论文标题
欧几里得球上的随机几何图
Random Geometric Graphs on Euclidean Balls
论文作者
论文摘要
我们考虑了随机图的潜在空间模型，其中节点$ i $与欧几里得单元球上的随机潜在点$ x_i $相关联。两个节点之间存在边缘的概率由``链接''函数确定，该功能对应于点乘积内核。对于$ x_i $的球形对称分布的给定类$ \ f $，我们考虑了两个估计问题：潜在的norm恢复和潜在的革兰氏矩阵估计。在模型的情况下，我们基于观测图的节点的程度构建一个潜在规范的估计量，其中$ f（\ langle x_i，x_j \ rangle）= \ mathbbm {1} _ {\ langle x_i x_i x_j \ x_j \ rangle x_ \ rangle \ geq $ 0 <我们根据观察到的图形的特征向量引入了革兰氏矩阵的估计器，并为误差建立了frobenius型的保证，前提是链接函数在Sobolev的意义上是足够规则的，并且光谱间隙型型条件持有。我们证明，对于某些链接函数，此处考虑的模型生成的图形分布具有带有幂律型分布的尾巴，这可以看作是相对于欧几里得球体上经典的随机随机几何图模型，此处介绍的模型的优势。我们通过数值实验说明了结果。
We consider a latent space model for random graphs where a node $i$ is associated to a random latent point $X_i$ on the Euclidean unit ball. The probability that an edge exists between two nodes is determined by a ``link'' function, which corresponds to a dot product kernel. For a given class $\F$ of spherically symmetric distributions for $X_i$, we consider two estimation problems: latent norm recovery and latent Gram matrix estimation. We construct an estimator for the latent norms based on the degree of the nodes of an observed graph in the case of the model where the edge probability is given by $f(\langle X_i,X_j\rangle)=\mathbbm{1}_{\langle X_i,X_j\rangle\geq τ}$, where $0<τ<1$. We introduce an estimator for the Gram matrix based on the eigenvectors of observed graph and we establish Frobenius type guarantee for the error, provided that the link function is sufficiently regular in the Sobolev sense and that a spectral-gap-type condition holds. We prove that for certain link functions, the model considered here generates graphs with degree distribution that have tails with a power-law-type distribution, which can be seen as an advantage of the model presented here with respect to the classic Random Geometric Graph model on the Euclidean sphere. We illustrate our results with numerical experiments.
