随机超图上的高阶三元渗流
Higher-order triadic percolation on random hypergraphs
AI总结：
本文提出超图上的高阶三元渗流框架，结合渗流理论与非线性动力学，证明巨分量序参量可经逻辑斯蒂映射普适类通向混沌，并推广到分层嵌套三元相互作用及相互依赖与节点调控情形。
摘要：
在这项工作中，我们提出了一个将渗流理论与非线性动力学相结合的综合理论框架，以研究具有时变巨分量的超图。我们特别考虑具有高阶三元相互作用的超图，这些相互作用可以上调或下调超边。三元相互作用是一种一般类型的有符号调控相互作用，当一个第三节点调控另外两个节点之间的相互作用时发生。例如，在大脑网络中，胶质细胞可以促进或抑制神经元之间的突触相互作用。然而，调控相互作用不仅可能发生在调控节点与成对相互作用之间，也可能发生在调控节点与高阶相互作用（超边）之间，从而产生高阶三元相互作用。例如，在生化反应网络中，酶调控涉及多个反应物的反应。在此，我们提出并研究了超图上的高阶三元渗流，表明巨分量可以具有非平凡的动力学。具体而言，我们证明在适当条件下，该渗流问题的序参量，即巨分量中节点的比例，会经历一条属于逻辑斯蒂映射普适类的通向混沌的道路。在分层高阶三元渗流中，我们扩展了这一范式以处理分层嵌套的三元相互作用，证明了其增加的组合复杂性对临界现象和过程动力学性质的非平凡影响。最后，我们考虑该模型的其他推广，研究考虑相互依赖性和节点调控而非超边调控的效果。
英文摘要：
In this work, we propose a comprehensive theoretical framework combining percolation theory with nonlinear dynamics in order to study hypergraphs with a time-varying giant component. We consider in particular hypergraphs with higher-order triadic interactions that can upregulate or downregulate the hyperedges. Triadic interactions are a general type of signed regulatory interaction that occurs when a third node regulates the interaction between two other nodes. For example, in brain networks, the glia can facilitate or inhibit synaptic interactions between neurons. However, the regulatory interactions may not only occur between regulator nodes and pairwise interactions but also between regulator nodes and higher-order interactions (hyperedges), leading to higher-order triadic interactions. For instance, in biochemical reaction networks, the enzymes regulate the reactions involving multiple reactants. Here we propose and investigate higher-order triadic percolation on hypergraphs showing that the giant component can have a non-trivial dynamics. Specifically, we demonstrate that, under suitable conditions, the order parameter of this percolation problem, i.e., the fraction of nodes in the giant component, undergoes a route to chaos in the universality class of the logistic map. In hierarchical higher-order triadic percolation, we extend this paradigm in order to treat hierarchically nested triadic interactions demonstrating the non-trivial effect of their increased combinatorial complexity on the critical phenomena and the dynamical properties of the process. Finally, we consider other generalizations of the model studying the effect of considering interdependencies and node regulation instead of hyperedge regulation.
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