  People's opinions change with time as they interact with each other. In a
bounded-confidence model (BCM) of opinion dynamics, individuals (which are
represented by the nodes of a network) have continuous-valued opinions and are
influenced by neighboring nodes whose opinions are sufficiently similar to
theirs (i.e., are within a confidence bound). In this paper, we formulate and
analyze discrete-time BCMs with heterogeneous and adaptive confidence bounds.
We introduce two new models: (1) a BCM with synchronous opinion updates that
generalizes the Hegselmann--Krause (HK) model and (2) a BCM with asynchronous
opinion updates that generalizes the Deffuant--Weisbuch (DW) model. We
analytically and numerically explore our adaptive BCMs' limiting behaviors,
including the confidence-bound dynamics, the formation of clusters of nodes
with similar opinions, and the time evolution of an "effective graph", which is
a time-dependent subgraph of a network with edges between nodes that {are
currently receptive to each other.} For a variety of networks and a wide range
of values of the parameters that control the increase and decrease of
confidence bounds, we demonstrate numerically that our adaptive BCMs result in
fewer major opinion clusters and longer convergence times than the baseline
(i.e., nonadaptive) BCMs. We also show that our adaptive BCMs can have adjacent
nodes that converge to the same opinion but are not {receptive to each other.}
This qualitative behavior does not occur in the associated baseline BCMs.
