Abstract We propose a setting for two-phase opinion dynamics in social networks, where a node’s final opinion in the first phase acts as its initial biased opinion in the second phase. In this setting, we study the problem of two camps aiming to maximize adoption of their respective opinions, by strategically investing on nodes in the two phases. A node’s initial opinion in the second phase naturally plays a key role in determining the final opinion of that node, and hence also of other nodes in the network due to its influence on them. However, more importantly, this bias also determines the effectiveness of a camp’s investment on that node in the second phase. In order to formalize this two-phase investment setting, we propose an extension of Friedkin–Johnsen model, and hence formulate the utility functions of the camps. We arrive at a decision parameter which can be interpreted as two-phase Katz centrality. There is a natural tradeoff while splitting the available budget between the two phases. A lower investment in the first phase results in worse initial biases in the network for the second phase. On the other hand, a higher investment in the first phase spares a lower available budget for the second phase, resulting in an inability to fully harness the influenced biases. We first analyze the non-competitive case where only one camp invests, for which we present a polynomial time algorithm for determining an optimal way to split the camp’s budget between the two phases. We then analyze the case of competing camps, where we show the existence of Nash equilibrium and that it can be computed in polynomial time under reasonable assumptions. We conclude our study with simulations on real-world network datasets, in order to quantify the effects of the initial biases and the weightage attributed by nodes to their initial biases, as well as that of a camp deviating from its equilibrium strategy. Our main conclusion is that, if nodes attribute high weightage to their initial biases, it is advantageous to have a high investment in the first phase, so as to effectively influence the biases to be harnessed in the second phase.
The topic of opinion dynamics has received significant attention in the social networks community. Xia, Wang, and Xuan @cite_27 give a multidisciplinary review of the field of opinion dynamics as a combination of the social processes and the analytical and computational tools. A line of work deals with opinion diffusion in social networks under popular models such as the independent cascade and linear threshold @cite_2 @cite_39 @cite_46 . Lorenz @cite_25 surveys several modeling frameworks concerning continuous opinion dynamics. Acemoglu and Ozdaglar @cite_42 review several other models of opinion dynamics, some noteworthy ones being DeGroot @cite_34 , Voter @cite_38 , Friedkin-Johnsen @cite_40 @cite_8 , bounded confidence @cite_9 , etc. In Friedkin-Johnsen model, each node updates its opinion using a weighted combination of its initial bias and its neighbors' opinions. In this paper, we generalize this model to multiple phases, while also incorporating the camps' investments.
@cite_38 @cite_8 @cite_9 @cite_42 @cite_39 @cite_27 @cite_40 @cite_2 @cite_46 @cite_34 @cite_25
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In this article we derive implications about social positions from a formal theory of social influence. The formal theory describes how, in a group of actors with heterogeneous initial opinions, a network of interpersonal influences enters into the formation of actors' settled opinions. We derive the following conclusions about a special form of structural equivalence. If actors are structurally equivalent in the network of interpersonal influences, then any dissimilarity of their initial opinions is reduced by the social influence process. If the social positions of actors are identical, i.e. if they have identical initial opinions and are structurally equivalent in the influence network, then they have identical opinions at equilibrium. If actors are not structurally equivalent in the network of interpersonal influences, then the social influence process does not necessarily reduce dissimilarities of initial opinions. We extend our analysis to consider automorphic equivalence. Consensus formation among n experts is modeled as a positive discrete dynamical system in n dimensions. The well–known linear but non–autonomous model is extended to a nonlinear one admitting also various kinds of averaging beside the weighted arithmetic mean. For this model a sufficient condition for reaching a consensus is presented. As a special case consensus formation under bounded confidence is analyzed. We provide an overview of recent research on belief and opinion dynamics in social networks. We discuss both Bayesian and non-Bayesian models of social learning and focus on the implications of the form of learning (e.g., Bayesian vs. non-Bayesian), the sources of information (e.g., observation vs. communication), and the structure of social networks in which individuals are situated on three key questions: (1) whether social learning will lead to consensus, i.e., to agreement among individuals starting with different views; (2) whether social learning will effectively aggregate dispersed information and thus weed out incorrect beliefs; (3) whether media sources, prominent agents, politicians and the state will be able to manipulate beliefs and spread misinformation in a society. As a key sub-field of social dynamics and sociophysics, opinion dynamics utilizes mathematical and physical models and the agent-based computational modeling tools, to investigate the spreading of opinions in a collection of human beings. This research field stems from various disciplines in social sciences, especially the social influence models developed in social psychology and sociology. A multidisciplinary review is given in this paper, attempting to keep track of the historical development of the field and to shed light on its future directions. In the review, the authors discuss the disciplinary origins of opinion dynamics, showing that the combination of the social processes, which are conventionally studied in social sciences, and the analytical and computational tools, which are developed in mathematics, physics and complex system studies, gives birth to the interdisciplinary field of opinion dynamics. The current state of the art of opinion dynamics is then overviewed, with the research progresses on the typical models like the voter model, the Sznajd model, the culture dissemination model, and the bounded confidence model being highlighted. Correspondingly, the future directions of this academic field are envisioned, with an advocation for closer synthesis of the related disciplines. In this paper we describe an approach to the relationship between a network of interpersonal influences and the content of individuals’ opinions. Our work starts with the specification of social process rather than social equilibrium. Several models of social influence that have appeared in the literature are derived as special cases of the approach. Some implications for theories on social conflict and conformity also are developed in this paper. Are all film stars linked to Kevin Bacon? Why do the stock markets rise and fall sharply on the strength of a vague rumour? How does gossip spread so quickly? Are we all related through six degrees of separation? There is a growing awareness of the complex networks that pervade modern society. We see them in the rapid growth of the internet, the ease of global communication, the swift spread of news and information, and in the way epidemics and financial crises develop with startling speed and intensity. This introductory book on the new science of networks takes an interdisciplinary approach, using economics, sociology, computing, information science and applied mathematics to address fundamental questions about the links that connect us, and the ways that our decisions can have consequences for others. Abstract Consider a group of individuals who must act together as a team or committee, and suppose that each individual in the group has his own subjective probability distribution for the unknown value of some parameter. A model is presented which describes how the group might reach agreement on a common subjective probability distribution for the parameter by pooling their individual opinions. The process leading to the consensus is explicitly described and the common distribution that is reached is explicitly determined. The model can also be applied to problems of reaching a consensus when the opinion of each member of the group is represented simply as a point estimate of the parameter rather than as a probability distribution. Models of continuous opinion dynamics under bounded confidence have been presented independently by Krause and Hegselmann and by in 2000. They have raised a fair amount of attention in the communities of social simulation, sociophysics and complexity science. The researchers working on it come from disciplines such as physics, mathematics, computer science, social psychology and philosophy. In these models agents hold continuous opinions which they can gradually adjust if they hear the opinions of others. The idea of bounded confidence is that agents only interact if they are close in opinion to each other. Usually, the models are analyzed with agent-based simulations in a Monte Carlo style, but they can also be reformulated on the agent's density in the opinion space in a master equation style. The contribution of this survey is fourfold. First, it will present the agent-based and density-based modeling frameworks including the cases of multidimensional opinions and heterogeneous bounds o...