We propose the model of a random polymer network, formed on the base on Erd\"os-R\'enyi random graph. In the language of mathematical graphs, the chemical bonds between monomers can be treated as vertices, and their chemical functionalities as degrees of these vertices. We consider graphs with fixed number of vertices $N=5$ and variable parameter $c$ (connectedness), defining the total number of links $L=cN(N-1)/2$ between vertices. Each link in such graphs is treated as a Gaussian polymer chain. The universal rotationally invariant size and shape characteristics, such as averaged asphericity $\langle A_3 \rangle$ and size ratio $g$ of such structures are obtained both numerically by application of Wei's method and analytically within the continuous chain model. In particular, our results quantitatively indicate an increase of asymmetry of polymer network structure when its connectedness $c$ decreases.

中文翻译：

---

无规聚合物网络的形状分析

我们提出了一个基于 Erd\"os-R\'enyi 随机图的随机聚合物网络模型。在数学图的语言中，单体之间的化学键可以被视为顶点，它们的化学功能为这些顶点的度数。我们考虑具有固定顶点数 $N=5$ 和可变参数 $c$（连通性）的图，定义顶点之间的链接总数 $L=cN(N-1)/2$。每个此类图中的链接被视为高斯聚合物链。通用旋转不变的尺寸和形状特征，例如这种结构的平均非球面度 $\lange A_3 \rangle$ 和尺寸比 $g$ 均通过应用魏的方法在数值上获得并在连续链模型中进行分析。特别是，我们的结果定量地表明，当聚合物网络结构的连通性 $c$ 降低时，其不对称性会增加。