  In real-world networks the interactions between network elements are
inherently time-delayed. These time-delays can not only slow the network but
can have a destabilizing effect on the network's dynamics leading to poor
performance. The same is true in computational networks used for machine
learning etc. where time-delays increase the network's memory but can degrade
the network's ability to be trained. However, not all networks can be
destabilized by time-delays. Previously, it has been shown that if a network or
high-dimensional dynamical system is intrinsically stable, which is a stronger
form of the standard notion of global stability, then it maintains its
stability when constant time-delays are introduced into the system. Here we
show that intrinsically stable systems, including intrinsically stable networks
and a broad class of switched systems, i.e. systems whose mapping is
time-dependent, remain stable in the presence of any type of time-varying
time-delays whether these delays are periodic, stochastic, or otherwise. We
apply these results to a number of well-studied systems to demonstrate that the
notion of intrinsic stability is both computationally inexpensive, relative to
other methods, and can be used to improve on some of the best known stability
results. We also show that the asymptotic state of an intrinsically stable
switched system is exponentially independent of the system's initial
conditions.
