We analyze maximum entropy random graph ensembles with constrained degrees, drawn from arbitrary degree distributions, and a tuneable number of three-cycles (triangles). We find that such ensembles generally exhibit two transitions, a clustering and a shattering transition, separating three distinct regimes. At the clustering transition, the graphs change from typically having only isolated cycles to forming cycle clusters. At the shattering transition the graphs break up into many small cliques to achieve the desired three-cycle density. The locations of both transitions depend nontrivially on the system size. We derive a general formula for the three-cycle density in the regime of isolated cycles, for graphs with degree distributions that have finite first and second moments. For bounded degree distributions we present further analytical results on cycle densities and phase transition locations, which, while non-rigorous, are all validated via MCMC sampling simulations. We show that the shattering transition is of an entropic nature, occurring for all three-cycle density values, provided the system is large enough.

我们分析具有约束度的最大熵随机图集合，从任意度分布中提取，以及可调数量的三循环（三角形）。我们发现这样的集合通常表现出两个过渡，一个聚类和一个破碎过渡，将三个不同的制度分开。在聚类转换时，图从通常只有孤立的循环变为形成循环簇。在破碎过渡处，图形分解成许多小团以实现所需的三循环密度。两种转换的位置都非常依赖于系统大小。对于具有有限一阶和二阶矩的度分布的图，我们推导出了孤立循环范围内三循环密度的一般公式。对于有界度分布，我们提供了关于循环密度和相变位置的进一步分析结果，虽然不严格，但都通过 MCMC 采样模拟进行了验证。我们表明，只要系统足够大，破碎转变具有熵性质，发生在所有三循环密度值上。