论文标题
循环随机图的过渡，具有固定度和任意度分布
Transitions in loopy random graphs with fixed degrees and arbitrary degree distributions
论文作者
论文摘要
我们分析了最大的熵随机图，以约束程度，由任意程度分布绘制，以及可调数的3循环（三角形）。我们发现，这样的合奏通常表现出两个过渡，一个聚类和一个破碎的过渡，分开了三个不同的政权。在聚类过渡时，这些图从通常只有隔离环变成形成循环簇。在破碎的过渡时，图将大量分解为许多小集团，以达到所需的环密度。两个过渡的位置都非凡地取决于系统大小。我们得出了一个隔离环状态中环密度的通用公式，该循环密度具有有限和第二矩的度分布的图。对于有界程度的分布，我们在循环密度和相变位置提出了进一步的分析结果，尽管循环密度和相变位置都通过MCMC采样模拟进行了验证。我们表明，如果系统足够大，那么粉碎的过渡是熵性质，它是针对所有循环密度值出现的。
We analyze maximum entropy random graph ensembles with constrained degrees, drawn from arbitrary degree distributions, and a tuneable number of 3-loops (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 loops to forming loop clusters. At the shattering transition the graphs break up into extensively many small cliques to achieve the desired loop density. The locations of both transitions depend nontrivially on the system size. We derive a general formula for the loop density in the regime of isolated loops, for graphs with degree distributions that have finite and second moments. For bounded degree distributions we present further analytical results on loop 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 loop density values, provided the system is large enough.
