Empirical evidence reveals that contagion processes often occur with
competition of simple and complex contagion, meaning that while some agents
follow simple contagion, others follow complex contagion. Simple contagion
refers to spreading processes induced by a single exposure to a contagious
entity while complex contagion demands multiple exposures for transmission.
Inspired by this observation, we propose a model of contagion dynamics with a
transmission probability that initiates a process of complex contagion. With
this probability nodes subject to simple contagion get adopted and trigger a
process of complex contagion. We obtain a phase diagram in the parameter space
of the transmission probability and the fraction of nodes subject to complex
contagion. Our contagion model exhibits a rich variety of phase transitions
such as continuous, discontinuous, and hybrid phase transitions, criticality,
tricriticality, and double transitions. In particular, we find a double phase
transition showing a continuous transition and a following discontinuous
transition in the density of adopted nodes with respect to the transmission
probability. We show that the double transition occurs with an intermediate
phase in which nodes following simple contagion become adopted but nodes with
complex contagion remain susceptible.Comment: 9 pages, 4 figure