Mapping of a Diffusion Model on an Online Social Network to a Non-hermitian Quantum Chain

Abstract

After the success of Twitter and Facebook, understanding of structure and dynamics on online social networks has become extremely significant. There have been many statistical analyses done for online social networks in the literature, mainly in the context of computer science and social science. Instead of building phenomenological models from big data, our goal here is to understand the dynamics on online social networks from fundamental laws. Our interest of this article lies in the information diffusion on the network. In some online social networks such as Digg, the information diffusion mainly occurs owing to the fact that some featured contents attract many users on the network [1]; such a phenomenon is observed in other networks including Wikipedia and other web contents [2] as well. However, this is not a kind of information diffusion that we will discuss below. On Twitter and Facebook, there is a mechanism to let users spread information, i.e. retweet on Twitter and share on Facebook, and therefore the information diffuses because of the activity of many users. In this article, we will focus on this type of diffusion. The starting point of our discussion is a simple stochastic model which one of the authors introduced recently [3] in order to describe the diffusion phenomenon on the Twitter network. The aim of the present article is to consider a generalization of the model and try to express it in a physically tractable way, namely, mapping to a non-hermitian quantum chain. The original paper focused on the statistics of a stochastic variable in the model and did not go into details such as the structure of the underlying network and complex dynamics. Using a simplified network as a toy model, we have also analyzed the possibility of the viral diffusion on the network [4] in the case of a correlated stochastic process. While the extension to a generalized case seems very hard in a straight-forward treatment, because there are many mathematical techniques and properties explored in the context of non-hermitian quantum mechanics [5, 6], we expect that the mapping would give us a new insight. In outline, the article is as follows. We first review the basic stochastic model of the information diffusion and then discuss its generalization as we introduce the mapping to a non-hermitian quantum chain.