)基于粒子网络方案的简化建模框架示意图。在宏观层面上，系统是由一个异质性的亚种群网络组成。由于粒子在亚群间的扩散，一个亚群内的传染过程可以传播到其他亚群。(b)在微观水平上，每个亚种群包含一个个体种群。例如，动态过程，一种传染现象，可以用简单的划分来描述(图中不同颜色的点表示这些划分)。在每个子种群中，个体可以均匀地或根据一个子网络进行混合，并且可以沿着网络的边缘以概率<math>p</math>的方式从一个子种群扩散到另一个子种群。(c)个体或粒子扩散的临界值<math>p_c</math>确定了在传染影响系统的大部分和只影响一小部分的状态之间的相变。]]  

[[File:Snipaste_2020-10-23_22-40-33.png|thumb|right|(a)基于粒子-网络方案的简化建模框架示意图。在宏观层面上，系统是由一个异质性的亚种群网络组成。由于粒子在亚种群间的扩散，一个亚种群内的传染过程可以传播到其他亚种群。(b)在微观水平上，每个亚种群包含一个个体种群。例如，一种传染现象的动态过程，可以用简单的划分区间来描述(图中不同颜色的点表示这些划分区间)。在每个亚种群中，个体可以均匀地混合扩散，或者基于一个子网络进行混合，在网络混合的方式中个体可以沿着网络的边缘以概率<math>p</math>的方式从一个亚种群扩散到另一个亚种群。(c)个体或粒子扩散的临界值<math> p_c </math>确定了在传染影响系统内大部分节点和只影响系统内一小部分节点这两种状态之间的相变。]]  

 

一般而言，社会行为和人类流动性的网络模型通常被构建为反应-扩散过程<font color="#ff8000"> Reaction-Diffusion Processes</font> ,在这个网络模型框架中，每个节点<math>i</math>可以容纳任何非负整数个粒子<math>\mathcal{N}(i)</math>，因此系统的总粒子数为<math>\mathcal{N}=\sum_i\mathcal{N}(i)</math>。该粒子-网络框架中节点内的每个粒子只能沿着网络中连接节点的连边进行扩散，并且扩散系数取决于节点的度<font color="#ff8000"> Node degree</font>或者节点的其他属性。对于不同的系统，节点内粒子的反应规则也不同。这样的粒子-网络框架模型的简单示意如图所示。

一般而言，社会行为和人类流动性的网络模型通常被构建为反应-扩散过程<font color="#ff8000"> Reaction-Diffusion Processes</font> ,在这个网络模型框架中，每个节点<math>i</math>可以容纳任何非负整数个粒子<math>\mathcal{N}(i)</math>，因此系统的总粒子数为<math>\mathcal{N}=\sum_i\mathcal{N}(i)</math>。该粒子-网络框架中节点内的每个粒子只能沿着网络中连接节点的连边进行扩散，并且扩散系数取决于节点的度<font color="#ff8000"> Node degree</font>或者节点的其他属性。对于不同的系统，节点内粒子的反应规则也不同。这样的粒子-网络框架模型的简单示意如图所示。

 

 

 

In order to have an analytic description of reaction-diffusion systems in networks one has to allow the possibility of heterogeneous connectivity patterns among nodes. A first analytical approach to these systems considers the extension of the degree-based mean-field approach to reaction-diffusion systems in networks with arbitrary degree distribution. For simplicity, we first consider the DBMF approach to the case of a simple system in which noninteracting particles (individuals) diffuse on a network with arbitrary topology. A convenient representation of the system is therefore provided by quantities defined in terms of the degree k:

In order to have an analytic description of reaction-diffusion systems in networks one has to allow the possibility of heterogeneous connectivity patterns among nodes. A first analytical approach to these systems considers the extension of the degree-based mean-field approach to reaction-diffusion systems in networks with arbitrary degree distribution. For simplicity, we first consider the DBMF approach to the case of a simple system in which noninteracting particles (individuals) diffuse on a network with arbitrary topology. A convenient representation of the system is therefore provided by quantities defined in terms of the degree k:

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度的平均场理论方法扩展到具有任意度分布的网络中的反应扩散系统。简单起见，首先考虑一个任意网络结构的简单系统，其中粒子间没有相互作用或反应，粒子只会在网络上扩散的的情况。因此，通过以度$k$来分类的话，可以将系统简单表示为：

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度的平均场理论方法扩展到具有任意度分布的网络中的反应扩散系统。简单起见，首先考虑一个任意网络结构的简单系统，其中粒子间没有相互作用或反应，粒子只会在网络上扩散的的情况。因此，通过以度<math> k </math>来分类的话，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)

where $N_k=NP(k)$ is the number of nodes with degree $k$ and the sum runs over the set of nodes $\mathcal{V}(k)$ having degree equal to k. The degree block variable $\mathcal{N}_k$ represents the average number of particles in nodes with degree k. The use of the DBMF approach amounts to the assumption that nodes with degree k, and thus the particles in those nodes, are statistically equivalent. In this approximation the dynamics of particles randomly diffusing on the network is given by a mean-field dynamical equation expressing the variation in time of the particle subpopulation N kðtÞ in each degree block k. This can be easily written as

where $N_k=NP(k)$ is the number of nodes with degree $k$ and the sum runs over the set of nodes $\mathcal{V}(k)$ having degree equal to k. The degree block variable $\mathcal{N}_k$ represents the average number of particles in nodes with degree k. The use of the DBMF approach amounts to the assumption that nodes with degree k, and thus the particles in those nodes, are statistically equivalent. In this approximation the dynamics of particles randomly diffusing on the network is given by a mean-field dynamical equation expressing the variation in time of the particle subpopulation N kðtÞ in each degree block k. This can be easily written as

其中$N_k=NP(k)$表示度为k的节点数量，$\mathcal{V}(k)$表示度为$k$的节点集合，$\mathcal{N}_k$表示度为$k$的节点的平均粒子数。基于度的平均场理论假设度为$k$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： 

\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)

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>

<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）：

<nowiki>方程中的第一项表示每单位时间内有<math> d_k </math>部分的粒子从度为<math> k </math>的节点中离开，第二项表示粒子从邻居节点扩散到度为<math> k </math>的节点中，该项与连边数<math> k </math>成正比。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}

\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}