  Understanding the impact of messages on agents' opinions over social networks
is important. However, to our best knowledge, there has been limited
quantitative investigation into this phenomenon in the prior works. To address
this gap, this paper proposes the Message-Enhanced DeGroot model. The Bounded
Brownian Message model provides a quantitative description of the message
evolution, jointly considering temporal continuity, randomness, and
polarization from mass media theory. The Message-Enhanced DeGroot model,
combining the Bounded Brownian Message model with the traditional DeGroot
model, quantitatively describes the evolution of agents' opinions under the
influence of messages. We theoretically study the probability distribution and
statistics of the messages and agents' opinions and quantitatively analyze the
impact of messages on opinions. We also conduct simulations to validate our
analyses.
