Explosive percolation transitions in growing networks

Recent extensive studies of the explosive percolation (EP) model revealed that the EP transition is second order with an extremely small value of the critical exponent beta associated with the order parameter. This result was obtained from static networks, in which the number of nodes in the system remains constant during the evolution of the network. However, explosive percolating behavior of the order parameter can be observed in social networks, which are often growing networks, where the number of nodes in the system increases as dynamics proceeds. However, extensive studies of the EP transition in such growing networks are still missing. Here we study the nature of the EP transition in growing networks by extending an existing growing network model to a general case in which m node candidates are picked up in the Achiloptas process. When m = 2, this model reduces to the existing model, which undergoes an infinite-order transition. We show that when m ??= 3, the transition becomes second order due to the suppression effect against the growth of large clusters. Using the rate-equation approach and performing numerical simulations, we also show that the exponent beta decreases algebraically with increasing m, whereas it does exponentially in a corresponding static random network model. Finally, we find that the hyperscaling relations hold but in different forms.This work was supported by the National Research Foundation of Korea (NRF) through Grants No. NRF-2014R1A3A2069005 (B.K.) and No. NRF-2014R1A1A2057396 (S.-W.S), SNU R&D program (B.K.), and the TJ Park Science Fellowship of POSCO TJ Park Foundation (S.-W.S.)