A Mathematical Foundation for Stochastic Opinion Dynamics

Introduction

The ability to track how a person or a group’s opinions about a topic evolves is vital in several domains such as marketing, finance, and public policy. This ability enables researchers and practitioners to identify emerging opinions, opinion leaders, influencers, and stubborn agents¹. The ability to collate such opinions coming from multiple sources and identify consensus solutions is also important in domains such as engineering, decision sciences, and information sciences where problems such as multi-agent coordination, group decision making, and information fusion abound (Ren, Beard, & Atkins, 2005). Consequently, the literature on modeling opinion and consensus dynamics is evolving at a rapid pace. However, the multi-disciplinary nature of the topic and the fast pace of development has left several gaps in the theory and between theory and practice. As an example, it is unclear as to how an opinion can be defined or measured, or how the heterogeneity in the agent behavior impacts opinion dynamics, or how an external observer could identify the degree of influence that one agent has over the other in real systems such as Twitter (Bihl, Young, & Weckman, 2016; Sobkowicz, 2009; Sobkowicz, Kaschesky, & Bouchard, 2012).

The growth and popularity of large online social systems such as Twitter, review platforms such as Yelp, and user-generated content in the form of blogs and product reviews have made the need for means to track opinion dynamics on large systems using real data imperative. These systems also provide data and system information to researchers to test new theories and devise methods to bridge the gap between theory and practice (Sobkowicz et al., 2012). This paper aims at bridging the gap between theory and practice by proposing a probabilistic representation of an opinion and providing a mathematical foundation for tracking opinions on real online systems.

The paper makes two contributions to the extant literature on opinion dynamics. First, it proposes the modeling of a person’s opinion as a probability distribution as against a point object. This probability distribution based representation is aligned with how opinions could actually be extracted from real online systems using natural language processing (Dakota & Kübler, 2016; Gupta & Gupta, 2016; Wickramarathne, Premaratne, Murthi, & Chawla, 2014) as well as recent research in the areas of economics, psychology, and sociology that shows that an opinion is not a fixed point (Budescu & Rantilla, 2000; Budescu & Yu, 2007; Jackson & López-Pintado, 2013; Li, Myaeng, & Kim, 2007; Nakata, 2003). The use of a probability distribution to represent an agent’s opinion also enables us to observe agreement regions, i.e., regions where the opinions of two agents overlap when global consensus is absent. These agreement regions yield a compromise solution.

The paper also extends the concept of consensus into the probabilistic framework by building on the concept of stability in dynamic linear systems (Kovalev, Kolmanovskii, & Shaikhet, 1998; Paternoster & Shaikhet, 2000). The proposed definition is less restrictive as compared to the notion of strong and weak consensus used in extant literature (Hegselmann & Krause, 2002; Li, Scaglione, Swami, & Zhao, 2013; Li, Braunstein, Wang, Shao, Stanley, & Havlin, 2013; Olfati-Saber & Murray, 2004; Ren et al., 2005).