We present a new method for assessing homophily in networks whose vertices have categorical attributes, namely when the vertices of networks come partitioned into classes. We apply this method to Protein- Protein Interaction networks, where vertices correspond to proteins, partitioned according to they functional role, and edges represent potential interactions between proteins. Similarly to other classical and well consolidated approaches, our method compares the relative edge density of the subgraphs induced by each class with the corresponding expected relative edge density under a null model. The novelty of our approach consists in prescribing an endogenous null model, namely, the sample space of the null model is built on the input network itself. This allows us to give exact explicit expression for the z-score of the relative edge density of each class as well as other related statistics. The z-scores directly quantify the statistical significance of the observed homophily via \v{C}eby\v{s}\"ev inequality. The expression of each z-score is entered by the network structure through basic combinatorial invariant such as the number of subgraphs with two spanning edges. Each z-score is computed in O(n 3 ) worst-case time for a network with n vertices. This leads to an overall effective computational method for assesing homophily. Theoretical results are then exploited to prove that Protein-Protein Interaction networks networks are significantly homophillous.

中文翻译：

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微生物蛋白质-蛋白质相互作用网络的功能同质性

我们提出了一种新方法，用于评估顶点具有分类属性的网络中的同质性，即当网络的顶点划分为类时。我们将此方法应用于蛋白质-蛋白质相互作用网络，其中顶点对应于蛋白质，根据它们的功能作用进行划分，边代表蛋白质之间的潜在相互作用。与其他经典的和整合良好的方法类似，我们的方法将每个类诱导的子图的相对边缘密度与空模型下相应的预期相对边缘密度进行比较。我们方法的新颖之处在于规定了一个内生空模型，即空模型的样本空间建立在输入网络本身上。这使我们能够为每个类别的相对边缘密度的 z​​ 分数以及其他相关统计数据提供精确的显式表达式。z-scores 通过 \v{C}eby\v{s}\"ev 不等式直接量化观察到的同质性的统计显着性。每个 z-score 的表达式是通过基本的组合不变量由网络结构输入的，例如具有两个生成边的子图的数量。对于具有 n 个顶点的网络，每个 z 分数在 O(n 3 ) 最坏情况下计算。这导致了用于评估同质性的整体有效计算方法。然后利用理论结果来证明蛋白质-蛋白质相互作用网络网络是显着同质的。对于具有 n 个顶点的网络，每个 z-score 的计算时间为 O(n 3 ) 最坏情况。这导致了一种用于评估同质性的整体有效的计算方法。然后利用理论结果来证明蛋白质-蛋白质相互作用网络网络具有显着的同质性。对于具有 n 个顶点的网络，每个 z-score 的计算时间为 O(n 3 ) 最坏情况。这导致了一种用于评估同质性的整体有效的计算方法。然后利用理论结果来证明蛋白质-蛋白质相互作用网络网络具有显着的同质性。