Hybrid phase transition when connecting the disconnected

Consider connecting randomly individuals and imagine a law that imposes every new connection must at least involve one individual belonging to g fraction of the most disconnected population an individual being the more disconnected, the smaller the cluster of connected individuals it belongs to. Here we show that this simple strategy to improve connection exhibits a phase transition never described before, namely a hybrid phase transition with continuously varying exponents. A hybrid phase transition includes properties of both continuous and discontinuous phase transitions. Using the generating function method we evaluate the cluster size distribution of the restricted ErdősRényi model and obtain an analytic expression for the dependence of its critical exponent on g. We also obtain a susceptibility exhibiting asymmetric criticality, i.e. diverging with different exponents above and below the transition point.