  There is growing recognition that the network structures arising from
interactions between different entities in physical, social and biological
systems fundamentally alter the evolutionary outcomes. Previous paradigm
exploring evolutionary game dynamics has assumed that individuals update their
strategies at an identical rate, reporting that structurally heterogeneous
networks -- despite their ubiquity in real systems -- generally hinder the
emergence of collective cooperation compared to their homogeneous counterparts.
Here we solve this paradox by creating a new paradigm where individuals on
arbitrary networks are allowed to update strategies at arbitrary, personalized
rates, and provide the precise condition under which universal collective
cooperation is favored. We find that when individuals' update rates vary
inversely with their number of connections, heterogeneous networks actually
outperform homogeneous ones in promoting cooperation. This surprising property
of such "doubly heterogeneous" networks cautions against the conventional
wisdom that heterogeneous networks are antagonistic to cooperation. We further
develop an efficient protocol for optimizing the promotion of cooperation by
tuning individuals' update rates in any structure. Our findings highlight that
personalized interaction dynamics, beyond structure, in complex networks are
fundamental to understanding and promoting collective cooperation.
