We address formally the problem of opinion dynamics when the agents of a social network (e.g., consumers) are not only influenced by their neighbors but also by an external influential entity referred to as a marketer. The influential entity tries to sway the overall opinion as close as possible to a desired opinion by using a specific influence budget. We assume that the exogenous influences of the entity happen during discrete-time advertising campaigns; consequently, the overall closed-loop opinion dynamics becomes a linear-impulsive (hybrid) one. The main technical issue addressed is finding how the marketer should allocate its budget over time (through marketing campaigns) and over space (among the agents) such that the agents' opinion be as close as possible to the desired opinion. Our main results show that the marketer has to prioritize certain agents over others based on their initial condition, their influence power in the social graph and the size of the cluster they belong to. The corresponding space-time allocation problem is formulated and solved for several special cases of practical interest. Valuable insights can be extracted from our analysis. For instance, for most cases, we prove that the marketer has an interest in investing most of its budget at the beginning of the process and that budget should be shared among agents according to the famous water-filling allocation rule. Numerical examples illustrate the analysis.

中文翻译：

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病毒式营销的时空预算分配策略设计

当社交网络的代理（例如，消费者）不仅受到其邻居的影响，而且还受到称为营销人员的外部影响实体的影响时，我们正式解决了意见动态问题。有影响力的实体试图通过使用特定的影响力预算，使总体意见尽可能接近所需意见。我们假设实体的外生影响发生在离散时间的广告活动中；因此，整个闭环意见动态变成了线性脉冲（混合）动态。解决的主要技术问题是寻找营销人员应如何随时间（通过营销活动）和空间（在代理之间）分配预算，以便代理的意见尽可能接近所需的意见。我们的主要结果表明，营销人员必须根据某些代理的初始条件、他们在社交图中的影响力以及他们所属集群的大小来优先考虑某些代理。针对几个具有实际意义的特殊情况，制定并解决了相应的时空分配问题。可以从我们的分析中提取有价值的见解。例如，在大多数情况下，我们证明营销人员有兴趣在流程开始时投资其大部分预算，并且该预算应根据著名的注水分配规则在代理之间共享。数值例子说明了分析。针对几个具有实际意义的特殊情况，制定并解决了相应的时空分配问题。可以从我们的分析中提取有价值的见解。例如，在大多数情况下，我们证明营销人员有兴趣在流程开始时投资其大部分预算，并且该预算应根据著名的注水分配规则在代理之间共享。数值例子说明了分析。针对几个具有实际意义的特殊情况，制定并解决了相应的时空分配问题。可以从我们的分析中提取有价值的见解。例如，在大多数情况下，我们证明营销人员有兴趣在流程开始时投资其大部分预算，并且该预算应根据著名的注水分配规则在代理之间共享。数值例子说明了分析。