  There has been considerable recent interest in the properties of networks,
such as citation networks and the worldwide web, that grow by the addition of
vertices, and a number of simple solvable models of network growth have been
studied. In the real world, however, many networks, including the web, not only
add vertices but also lose them. Here we formulate models of the time evolution
of such networks and give exact solutions for a number of cases of particular
interest. For the case of net growth and so-called preferential attachment --
in which newly appearing vertices attach to previously existing ones in
proportion to vertex degree -- we show that the resulting networks have
power-law degree distributions, but with an exponent that diverges as the
growth rate vanishes. We conjecture that the low exponent values observed in
real-world networks are thus the result of vigorous growth in which the rate of
addition of vertices far exceeds the rate of removal. Were growth to slow in
the future, for instance in a more mature future version of the web, we would
expect to see exponents increase, potentially without bound.
