Exponential Random Graphs (ERGs) are among the most widely used network models, derived as principled least-bias graph ensembles that maximize Shannon entropy under constraints on the expected values of given structural properties. However, it has been recently (re)discovered that, in the absence of additional information privileging Shannon entropy, the most agnostic inferential construction should maximize the broader class of Uffink entropies. The resulting entropy-maximizing distribution changes from the exponential (Boltzmann-Gibbs) to the so-called q-exponential one. Since maximizing Shannon entropy may produce an unjustified independence between degrees of freedom, here we investigate how the most popular ERGs with independent edges (namely, the Erdos-Renyi and configuration models) generalize to higher-order q-Exponential Random Graphs with dependent edges in the non-Shannon case, while keeping their defining constraints (number of links and degree sequence, respectively) unchanged. We find features, such as a phase transition between sparse and dense regimes, that are absent in the original ERGs but typical of higher-order networks, plus novel phenomena such as richer assortativity and clustering profiles, which allow for the coexistence of link sparsity and triadic closure. These results show that higher-order networks do not necessarily require higher-order constraints, as they naturally arise from simpler ones in a framework that is even more agnostic than Shannon's.

翻译：指数随机图（ERGs）是最广泛使用的网络模型之一，其作为有原则的最小偏倚图系综而导出，在给定结构性质的期望值约束下最大化香农熵。然而，近期（重新）发现，在缺乏优先考虑香农熵的额外信息时，最不可知论的推断构建应最大化更广泛的Uffink熵类。由此产生的熵最大化分布从指数分布（玻尔兹曼-吉布斯）转变为所谓的q-指数分布。由于最大化香农熵可能导致自由度之间不合理的独立性，本文研究在非香农情形下，最流行的具有独立边的ERGs（即厄多斯-雷尼模型与配置模型）如何泛化为具有依赖边的高阶q-指数随机图，同时保持其定义约束（分别为链接数和度序列）不变。我们发现了原始ERGs中不存在但高阶网络典型特征的现象，例如稀疏与稠密区域之间的相变，以及更丰富的同配性与聚类轮廓等新现象，这些允许链接稀疏性与三角闭合的共存。这些结果表明，高阶网络并非必然需要高阶约束，因为它们可以在比香农框架更不可知论的体系中从简单约束自然产生。