论社交网络中影响力的子模块性
On the Submodularity of Influence in Social Networks
摘要：
我们证明并扩展了Kempe，Kleinberg，and Tardos（KKT）关于社交网络中影响力传播的一个猜想。社交网络可以由有向图表示，其中节点是个体，边指示社交关系的形式。一个简单的方法来模拟思想的传播，创新行为，或"口碑“效应在这样一个图是考虑一个增过程的"感染”（或活动）节点：每个节点成为感染一旦激活函数的一组感染的邻居超过一定的门限值。这种模型是由KKT在\cite{KeKlTa:03,KeKlTa:05}中引入的，作者还强加了几个自然假设：门限值是（一致）随机的;激活函数是单调的和次模的。对于活动节点$S$的初始集合，令$σ(S)$表示终止时活动节点的预期数量。本文证明了KKT的一个猜想：在上述假设下，证明了函数$σ(S)$是次模的。我们证明了相同的结果为期望值的任何单调，次模函数的一组活动节点在终止。
英文摘要：
We prove and extend a conjecture of Kempe, Kleinberg, and Tardos (KKT) on the spread of influence in social networks. A social network can be represented by a directed graph where the nodes are individuals and the edges indicate a form of social relationship. A simple way to model the diffusion of ideas, innovative behavior, or ``word-of-mouth'' effects on such a graph is to consider an increasing process of ``infected'' (or active) nodes: each node becomes infected once an activation function of the set of its infected neighbors crosses a certain threshold value. Such a model was introduced by KKT in \cite{KeKlTa:03,KeKlTa:05} where the authors also impose several natural assumptions: the threshold values are (uniformly) random; and the activation functions are monotone and submodular. For an initial set of active nodes $S$, let $σ(S)$ denote the expected number of active nodes at termination. Here we prove a conjecture of KKT: we show that the function $σ(S)$ is submodular under the assumptions above. We prove the same result for the expected value of any monotone, submodular function of the set of active nodes at termination.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
