论文标题
随机块模型框架下的随机反应网络的缺陷为零
Deficiency zero for random reaction networks under a stochastic block model framework
论文作者
论文摘要
缺陷零是一个重要的网络结构，并且一直是反应网络理论中许多著名结果的重点。在我们以前的论文\ textIt {缺陷率零反应网络中的零反应网络中的流行率，我们提供了一个框架，以量化随机生成的反应网络中缺陷零的流行率。具体而言，给定一个带有$ n $种的随机生成的二进制反应网络，在两个任意的顶点之间具有独立发生的概率$ p_n $之间的优势，我们确定了阈值函数$ r（n）= \ frac {1} {1} {n^3} $，因此缺乏网络的概率是，缺乏率零是$ \ frac} n $ \ frac {p_ p_} p_ {p_} r {p_ {p_} r {p}如果$ \ frac {p_n} {r（n）} \ to \ infty $，则收敛到0，为$ n \ to \ infty $。 以ERD \ H OS-Rényi框架作为起点，当前的论文通过控制参数加权边缘概率$α_{i，j} $，以$ i，j \ in \ in \ in \ {0,1,1,2 \} $计数可能的Vertices（Zeroth，第一个或第二个订单）。可以选择控制参数以生成具有特定基础结构的随机反应网络，例如几乎没有流入和流出反应的“封闭”网络，或具有丰富流入和流出的“开放”网络。在此新框架下，对于每种控制参数的选择$ \ frac {p_n} {r（n，\ {α_{i，j} \}）} \ 0 $ to 0 $，并收敛至0，如果$ \ frac {p_n} {r（n，\ {n，\ {α_{α_{i，j} \} \} \} \}）}}}}
Deficiency zero is an important network structure and has been the focus of many celebrated results within reaction network theory. In our previous paper \textit{Prevalence of deficiency zero reaction networks in an Erd\H os-Rényi framework}, we provided a framework to quantify the prevalence of deficiency zero among randomly generated reaction networks. Specifically, given a randomly generated binary reaction network with $n$ species, with an edge between two arbitrary vertices occurring independently with probability $p_n$, we established the threshold function $r(n)=\frac{1}{n^3}$ such that the probability of the random network being deficiency zero converges to 1 if $\frac{p_n}{r(n)}\to 0$ and converges to 0 if $\frac{p_n}{r(n)}\to\infty$, as $n \to \infty$. With the base Erd\H os-Rényi framework as a starting point, the current paper provides a significantly more flexible framework by weighting the edge probabilities via control parameters $α_{i,j}$, with $i,j\in \{0,1,2\}$ enumerating the types of possible vertices (zeroth, first, or second order). The control parameters can be chosen to generate random reaction networks with a specific underlying structure, such as "closed" networks with very few inflow and outflow reactions, or "open" networks with abundant inflow and outflow. Under this new framework, for each choice of control parameters $\{α_{i,j}\}$, we establish a threshold function $r(n,\{α_{i,j}\})$ such that the probability of the random network being deficiency zero converges to 1 if $\frac{p_n}{r(n,\{α_{i,j}\})}\to 0$ and converges to 0 if $\frac{p_n}{r(n,\{α_{i,j}\})}\to \infty$.
