  The Hegselmann-Krause (HK) model allows one to characterize the continuous
change of agents' opinions with the bounded confidence threshold $\varepsilon$.
To consider the heterogeneity of agents in characteristics, we study the HK
model on homogeneous and heterogeneous networks by introducing a kind of smart
agent. Different from the averaging rule in opinion update of HK model, smart
agents will consider, in updating their opinions, the environmental influence
following the fact that an agent's behavior is often coupled with environmental
changes. The environment is characterized by a parameter that represents the
biased resource allocation between different cliques. We focus on the critical
behavior of the underlying system. A phase transition point separating a
complete consensus from the coexistence of different opinions is identified,
which occurs at a critical value $\varepsilon_c$ for the bounded confidence
threshold. We state analytically that $\varepsilon_c$ can take only one of two
possible values, depending on the behavior of the average degree $k_a$ of a
social graph, when agents are homogeneous in characteristics. Results also
suggest that the phase transition point weakly depends on the network structure
but is strongly correlated with the fraction of smart agents and the
environmental parameter. We finally establish the finite size scaling law that
stresses the role that the system size has in the underlying opinion dynamics.
Meanwhile, introducing smart agents does not change the functional dependence
between the time to reach a complete consensus and the system size. However, it
can drive a complete consensus to be reached faster, for homogeneous networks
that are far from the mean field limit.
