  We study evolving networks based on the Barabasi-Albert scale-free network
model with vertices sensitive to overload breakdown. The load of a vertex is
defined as the betweenness centrality of the vertex. Two cases of load
limitation are considered, corresponding to that the average number of
connections per vertex is increasing with the network's size ("extrinsic
communication activity"), or that it is constant ("intrinsic communication
activity"). Avalanche-like breakdowns for both load limitations are observed.
In order to avoid such avalanches we argue that the capacity of the vertices
has to grow with the size of the system. An interesting irregular dynamics of
the formation of the giant component (for the intrinsic communication activity
case is also studied). Implications on the growth of the Internet is discussed.
