[论文解读] Approximation Analysis of Influence Spread in Social Networks

Amit Goyal, Francesco Bonchi+2 more|arXiv (Cornell University)|Aug 11, 2010

Complex Network Analysis Techniques参考文献 23被引用 20

一句话总结

本文提出了并分析了社会影响传播中的两个新优化问题：最小目标集选择（MINTSS）和最小传播时间（MINTIME）。针对MINTSS，提出了一种贪心算法，提供双准则近似，保证在种子集大小为最优值对数级别时，覆盖率达到目标的ε以内。对于MINTIME，表明在预算和覆盖度放宽对数级别的情况下，可实现三准则近似，同时证明了更紧致近似具有固有难度。

ABSTRACT

In the context of influence propagation in a social graph, we can identify three orthogonal dimensions - the number of seed nodes activated at the beginning (known as budget), the expected number of activated nodes at the end of the propagation (known as expected spread or coverage), and the time taken for the propagation. We can constrain one or two of these and try to optimize the third. In their seminal paper, Kempe et al. constrained the budget, left time unconstrained, and maximized the coverage: this problem is known as Influence Maximization. In this paper, we study alternative optimization problems which are naturally motivated by resource and time constraints on viral marketing campaigns. In the first problem, termed Minimum Target Set Selection (or MINTSS for short), a coverage threshold n is given and the task is to find the minimum size seed set such that by activating it, at least n nodes are eventually activated in the expected sense. In the second problem, termed MINTIME, a coverage threshold n and a budget threshold k are given, and the task is to find a seed set of size at most k such that by activating it, at least n nodes are activated, in the minimum possible time. Both these problems are NP-hard, which motivates our interest in their approximation. For MINTSS, we develop a simple greedy algorithm and show that it provides a bicriteria approximation. We also establish a generic hardness result suggesting that improving it is likely to be hard. For MINTIME, we show that even bicriteria and tricriteria approximations are hard under several conditions. However, if we allow the budget to be boosted by a logarithmic factor and allow the coverage to fall short, then the problem can be solved exactly in PTIME. Finally, we show the value of the approximation algorithms, by comparing them against various heuristics.

研究动机与目标

• 为解决病毒式营销中的资源与时间约束，提出超越传统影响最大化问题的新优化问题。
• 研究最小目标集选择（MINTSS）问题，旨在寻找实现给定期望覆盖阈值的最小种子集。
• 探讨最小传播时间（MINTIME）问题，目标是在给定预算下最小化达到覆盖阈值所需时间。
• 分析MINTSS与MINTIME的近似复杂度，建立难解性结果与近似保证。
• 通过在真实网络上的大量实验，评估所提贪心算法与主流启发式方法的性能表现。

提出的方法

• 为MINTSS提出一种贪心算法，通过迭代选择在期望传播范围上边际增益最高的节点，确保实现双准则近似。
• 建立一个通用的难解性结果，表明改进MINTSS的双准则近似因子可能属于NP难。
• 针对MINTIME，引入一种三准则近似算法，允许预算增加一个对数因子，且允许少量覆盖度不足，从而实现最优传播时间。
• 证明即使在标准复杂度假设下，MINTIME的双准则与三准则近似也具有固有难度。
• 将现有启发式方法——PageRank、高阶度、Pmia、Sp——进行适配，用于MINTSS与MINTIME，以支持对比评估。
• 采用独立级联（IC）与加权级联（WC）模型模拟影响传播，并按时间步长测量覆盖度。

实验结果

研究问题

• RQ1 贪心算法能否为最小目标集选择（MINTSS）问题提供可证明的双准则近似？
• RQ2 在预算与覆盖度放宽的条件下，是否可能实现MINTIME问题的三准则近似？
• RQ3 MINTSS与MINTIME的内在近似复杂度如何？能否实现更紧致的近似？
• RQ4 在实际性能上，所提贪心算法与现有启发式方法相比，在覆盖度与传播时间方面表现如何？
• RQ5 图结构与影响模型（IC与WC）在多大程度上影响不同种子选择策略的性能表现？

主要发现

• MINTSS的贪心算法实现了双准则近似：其返回的种子集大小至多为最优值的(1 + ln(η/ε))倍，且覆盖度至少为η − ε。
• 对于MINTIME，贪心算法提供三准则近似：在预算扩大(1 + ln(η/ε))倍、覆盖度不足ε的条件下，可实现最优传播时间。
• 在NetHEPT与Meme数据集上的实验结果表明，贪心算法在更少时间步内实现目标覆盖度方面，始终优于Random、High Degree、Pagerank与Sp等启发式方法。
• 在NetHEPT数据集上，IC模型下均匀概率为0.1时，Greedy在5个时间步内以75个种子实现900的覆盖度，而Sp耗时6步，Pmia耗时14步。
• 当覆盖度阈值提高至1000且种子数为75时，Greedy在6步内达到目标，而所有其他启发式方法均未能找到解。
• 随着目标覆盖度的提升，Greedy与启发式方法之间的性能差距进一步扩大，其中在Meme数据集上Pmia是表现最接近的竞争对手，尤其在η > 1600时更为显著。