  This research note is organized with respect to a novel approach to solving
problems related to the spread of fake news and effective fact-checking.
Focusing on the least-cost routing problem, the discussion is organized with
respect to the use of Metzler functions and Metzler matrices to model the
dynamics of information propagation among news providers. With this approach,
we designed a strategy to minimize the spread of fake news, which is
detrimental to informational health, while at the same time maximizing the
spread of credible information. In particular, through the punitive dominance
problem and the maximum compensation problem, we developed and examined a path
to reassess the incentives of news providers to act and to analyze their impact
on the equilibrium of the information market. By applying the concept of
entanglement to the context of information propagation, we shed light on the
complexity of interactions among news providers and contribute to the
formulation of more effective information management strategies. This study
provides new theoretical and practical insights into issues related to fake
news and fact-checking, and will be examined against improving informational
health and public digital health.This paper is partially an attempt to utilize
"Generative AI" and was written with educational intent. There are currently no
plans for it to become a peer-reviewed paper.
