  A generalization of the de Gennes-Alexander micronetworks theory is
presented. In this framework, the phase transition of synthetic networks of
superconducting islands is described by means of a Ginzburg-Landau approach
adapted to the case of granular systems. The general implications of the theory
are carefully explained. As a specific example, we demonstrate that star
networks support the exponential localization of the order parameter
accompanied by an enhancement of the critical temperature of the system. These
findings contribute to clarify the physics of the phase transitions in
synthetic networks of Josephson-coupled superconducting islands.

Superconductivity;Mesoscale and Nanoscale Physics