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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We introduce a statistical-physics model for opinion dynamics on random networks where agents adopt the opinion held by the majority of their direct neighbors only if the fraction of these neighbors exceeds a certain threshold, p(u). We find a transition from total final consensus to a mixed phase where opinions coexist amongst the agents. The relevant parameters are the relative sizes in the initial opinion distribution within the population and the connectivity of the underlying network. As the order parameter we de. ne the asymptotic state of opinions. In the phase diagram we find regions of total consensus and a mixed phase. As the "laggard parameter" pu increases the regions of consensus shrink. In addition we introduce rewiring of the underlying network during the opinion formation process and discuss the resulting consequences in the phase diagram. Copyright (c) EPLA, 2008. 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... 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. We study the problem of two competing camps aiming to maximize the adoption of their respective opinions, by optimally investing in nodes of a social network in multiple phases. The final opinion of a node in a phase acts as its biased opinion in the following phase. Using an extension of Friedkin-Johnsen model, we formulate the camps' utility functions, which we show to involve what can be interpreted as multiphase Katz centrality. We hence present optimal investment strategies of the camps, and the loss incurred if myopic strategy is employed. Simulations affirm that nodes attributing higher weightage to bias necessitate higher investment in initial phase. The extended version of this paper analyzes a setting where a camp's influence on a node depends on the node's bias; we show existence and polynomial time computability of Nash equilibrium.