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：友谊悖论——“你的朋友比你拥有更多朋友”——是图中度数的组合事实；但鉴于许多基于网络的社交活动与用户的度数相关，这一事实被更广泛地引申为经验原则：“你的朋友也比你更活跃。”这种广义友谊悖论认为，任何与度数正相关的属性都服从友谊悖论，并在网络层面的版本中从数学上得到确立，该版本本质上是对网络所有边的统一聚合。然而，我们在此证明，自然存在的基于节点的广义友谊悖论版本——即对节点而非边进行聚合——可能不成立，即使度数-属性相关性接近1。该版本是否成立不仅取决于度数-属性相关性，还依赖于底层网络结构，因此不能被视为普遍现象。我们针对广义友谊悖论的这个基于节点的版本确立了正反两方面的结果，并探讨了其对社交网络数据的启示。