[论文解读] Selecting nodes and buying links to maximize the information diffusion in a network

Gianlorenzo D’Angelo, Lorenzo Severini+1 more|arXiv (Cornell University)|Jun 19, 2017

Complex Network Analysis Techniques参考文献 18被引用 8

一句话总结

本文提出了一种新颖的预算影响最大化问题（BIMA），通过允许购买与种子节点相连的新边来扩展经典的种子选择问题，从而在独立级联模型下增强信息传播。该文提出了两种近似算法：一种具有依赖成本的近似比，另一种具有恒定的近似因子0.0878，在边成本较低时优于先前的工作。

ABSTRACT

The Independent Cascade Model (ICM) is a widely studied model that aims to capture the dynamics of the information diffusion in social networks and in general complex networks. In this model, we can distinguish between active nodes which spread the information and inactive ones. The process starts from a set of initially active nodes called seeds. Recursively, currently active nodes can activate their neighbours according to a probability distribution on the set of edges. After a certain number of these recursive cycles, a large number of nodes might become active. The process terminates when no further node gets activated. Starting from the work of Domingos and Richardson, several studies have been conducted with the aim of shaping a given diffusion process so as to maximize the number of activated nodes at the end of the process. One of the most studied problems has been formalized by Kempe et al. and consists in finding a set of initial seeds that maximizes the expected number of active nodes under a budget constraint. In this paper we study a generalization of the problem of Kempe et al. in which we are allowed to spend part of the budget to create new edges incident to the seeds. That is, the budget can be spent to buy seeds or edges according to a cost function. The problem does not admin a PTAS, unless $P=NP$. We propose two approximation algorithms: the former one gives an approximation ratio that depends on the edge costs and increases when these costs are high; the latter algorithm gives a constant approximation guarantee which is greater than that of the first algorithm when the edge costs can be small.

研究动机与目标

• 为解决经典影响最大化仅选择种子节点的局限性，通过允许对与种子节点相连的新边进行投资来扩展问题。
• 建模现实场景中影响者可主动建立连接以扩大影响范围的情形。
• 在独立级联模型下，针对种子节点和边成本的预算约束，开发BIMA的近似算法。
• 分析边成本与影响传播之间的权衡，特别是在边成本较低时。
• 弥合已知的1−1/e ≈ 0.6321近似下界与当前最佳恒定近似因子0.0878之间的差距。

提出的方法

• 将BIMA问题形式化为：在预算约束下，选择一组种子节点及其关联的新边，以最大化期望激活节点数。
• 提出一种两阶段贪心算法：首先基于单位成本的边际增益选择种子和边，然后通过成本感知的近似策略优化解。
• 采用成本敏感分析以界定近似比，推导出依赖于最小边成本和控制搜索深度的参数b的近似比。
• 使用递归近似框架，每一步评估添加种子节点及其关联边的边际增益，受限于预算阈值。
• 应用已知的(1−1/e)-近似算法求解CostIMA子问题（在固定种子和边预算下最大化影响传播），以获得基础近似因子。
• 将子问题结果与全局近似比分析相结合，得出R CostIMA子问题的最终近似保证为(1−1/e)/(b+1)。

实验结果

研究问题

• RQ1 我们能否通过允许购买与种子节点相连的新边，而不仅限于选择种子节点，来改进影响最大化？
• RQ2 当边成本较低时，可实现的近似比是多少，与已知的(1−1/e)下界相比如何？
• RQ3 所提算法的性能如何随不同边成本分布变化，特别是在成本最小时？
• RQ4 尽管BIMA问题是NP难的且不存在PTAS，我们能否在边成本较低时实现恒定因子近似？
• RQ5 在预算分配与影响传播之间，种子选择与边购买的最优权衡是什么？

主要发现

• 除非P=NP，否则BIMA问题不具有PTAS，表明其具有固有的计算难度。
• 第一种算法的近似比为(1−1/e^((bα)/(bα+b+α+2)))，随着边成本增加而提升。
• 第二种算法保证了恒定的近似因子0.0878，这是在当前子问题近似框架下的最优结果。
• 当参数b被优化时，恒定因子算法的近似比达到最大值0.0878。
• 算法性能对最小边成本极为敏感：当c_min < 0.1011时，恒定因子算法优于依赖成本的算法。
• 性能曲线的交点出现在c_min ≈ 0.1011和c_min ≈ 0.3821，标志着应切换算法的临界点。