标题： 随机图的最小作用量原理 显示英文标题

标题： The principle of least action for random graphs

摘要： 我们研究了随机图的物理作用量$S$的统计性质，通过将图中每个顶点的邻居数量（度数）视为标量场。 对于图的每个配置（运行），我们使用格点量子场论（LQFT）方法计算度数场的拉格朗日量。 然后通过将拉格朗日量在整个图的顶点上积分来计算相应的作用量。 我们通过每基本量子时间移除一条边来实现图的演化机制，这导致了不同的演化路径，这些路径基于在每次演化步骤中选择的运行而有所不同。 我们沿着这些演化路径计算作用量，这使我们能够计算$S$的概率分布。 我们发现，随着图变得越来越密集，即在顶点之间添加更多边时，该分布趋近于正态（高斯）形式。 作用量的概率分布的最大值对应于$S$值之间的间隔为零$\Delta S=0$的图配置，这对应于最小作用量（哈密顿）原理，该原理给出了物理系统经典遵循的路径。 此外，我们计算了度数场的波动（方差），结果显示对应于$S$最大概率的图配置，遵循哈密顿原理，其结构在规则图和不规则图之间达到平衡。 显示英文摘要

摘要： We study the statistical properties of the physical action $S$ for random graphs, by treating the number of neighbors at each vertex of the graph (degree), as a scalar field. For each configuration (run) of the graph we calculate the Lagrangian of the degree field by using a lattice quantum field theory(LQFT) approach. Then the corresponding action is calculated by integrating the Lagrangian over all the vertices of the graph. We implement an evolution mechanism for the graph by removing one edge per a fundamental quantum of time, resulting in different evolution paths based on the run that is chosen at each evolution step. We calculate the action along each of these evolution paths, which allows us to calculate the probability distribution of $S$. We find that the distribution approaches the normal(Gaussian) form as the graph becomes denser, by adding more edges between its vertices. The maximum of the probability distribution of the action corresponds to graph configurations whose spacing between the values of $S$ becomes zero $\Delta S=0$, corresponding to the least-action (Hamilton) principle, which gives the path that the physical system follows classically. In addition, we calculate the fluctuations(variance) of the degree field showing that the graph configurations corresponding to the maximum probability of $S$, which follow the Hamilton's principle, have a balanced structure between regular and irregular graphs.