[论文解读] How Complex Contagions Spread and Spread Quickly.

Roozbeh Ebrahimi, Jie Gao+2 more|arXiv (Cornell University)|Apr 10, 2014

Complex Network Analysis Techniques参考文献 46被引用 1

一句话总结

本文证明，$k$-复杂传播——即节点需有 $k$ 个感染邻居才会被感染——在时间演化网络（如 preferential attachment 和 copy 模型）中可迅速传播，当种子为最老节点时，可在 $O(\log n)$ 步内实现全网感染。关键洞见在于，演化网络结构（而不仅是幂律度分布）使传播加速，而随机种子会导致传播过早停止。

ABSTRACT

In this paper, we study the spreading speed of complex contagions in a social network. A $k$-complex contagion starts from a set of initially infected seeds such that any node with at least $k$ infected neighbors gets infected. Simple contagions, i.e., $k=1$, quickly spread to the entire network in small world graphs. However, fast spreading of complex contagions appears to be less likely and more delicate; the successful cases depend crucially on the network structure~\cite{G08,Ghasemiesfeh:2013:CCW}. Our main result shows that complex contagions can spread fast in a general family of time-evolving networks that includes the preferential attachment model~\cite{barabasi99emergence}. We prove that if the initial seeds are chosen as the oldest nodes in a network of this family, a $k$-complex contagion covers the entire network of $n$ nodes in $O(\log n)$ steps. We show that the choice of the initial seeds is crucial. If the initial seeds are uniformly randomly chosen in the PA model, even with a polynomial number of them, a complex contagion would stop prematurely. The oldest nodes in a preferential attachment model are likely to have high degrees. However, we remark that it is actually not the power law degree distribution per se that facilitates fast spreading of complex contagions, but rather the evolutionary graph structure of such models. Some members of the said family do not even have a power-law distribution. We also prove that complex contagions are fast in the copy model~\cite{KumarRaRa00}, a variant of the preferential attachment family. Finally, we prove that when a complex contagion starts from an arbitrary set of initial seeds on a general graph, determining if the number of infected vertices is above a given threshold is $\mathbf{P}$-complete. Thus, one cannot hope to categorize all the settings in which complex contagions percolate in a graph.

研究动机与目标

• 理解在何种网络结构下，需要多个感染邻居的复杂传播能够快速扩散。
• 研究网络演化与种子选择在促进复杂传播快速扩散中的作用。
• 确定快速传播的驱动力是幂律度分布，还是如 preferential attachment 模型中的演化图结构。
• 建立预测复杂传播是否能感染阈值数量节点的计算困难性结果。

提出的方法

• 分析 preferential attachment 和 copy 模型中的时间演化网络，这些模型通过优先连接和复制机制增长。
• 证明选择最老节点作为初始种子时，$k$-复杂传播可在 $O(\log n)$ 步内实现全网感染。
• 证明即使使用多项式数量的随机种子，也会在 preferential attachment 模型中导致传播过早停止。
• 表明演化结构（而非仅幂律度分布）是实现快速传播的关键，因为该家族中的某些模型并不具备幂律行为。
• 利用计算复杂性理论证明，在一般图上判断复杂传播是否超过阈值感染规模是 $\mathbf{P}$-完全的。

实验结果

研究问题

• RQ1 在时间演化网络（如 preferential attachment 模型）中，$k$-复杂传播能否快速扩散？
• RQ2 为何种子选择（特别是选择最老节点）能导致快速传播，而随机种子会失败？
• RQ3 幂律度分布是快速传播的关键驱动因素，还是演化网络结构才是关键？
• RQ4 我们能否对所有图结构设定进行分类，以判断复杂传播是否能蔓延？其预测的计算复杂度如何？

主要发现

• 当初始种子为网络中最老节点时，$k$-复杂传播可在 $O(\log n)$ 步内传播至 $n$ 个节点的整个网络。
• 在 preferential attachment 模型中，即使使用多项式数量的随机种子，传播也会过早停止。
• 时间演化网络的演化图结构比幂律度分布对复杂传播的快速扩散更为关键。
• copy 模型（preferential attachment 家族的变体）在相同种子条件下，同样支持 $k$-复杂传播的快速扩散。
• 在一般图上，判断复杂传播是否感染超过给定阈值数量的节点是 $\mathbf{P}$-完全的，意味着在一般情况下无法高效分类。