有界可交换图
Boundedly exchangeable graphs
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中文总结 AI 辅助
该研究针对经典可交换性无法描述稀疏图的问题，引入有界可交换性，证明泊松图过程边测度联合可交换的条件，并分析齐次黎曼流形上阈值图模型的性质。
中文摘要 AI 辅助
无限联合可交换邻接矩阵是图模型的混合，其对应的图几乎必然为空或稠密；因此经典矩阵可交换性排除了稀疏非空图。我们研究一种替代点过程的图过程，其顶点与波兰潜空间上的局部有限点过程的原子对应，边则基于对称连接核W条件独立地绘制。对于具有扩散σ有限强度的泊松图过程，我们证明其边测度作为随机测度是联合可交换的当且仅当W本质上是常数。因此非恒定核需要不同的可交换性概念。我们引入有界可交换性，在此概念下，每个非平凡有界限制确定一个在分布上唯一的无限联合可交换邻接矩阵，具有相关的局部图表示。随后我们研究齐次黎曼流形上的阈值模型，其中泊松分布的顶点在其内在距离不超过固定阈值时相连。在该类的每个无限体积流形上，该模型是稀疏的，其经验度分布收敛于泊松定律，且其极限聚类系数取决于可变化的附加几何参数，同时极限度定律保持固定。
英文摘要
An infinite jointly exchangeable adjacency matrix is a mixture of graphon models, and its graph is almost surely empty or dense; classical matrix exchangeability therefore excludes sparse nonempty graphs. We study a point-process alternative, graph processes, whose vertices are identified with the atoms of a locally finite point process on a Polish latent space and whose edges are drawn conditionally independently from a symmetric connection kernel $W$. For a Poisson graph process with diffuse $σ$-finite intensity, we show that its edge measure is jointly exchangeable as a random measure if and only if $W$ is essentially constant. Non-constant kernels therefore require a different exchangeability notion. We introduce bounded exchangeability, under which every nontrivial bounded restriction determines an infinite jointly exchangeable adjacency matrix that is unique in distribution, with an associated local graphon representation. We then study threshold models on homogeneous Riemannian manifolds, in which Poisson-distributed vertices are connected whenever their intrinsic distance is at most a fixed threshold. On every infinite-volume manifold in this class, the model is sparse, its empirical degree distribution converges to a Poisson law, and its limiting clustering coefficient depends on an additional geometric parameter that can vary while the limiting degree law is held fixed.
