  Deterministic dynamics is a mathematical model used to describe the temporal
evolution of a system, generally expressed as dx/dt = F(x), where x represents
the system's state, and F(x) determines its dynamics. It is employed to
understand long-term system behavior, including opinion formation and
polarization in online communities.
  Opinion dynamics models, like the Katz model and the logistic map, help
analyze how individual opinions are influenced within social networks and
exhibit chaotic behavior. These models are crucial for studying opinion
formation and collective behavior on social media, especially in conjunction
with branching theory.
  For instance, Galam's Ising model applies principles from physics to social
sciences, representing individual opinions as "spins" and illustrating how
local interactions influence consensus formation. The Bounding Confidence model
considers opinions within a confidence interval, showing how opinions converge
or polarize.
  These models effectively analyze opinion dynamics in online communities,
aiding in understanding trends and viral phenomena on social media. This
research aims to analyze discourse flow and opinion evolution, predicting
future trends in online communities and decoding digital-age human interaction
dynamics. Combining branching theory with opinion dynamics models enhances our
understanding of digital communication.
  In the modified opinion dynamics model, the weight parameter h for distance
introduces distance-based interaction terms. The update equation is adjusted to
control interaction strength based on distance, with hij calculated from the
distance dij. This customization allows for a more accurate representation of
opinion dynamics in specific scenarios, forming the basis for discussions in
this paper.
