Friedkin-Johnsen Model for Opinion Dynamics on Signed Graphs
Xiaotian Zhou, Haoxin Sun, Wanyue Xu, Wei Li, Zhongzhi Zhang
TL;DR
The paper addresses opinion dynamics on signed graphs under the Friedkin–Johnsen framework, introducing a random-walk interpretation on an augmented signed graph to express equilibrium opinions and developing a nearly-linear time signed Laplacian solver. It establishes a connection to an associated unsigned graph to enable fast approximations of social-phenomena measures and proposes ApproxQuan for fast evaluation. For opinion optimization, it presents an optimal O(n^3) solution and a nearly-linear time ApproxOpin with provable error guarantees, enabling scalable handling of large networks. Extensive experiments on real and synthetic networks demonstrate that the proposed methods are both accurate and scalable to graphs with tens of millions of nodes, offering practical tools for analysis and control of opinion dynamics on signed networks.
Abstract
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.