Sample Complexity of Learning Multi-value Opinions in Social Networks

Abstract

We consider how many users we need to query in order to estimate the extent to which multi-value opinions (information) have propagated in a social network. For example, if the launch date of a new product has changed many times, the company might want to know to which people the most current information has reached. In the propagation model we consider, the social network is represented as a directed graph, and an agent (node) updates its state if it receives a stronger opinion (updated information) and then forwards the opinion in accordance with the direction of its edges. Previous work evaluated opinion propagation in a social network by using the probably approximately correct (PAC) learning framework and considered only binary opinions. In general, PAC learnability, i.e., the finiteness of the number of samples needed, is not guaranteed when generalizing from a binary-value model to a multi-value model. We show that the PAC learnability of multi-value opinions propagating in a social network. We first prove that the number of samples needed in a multi-opinion model is sufficient for $(k-1)\log (k-1)$ times the number of samples needed in a binary-opinion model, when $k~(\ge 3)$ is the number of opinions. We next prove that the upper and lower bounds on the number of samples needed to learn a multi-opinion model can be determined from the Natarajan dimension, which is a generalization of the Vapnik-Chervonenkis dimension.