Given a social network of users with selection cost and a fixed budget, the problem of Budgeted Influence Maximization finds a subset of the nodes ( known as seed nodes) for initial activation to maximize the influence, such that the total selection cost is within the allocated budget. Existing solution methodologies for this problem make two assumptions, which are not applicable to real-life situations. First, an influenced node of the current time stamp can trigger only once in the next time stamp to its inactive neighbors and the other one is the diffusion process continues forever. To make the problem more practical, in this paper, we introduce the Budgeted Influence Maximization with Delay by relaxing the single time triggering constraint and imposing an additional constraint for maximum allowable diffusion time. For this purpose, we consider a delay distribution for each edge of the network, and consider a node is influenced, if it is so, within the allowable diffusion time. We first propose an incremental greedy strategy for solving this problem, which works based on the approximate computation of marginal gain in influence spread. Next, we make two subsequent improvements of this algorithm in terms of efficiency by exploiting the sub-modularity property of the time delayed influence function. We implement the proposed methodologies with three benchmark datasets. Reported results show that the seed set selected by the proposed methodologies can lead to more number of influenced nodes compared to that obtained by other baseline methods. We also observe that between the two improvised methodologies, the second one is more efficient for the larger datasets.

给定一个具有选择成本和固定预算的用户社交网络，预算影响最大化问题寻找初始激活的节点子集（称为种子节点）以最大化影响，使得总选择成本在分配的范围内预算。此问题的现有解决方案方法有两个假设，它们不适用于现实生活中的情况。首先，当前时间戳的受影响节点只能在下一个时间戳对其不活动的邻居触发一次，另一个是扩散过程永远持续下去。为了使问题更实际，在本文中，我们引入了带延迟的预算影响最大化通过放宽单次触发约束并对最大允许扩散时间施加附加约束。为此，我们考虑网络每个边缘的延迟分布，并考虑在允许的扩散时间内节点受到影响（如果是）。我们首先提出了一种解决这个问题的增量贪婪策略，该策略基于影响传播边际增益的近似计算。接下来，我们通过利用子模块性，在效率方面对该算法进行了两次后续改进时滞影响函数的性质。我们使用三个基准数据集实现了所提出的方法。报告的结果表明，与其他基线方法获得的相比，所提出的方法选择的种子集可以导致更多数量的受影响节点。我们还观察到，在两种临时方法之间，第二种方法对于较大的数据集更有效。