Continuous-time Opinion Dynamics on Multiple Interdependent Topics
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In this paper, and inspired by the recent discrete-time based works of [1,2], we study a continuous-time opinion dynamics model where the individuals discuss opinions on multiple logically interdependent topics. The logical interdependence between the different topics is captured by a matrix, which is distinct from the Laplacian matrix capturing interactions between individuals. We obtain a necessary and sufficient condition for the network to reach to a consensus on each separate topic. The condition involves both the interdependence matrix and Laplacian matrix together, and is thus distinct from the result of [1], where in the absence of stubborn individuals, separate regularity of the interdependence matrix and influence matrix (the discrete-time version of the Laplacian) is enough to ensure a consensus of opinions. Assuming that the interdependence matrix is fixed, we generate two sufficient conditions on the network, i.e. the Laplacian matrix, to ensure a consensus of opinions. For a class of interdependence matrices, we also establish the set of Laplacian matrices which guarantee consensus. The model is then expanded to include stubborn individuals, who remain attached to their initial opinions. Sufficient conditions are obtained for guaranteeing stability of the opinion dynamics system, with the final opinions being at a persistent disagreement as opposed to having reached a consensus. Simulations are provided to illustrate the results.
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