Kumar Vineet, Krackhardt David, Feld Scott
Yale School of Management, Yale University, New Haven, CT 06511.
John Heinz III College of Public Policy and Management, Carnegie Mellon University, Pittsburgh, PA 15213.
Proc Natl Acad Sci U S A. 2024 Jul 23;121(30):e2306412121. doi: 10.1073/pnas.2306412121. Epub 2024 Jul 19.
We provide the mathematical and empirical foundations of the friendship paradox in networks, often stated as "Your friends have more friends than you." We prove a set of network properties on friends of friends and characterize the concepts of ego-based and alter-based means. We propose a network property called inversity that quantifies the imbalance in degrees across edges and prove that the sign of inversity determines the ordering between ego-based or alter-based means for any network, with implications for interventions. Network intervention problems like immunization benefit from using highly connected nodes. We characterize two intervention strategies based on the friendship paradox to obtain such nodes, with the alter-based and ego-based strategy. Both strategies provide provably guaranteed improvements for any network structure with variation in node degrees. We demonstrate that the proposed strategies obtain several-fold improvement (100-fold in some networks) in node degree relative to a random benchmark, for both generated and real networks. We evaluate how inversity informs which strategy works better based on network topology and show how network aggregation can alter inversity. We illustrate how the strategies can be used to control contagion of an epidemic spreading across a set of village networks, finding that these strategies require far fewer nodes to be immunized (less than 50%, relative to random). The interventions do not require knowledge of network structure, are privacy-sensitive, are flexible for time-sensitive action, and only require selected nodes to nominate network neighbors.
我们提供了网络中友谊悖论的数学和实证基础，友谊悖论通常表述为“你的朋友比你有更多的朋友”。我们证明了关于朋友的朋友的一组网络属性，并刻画了基于自我和基于他人的均值概念。我们提出了一种称为逆性的网络属性，它量化了边之间度数的不平衡，并证明逆性的符号决定了任何网络中基于自我或基于他人的均值之间的排序，这对干预措施有影响。像免疫这样的网络干预问题受益于使用高度连接的节点。我们刻画了基于友谊悖论的两种干预策略来获取此类节点，即基于他人的策略和基于自我的策略。对于任何节点度数存在变化的网络结构，这两种策略都能提供可证明的保证改进。我们证明，对于生成网络和真实网络，相对于随机基准，所提出的策略在节点度数上能实现数倍的提升（在某些网络中可达100倍）。我们评估逆性如何根据网络拓扑结构告知哪种策略效果更好，并展示网络聚合如何改变逆性。我们说明了这些策略如何用于控制在一组村庄网络中传播的流行病的蔓延，发现这些策略所需免疫的节点要少得多（相对于随机情况不到50%）。这些干预措施不需要了解网络结构，对隐私敏感，对时间敏感的行动具有灵活性，并且只需要选定的节点提名网络邻居。
