发表机构
Nodes & Links Ltd（Nodes & Links 有限公司）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种自动生成函数计算方法，将复杂超图的渗流阈值与巨连通分量统一为同一张量的不同阶展开，并推广至相关层间结构，求解多种模型并验证均场理论失效原因。
AI 中文摘要
复杂超图（chygraphs）将图、超图、多重网络和交互网络作为特例包含在内，所有这些网络的渗流阈值都源自一个符号计算：由四个一阶矩矩阵构建的张量 $A$ 的谱。该计算止步于阈值。在此，我证明 $A$ 是一个非线性自洽映射在其平凡不动点处的雅可比矩阵，该映射的非平凡不动点即为巨连通分量占比，因此阈值和序参量是同一对象在同一指标集上的两个阶。将展开再推进一阶，即可得到 $S=B\Lambda+O(\Lambda^2)$ 中临界振幅 $B$ 的闭式表达式，并揭示出一个层级结构：一阶矩决定阈值，二阶矩决定振幅，而只有生成函数本身决定远离阈值处的序参量。放弃“一个复合体在不同层中的参与是独立的”这一假设，会再次推广张量，将无条件一阶矩替换为包含偏置的二阶矩：具有相同边际分布的分布具有不同的阈值和不同的序参量。随后，文献中的六个构造通过代入法求解，其中包括AND逻辑与OR逻辑超图渗流（它们共享一个复杂超图，仅在一个生成函数上不同），以及具有两个混合水平的SIR流行病（其家庭再生数从张量中得出）。已报道的均场理论在强聚类图上的失败，被证明是将其应用于错误对象所致：映射到树状骨干而非聚类图后，相同的计算重现了精确的阈值、序参量和临界指数。所有内容均在一个类中实现，在该类中，一个复杂超图通过其生成函数一次性指定，然后返回所有三个量，并与蒙特卡洛模拟进行了验证。
英文摘要
Complex hypergraphs (chygraphs) contain graphs, hypergraphs, multiplex and interacting networks as special cases, and the percolation threshold of all of them follows from one symbolic calculation: the spectrum of a tensor $A$ built from four matrices of first moments. That calculation stops at the threshold. Here I show that $A$ is the Jacobian, at its trivial fixed point, of a non-linear self-consistency map whose non-trivial fixed point is the giant component fraction, so threshold and order parameter are two orders of one object on one index set. Carrying the expansion one order further gives the critical amplitude $B$ in $S=BΛ+O(Λ^2)$ in closed form, and exposes a hierarchy: first moments fix the threshold, second moments the amplitude, and only the generating functions themselves the order parameter away from it. Dropping the assumption that a complex's participation in different layers is independent generalises the tensor again, replacing unconditional first moments by inclusion-biased second moments: distributions with identical marginals have different thresholds and different order parameters. Six constructions from the literature are then solved by substitution, among them AND- against OR-logic hypergraph percolation, which share a chygraph and differ in one generating function, and SIR epidemics with two levels of mixing, whose household reproduction number falls out of the tensor. A reported failure of mean-field theory on strongly clustered graphs is shown to be a failure of applying it to the wrong object: mapped onto the treelike backbone rather than the clustered graph, the same calculation reproduces the exact threshold, order parameter and critical exponents. Everything is implemented in one class, in which a chygraph is specified once by its generating functions and then returns all three, and validated against Monte Carlo simulation.
Comments13 pages, 6 figures
