  Transition points mark qualitative changes in the macroscopic properties of
large complex systems. Explosive transitions, exhibiting properties of both
continuous and discontinuous phase transitions, have recently been uncovered in
network growth processes. Real networks not only grow but often also
restructure, yet common network restructuring processes, such as small world
rewiring, do not exhibit phase transitions. Here, we uncover a class of
intrinsically discontinuous transitions emerging in network restructuring
processes controlled by \emph{adhesion} -- the preference of a chosen link to
remain connected to its end node. Deriving a master equation for the temporal
network evolution and working out an analytic solution, we identify genuinely
discontinuous transitions in non-growing networks, separating qualitatively
distinct phases with monotonic and with peaked degree distributions.
Intriguingly, our analysis of heuristic data indicates a separation between the
same two forms of degree distributions distinguishing abstract from
face-to-face social networks.
