# Reflecting on Social Influence in Networks

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## Abstract

In many social contexts, social influence seems to be inescapable: the behavior of others influences us to modify ours, and vice-versa. However, social psychology is full of examples of phenomena where individuals experience a discrepancy between their public behavior and their private opinion. This raises two central questions. First, how does an individual reason about the behavior of others and their private opinions in situations of social influence? And second, what are the laws of the resulting information dynamics? In this paper, we address these questions by introducing a formal framework for representing reasoning about an individual’s private opinions and public behavior under the dynamics of social influence in social networks. Moreover, we dig deeper into the involved information dynamics by modeling how individuals can learn about each other based on this reasoning. This compels us to introduce a new formal notion of *reflective social influence*. Finally, we initialize the work on proof theory and automated reasoning for our framework by introducing a sound and complete tableaux system for a fragment of our logic. Furthermore, this constitutes the first tableau system for the “Facebook logic” of J. Seligman, F. Liu, and P. Girard.

## Keywords

Social networks Modal logic Hybrid logic Opinion dynamics Social influence Tableau systems## Notes

### Acknowledgments

Zoé Christoff acknowledges support for this research from EPSRC (Grant EP/M015815/1, “Foundations of Opinion Formation in Autonomous Systems”), as well as from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013)/ERC Grant Agreement No. 283963. Carlo Proietti is sponsored by the Swedish Research Council (VR) through the project “Logical modelling of collective attitudes and their dynamics”.

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