作为超单纯性强化机制的超图优先连接
Preferential Attachment as a Simpliciality-Enforcing Mechanism in Hypergraphs
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中文总结 AI 辅助
该研究提出广义优先连接超图模型，推导其稳态超度分布规律，结合8个真实数据集验证优先连接可强化超图的超单纯性。
中文摘要 AI 辅助
以超图为代表的高阶网络可直接建模任意规模的多体相互作用。已观测到真实系统的超图表示呈现出高超单纯性——即超边的子集也会以超边形式出现的倾向，但这种结构的生成机制尚不明确。我们提出一种广义优先连接超图模型，其中超边大小Y_t和每步新增节点数X_t可来自任意分布；通过平均场近似主方程方法解析推导得出，稳态超度分布服从幂律，其指数仅取决于比值p = E[X_t]/E[Y_t]，与基础分布的形状无关。关键在于，X_t和Y_t均可通过反向步进过程直接从任意带时间戳的超图数据集中估计，使模型无需参数假设即可拟合。将该模型的非线性扩展应用于8个真实超图数据集，我们发现超单纯分数随优先连接强度单调递增，直至α>1的凝胶转变点，确立了优先连接作为超单纯性强化机制的作用。
英文摘要
Higher-order networks, represented as hypergraphs, enable direct modeling of multi-body interactions of arbitrary size. Hypergraph representations of real-world systems have been observed to exhibit high \emph{simpliciality} --- the tendency for subsets of hyperedges to also appear as hyperedges --- yet the generative mechanisms responsible for this structure are poorly understood. We introduce a generalized preferential attachment hypergraph model in which both hyperedge size $Y_t$ and the number of new nodes per step $X_t$ are drawn from arbitrary distributions, and derive analytically, using a mean-field approximate master equation approach, that the stationary hyperdegree distribution follows a power law whose exponent depends only on the ratio $p = E[X_t]/E[Y_t]$, independent of the shapes of the underlying distributions. Crucially, both $X_t$ and $Y_t$ can be estimated directly from any timestamped hypergraph dataset via a backward-stepping procedure, enabling the model to be fit without parametric assumptions. Applying a nonlinear extension of the model to eight real-world hypergraph datasets, we find that the simplicial fraction increases monotonically with the strength of preferential attachment up to the gelation transition at $α> 1$, establishing preferential attachment as a simpliciality-enforcing mechanism.
