  There has been a rich interplay in recent years between (i) empirical
investigations of real world dynamic networks, (ii) analytical modeling of the
microscopic mechanisms that drive the emergence of such networks, and (iii)
harnessing of these mechanisms to either manipulate existing networks, or
engineer new networks for specific tasks. We continue in this vein, and study
the deletion phenomenon in the web by following two different sets of web-sites
(each comprising more than 150,000 pages) over a one-year period. Empirical
data show that there is a significant deletion component in the underlying web
networks, but the deletion process is not uniform. This motivates us to
introduce a new mechanism of preferential survival (PS), where nodes are
removed according to a degree-dependent deletion kernel. We use the mean-field
rate equation approach to study a general dynamic model driven by Preferential
Attachment (PA), Double PA (DPA), and a tunable PS, where c nodes (c<1) are
deleted per node added to the network, and verify our predictions via
large-scale simulations. One of our results shows that, unlike in the case of
uniform deletion, the PS kernel when coupled with the standard PA mechanism,
can lead to heavy-tailed power law networks even in the presence of extreme
turnover in the network. Moreover, a weak DPA mechanism, coupled with PS, can
help make the network even more heavy-tailed, especially in the limit when
deletion and insertion rates are almost equal, and the overall network growth
is minimal. The dynamics reported in this work can be used to design and
engineer stable ad hoc networks and explain the stability of the power law
exponents observed in real-world networks.
