  Formation and fragmentation of networks is typically studied using
percolation theory, but most previous research has been restricted to studying
a phase transition in cluster size, examining the emergence of a giant
component. This approach does not study the effects of evolving network
structure on dynamics that occur at the nodes, such as the synchronization of
oscillators and the spread of information, epidemics, and neuronal excitations.
We introduce and analyze new link-formation rules, called Social Climber (SC)
attachment, that may be combined with arbitrary percolation models to produce a
previously unstudied phase transition using the largest eigenvalue of the
network adjacency matrix as the order parameter. This eigenvalue is significant
in the analyses of many network-coupled dynamical systems in which it measures
the quality of global coupling and is hence a natural measure of connectivity.
We highlight the important self-organized properties of SC attachment and
discuss implications for controlling dynamics on networks.
