The Friendship Paradox is a simple and powerful statement about node degrees in a graph (Feld 1991). However, it only applies to undirected graphs with no edge weights, and the only node characteristic it concerns is degree. Since many social networks are more complex than that, it is useful to generalize this phenomenon, if possible, and a number of papers have proposed different generalizations. Here, we unify these generalizations in a common framework, retaining the focus on undirected graphs and allowing for weighted edges and for numeric node attributes other than degree to be considered, since this extension allows for a clean characterization and links to the original concepts most naturally. While the original Friendship Paradox and the Weighted Friendship Paradox hold for all graphs, considering non-degree attributes actually makes the extensions fail around 50% of the time, given random attribute assignment. We provide simple correlation-based rules to see whether an attribute-based version of the paradox holds. In addition to theory, our simulation and data results show how all the concepts can be applied to synthetic and real networks. Where applicable, we draw connections to prior work to make this an accessible and comprehensive paper that lets one understand the math behind the Friendship Paradox and its basic extensions.

翻译：友谊悖论是关于图中节点度的一个简洁而有力的表述（Feld 1991）。然而，它仅适用于无向且无权重边的图，且所涉及的节点特征仅限于度。由于许多社交网络比此更为复杂，若有可能，对这一现象进行推广具有重要价值；已有若干论文提出了不同的推广形式。本文将这些推广形式统一于一个共同框架中，保持对无向图的关注，同时允许考虑带权边以及度以外的数值型节点属性，因为这种扩展能够提供清晰的刻画，并与原始概念建立最自然的联系。虽然原始友谊悖论与加权友谊悖论对所有图均成立，但在考虑非度属性时，给定随机属性分配，此类推广在大约50%的情况下并不成立。我们提出了基于相关性的简单规则，用以判断基于属性的悖论版本是否成立。除理论分析外，我们的仿真与数据结果展示了如何将这些概念应用于合成网络和真实网络。在适用的情况下，我们联系已有研究工作，使本文成为一篇易于理解且内容全面的论文，帮助读者深入理解友谊悖论及其基本推广背后的数学原理。