The Friendship Paradox--the principle that ``your friends have more friends than you do''--is a combinatorial fact about degrees in a graph; but given that many Web-based social activities are correlated with a user's degree, this fact has been taken more broadly to suggest the empirical principle that ``your friends are also more active than you are.'' This Generalized Friendship Paradox, the notion that any attribute positively correlated with degree obeys the Friendship Paradox, has been established mathematically in a network-level version that essentially aggregates uniformly over all the edges of a network. Here we show, however, that the natural node-based version of the Generalized Friendship Paradox--which aggregates over nodes, not edges--may fail, even for degree-attribute correlations approaching 1. Whether this version holds depends not only on degree-attribute correlations, but also on the underlying network structure and thus can't be said to be a universal phenomenon. We establish both positive and negative results for this node-based version of the Generalized Friendship Paradox and consider its implications for social-network data.
翻译：友谊-Paradox 原则是“你的朋友比你有更多的朋友 ” —— “你的朋友比你有更多的朋友 ” —— 是一个图表中关于度的组合式事实; 但是,鉴于许多基于网络的社会活动与用户的度相关, 这一事实被广泛地用来提出经验性原则, 即“你的朋友也比你更活跃。 ” 。 这个普遍化的友谊 —— 这个概念, 任何属性都与遵守度符合“友谊 paradox ” 具有积极关联性, 已经在网络层面的版本中以数学方式建立, 基本上对网络的所有边缘进行统一汇总。 然而, 我们在这里显示, 通用友谊 Paradox 的基于节点的自然节点版本 —— 集合超过节点, 而不是边缘- 可能失败, 甚至对于程度- 归属关系接近 1 。 这个版本是否不仅取决于程度- 归属关系, 而且还取决于基本的网络结构, 因而不能说是一个普遍现象。 我们为这个基于节点的版本为普遍友谊 Paradox 建立正和负结果, 并且考虑它对社会数据的影响 。
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