Homophily is the principle whereby "similarity breeds connections". We give a quantitative formulation of this principle within networks. We say that a network is homophillic with respect to a given labeled partition of its vertices, when the classes of the partition induce subgraphs that are significantly denser than what we expect under a random labeled partition into classes maintaining the same cardinalities (type). This is the recently introduced \emph{random coloring model} for network homophily. In this perspective, the vector whose entries are the sizes of the subgraphs induced by the corresponding classes, is viewed as the observed outcome of the random vector described by picking labeled partitions at random among partitions with the same type.\,Consequently, the input network is homophillic at the significance level $\alpha$ whenever the one-sided tail probability of observing an outcome at least as extreme as the observed one, is smaller than $\alpha$. Clearly, $\alpha$ can also be thought of as a quantifier of homophily in the scale $[0,1]$. Since, as we show, even approximating this tail probability is an NP-hard problem, we resort multidimensional extensions of classical Cantelli's inequality to bound $\alpha$ from above. This upper bound is the homophily index we propose. It requires the knowledge of the covariance matrix of the random vector, which was not previously known within the random coloring model. In this paper we close this gap by computing the covariance matrix of subgraph sizes under the random coloring model. Interestingly, the matrix depends on the input partition only through its type and on the network only through its degrees. Furthermore all the covariances have the same sign and this sign is a graph invariant. Plugging this structure into Cantelli's bound yields a meaningful, easy to compute index for measuring network homophily.

中文翻译：

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通过 Cantelli 不等式的多维扩展实现网络同质性

同质性是“相似产生联系”的原则。我们在网络中给出了这一原则的定量表述。我们说一个网络相对于其顶点的给定标记分区是同质的，当分区的类将比我们在随机标记分区下预期的要密集得多的子图划分为保持相同基数（类型）的类时。这是最近引入的用于网络同质性的 \emph{随机着色模型}。从这个角度来看，其条目是由相应类诱导的子图的大小的向量被视为通过在具有相同类型的分区中随机选择标记的分区来描述的随机向量的观察结果。\，因此，只要观察到至少与观察到的结果一样极端的结果的单边尾概率小于 $\alpha$，输入网络在显着性水平 $\alpha$ 上是同质的。显然，$\alpha$ 也可以被认为是在 $[0,1]$ 尺度上的同质性的量词。因为，正如我们所展示的，即使逼近这个尾概率也是一个 NP-hard 问题，我们将经典 Cantelli 不等式的多维扩展用于从上面限制 $\alpha$。这个上限是我们提出的同质指数。它需要了解随机向量的协方差矩阵，这在随机着色模型中是以前不知道的。在本文中，我们通过在随机着色模型下计算子图大小的协方差矩阵来弥补这一差距。有趣的是，矩阵仅通过其类型取决于输入分区，仅通过其度数取决于网络。此外，所有协方差都具有相同的符号，并且该符号是图不变量。将此结构插入 Cantelli 的界限会产生一个有意义的、易于计算的指数，用于测量网络同质性。