Optimal selection of the most informative nodes in Opinion Dynamics on Networks

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Finding the optimal subset to observe in a network system is a fundamental problem in science and engineering, with a wide range of applications like monitoring spatial phenomena, control of epidemic spread, feature selection in machine learning, or active surveying in social studies. The goal of this paper is to address the subset selection problem on an Opinion Dynamics model where the variable of interest Y is the average opinion of the community. We consider the opinion vector X to be updated according to a Friedkin-Johnsen opinion dynamics model where every agent i is equipped with an original unknown belief ui, which is assumed to be normally distributed, and a parameter λi describing its openness to interactions. The objective function of the optimization problem is the variance reduction from the observation of the steady-state opinions of a subset K ⊆ V of agents. We show how this functional can be rewritten in terms of the Bonacich centrality and the cycle centrality of the agents in social network when the subset selection is of cardinality 1, providing particular graph-theoretic interpretations related to the network itself. In addition, first exploratory simulations highlight a behaviour which deviates from the one of known centrality measures depending on the choice of model parameters. Finally, we show that the submodularity of the functional is not guaranteed in our case and thus results taken from known literature are non-enforceable. This paves the way for further analysis.