Consensus formation in opinion dynamics with online and offline interactions at complex networks

Nowadays, with the development of information communication technology and Internet, more and more people receive information and exchange their opinions with others via online environments (e.g. Twitter, Facebook, Weibo, and WeChat). According to eMarketer Report [Worldwide Internet and Mobile Users: eMarketer’s Updated Estimates and Forecast for 2015–2020 (eMarketer Report). Published October 11, 2016, https://www.emarketer.com/Report/Worldwide-Internet-Mobile-Users-eMarketers-Updated-Estimates-Forecast-20152020/2001897).], by the end of 2016, more than 3.2 billion individuals worldwide will use the Internet regularly, accounting for nearly 45% of the world population. By contrast, the other half of the global population still obtain information and regularly exchange their opinions in a more traditional way (e.g. face to face). Generally, the speed at which information spreads and opinions are exchanged and updated in an online environment is much faster than in an offline environment. This paper focuses on jointly investigating the challenge of consensus formation in opinion dynamics with online and offline interactions. Without loss of generality, we assume the speed at which information spreads and opinions are exchanged and updated in an online environment is (Formula presented.) times as fast as in an offline environment. We demonstrate that the update speed ratio in mixed online and offline environments (i.e. (Formula presented.)) strongly impacts the consensus formation at complex networks: a large update speed ratio of online and offline environments (i.e. (Formula presented.)) makes it difficult for all agents to reach consensus in opinion dynamics. Furthermore, these effects are often further intensified as the number of online participating agents increases.