发表机构
Institute for Biocomputation and Physics of Complex Systems (BIFI); Universidad de Zaragoza; University of Zaragoza; Instituto de Física Gleb Wataghin, Universidade Estadual de Campinas (UNICAMP)（生物计算与复杂系统物理研究所（BIFI）; 萨拉戈萨大学; 萨拉戈萨大学; 坎皮纳斯州立大学（UNICAMP）格列布·瓦塔金物理研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文区分结构投影、功能可表示性等概念，证明交互阶数是不变量，提出最小描述长度准则，表明成对与高阶模型无绝对优劣，需结合具体情境选择。
AI 中文摘要
关于成对模型与高阶模型的争论常常被表述为表达能力的竞赛，但这种框架具有误导性。具有任意多元节点函数的图可以模拟许多超图模型的节点级动力学，但这并不能消除超图编码的分组信息：它可能只是从结构转移到了动力学中。我们区分了四个经常被混淆的概念：结构投影、功能可表示性、统计可辨识性和机制充分性。我们证明有限状态映射的交互阶数是映射本身的不变量，因此表示的精确改变不能降低依赖的基本阶数。然后我们将比较重新表述为描述长度的问题。在无限制的算法层面，固定的编译器可以在结构和规则之间移动信息，仅需恒定的开销，因此仅凭表达能力并不能赋予图或超图特权。偏好仅在相对于明确的模型类、编码方案、正则性假设和数据时才会出现。这促使我们提出一个操作性的最小描述长度准则，结合结构成本、以结构为条件的规则成本和模型拟合度。对于大小为k的M个不相交组，我们证明团投影所需的边列表渐近地比相应的超边列表长k-1倍，使得投影成为对相同分组的更昂贵的编码。涵盖扩散、布尔动力学、生态相互作用和模糊投影的示例说明了图偏好、超图偏好和未解决的情况。由此产生的立场是对称的：不应仅从现象学推断高阶结构，但也不应将基于图的模拟视为图是最简约或科学上充分的描述的证据。
英文摘要
The debate over pairwise and higher-order models is often framed as a contest of expressive power. This framing is misleading. A graph with arbitrary multivariate node functions can reproduce the node-level dynamics of many hypergraph models, but the hypergraph's grouping information does not thereby disappear: it may simply move from the structure to the rule. We separate four notions frequently conflated here: structural representation, functional representability, statistical identifiability, and mechanistic adequacy. We then prove that interaction order, the number of variables that must act jointly in some term of any additive decomposition of a node's update, is the same in every exact representation: no change of structural language can lower it. As a description-length problem, at the unrestricted algorithmic level a fixed compiler redistributes information between structure and rule at constant overhead, so expressiveness alone cannot privilege either language. Preferences arise only relative to explicit model classes, code families, and data. This yields an operational minimum-description-length criterion combining structural cost, conditional rule cost, and imperfect fit. Within it, for $M$ disjoint groups of size $k$, the clique projection's edge list is asymptotically $k-1$ times longer than the hyperedge list it replaces: the projection encodes the same grouping at higher cost. Examples from diffusion, Boolean dynamics, ecology, and ambiguous projections give graph-preferred, hypergraph-preferred, and unresolved cases; for bipartite and multilayer lifts, cost, fit, and identifiability tie, and the choice turns on which entities are posited as primitive. The position is symmetric: higher-order structure should not be inferred from phenomenology alone, nor does a graph's ability to emulate a system make it the most parsimonious or adequate description.
Comments26 pages, no figures. Submitted for publication
