A signed graph offers richer information than an unsigned graph, since it describes both collaborative and competitive relationships in social networks. In this paper, we study opinion dynamics on a signed graph, based on the Friedkin-Johnsen model. We first interpret the equilibrium opinion in terms of a defined random walk on an augmented signed graph, by representing the equilibrium opinion of every node as a combination of all nodes' internal opinions, with the coefficient of the internal opinion for each node being the difference of two absorbing probabilities. We then quantify some relevant social phenomena and express them in terms of the $\ell_2$ norms of vectors. We also design a nearly-linear time signed Laplacian solver for assessing these quantities, by establishing a connection between the absorbing probability of random walks on a signed graph and that on an associated unsigned graph. We further study the opinion optimization problem by changing the initial opinions of a fixed number of nodes, which can be optimally solved in cubic time. We provide a nearly-linear time algorithm with error guarantee to approximately solve the problem. Finally, we execute extensive experiments on sixteen real-life signed networks, which show that both of our algorithms are effective and efficient, and are scalable to massive graphs with over 20 million nodes.

翻译：符号图比无符号图提供了更丰富的信息，因为它描述了社交网络中协作与竞争两种关系。本文基于Friedkin-Johnsen模型，研究了符号图上的意见动力学。我们首先通过在增广符号图上定义随机游走，对均衡意见进行解释：将每个节点的均衡意见表示为所有节点内在意见的组合，其中每个节点内在意见的系数为两个吸收概率之差。随后，我们量化了若干相关的社会现象，并用向量的$\ell_2$范数进行表达。通过建立符号图上随机游走吸收概率与对应无符号图上吸收概率之间的联系，我们设计了一种近似线性时间的符号拉普拉斯求解器，用于评估这些量。此外，我们通过改变固定数量节点的初始意见，研究了意见优化问题，该问题可在三次时间内求得最优解。我们提出了一种具有误差保证的近似线性时间算法来近似求解该问题。最后，我们在十六个真实符号网络上进行了大量实验，结果表明我们的两种算法均有效且高效，并能扩展到具有超过2000万个节点的大规模图。