  Complex networks with expanding dimensions are studied, where the networks
may be directed and weighted, and network nodes are varying in discrete time in
the sense that some new nodes may be added and some old nodes may be removed
from time to time. A model of such networks in computer data transmission is
discussed. Each node on the network has fixed dimensionality, while the
dimension of the whole network is defined by the total number of nodes. Based
on the spectacular properties of data transmission on computer networks, some
new concepts of stable and unstable networks differing from the classical
Lyapunov stability are defined. In particular, a special unstable network
model, called devil network, is introduced and discussed. It is further found
that a variety of structures and connection weights affects the network
stability substantially. Several criteria on stability, instability, and devil
network are established for a rather general class of networks, where some
conditions are actually necessary and sufficient. Mathematically, this paper
makes a first attempt to rigorously formulate a fundamental issue of modeling
discrete linear time-varying systems with expanding dimensions and study their
basic stability property.
