论文标题
随机超图的一类模型
A class of models for random hypergraphs
论文作者
论文摘要
尽管最近在各种过程中表现出高阶相互作用的重要性，但很少有灵活的模型可用。特别是，大多数有关超图的研究都集中在一小部分理论模型上。在这里，我们介绍了一类用于随机超图的模型，该模型显示了复杂网络模型的相似灵活性水平，主要成分是节点属于HyperEdge的概率。当这种概率是常数时，我们会以与Erdos-Renyi图相同的精神获得随机的超图。该框架还使我们能够引入不同的成分，例如超图的优先附件或空间随机超图。特别是，我们表明，对于Erdos-Renyi案例，有一个过渡阈值缩放为$ 1/\ sqrt {en} $，其中$ n $是节点的数量和$ e $ hyperedges的数量。我们还讨论了一个随机的几何超图，该图显示了阈值距离缩放为$ r_c^*\ sim 1/\ sqrt {e} $的渗透过渡。对于这些不同的模型，我们为最有趣的措施提供了结果，并在空间案例中引入了新的措施，以表征Hyperedges的几何特性。这些不同的模型可能是可用于分析经验数据的基准。
Despite the recently exhibited importance of higher-order interactions for various processes, few flexible (null) models are available. In particular, most studies on hypergraphs focus on a small set of theoretical models. Here, we introduce a class of models for random hypergraphs which displays a similar level of flexibility of complex network models and where the main ingredient is the probability that a node belongs to a hyperedge. When this probability is a constant, we obtain a random hypergraph in the same spirit as the Erdos-Renyi graph. This framework also allows us to introduce different ingredients such as the preferential attachment for hypergraphs, or spatial random hypergraphs. In particular, we show that for the Erdos-Renyi case there is a transition threshold scaling as $1/\sqrt{EN}$ where $N$ is the number of nodes and $E$ the number of hyperedges. We also discuss a random geometric hypergraph which displays a percolation transition for a threshold distance scaling as $r_c^*\sim 1/\sqrt{E}$. For these various models, we provide results for the most interesting measures, and also introduce new ones in the spatial case for characterizing the geometrical properties of hyperedges. These different models might serve as benchmarks useful for analyzing empirical data.
