Shadows of the susceptible-infectious-susceptible immortality transition in small networks.
Much of the research on the behavior of the SIS model on networks has concerned the infinite size limit; in particular the phase transition between a state where outbreaks can reach a finite fraction of the population, and a state where only a finite number would be infected. For finite networks, there is also a dynamic transition-the immortality transition-when the per-contact transmission probability λ reaches 1. If λ<1, the probability that an outbreak will survive by an observation time t tends to zero as t→∞; if λ=1, this probability is 1. We show that treating λ=1 as a critical point predicts the λ dependence of the survival probability also for more moderate λ values. The exponent, however, depends on the underlying network. This fact could, by measuring how a vertex's deletion changes the exponent, be used to evaluate the role of a vertex in the outbreak. Our work also confirms an extremely clear separation between the early die-off (from the outbreak failing to take hold in the population) and the later extinctions (corresponding to rare stochastic events of several consecutive transmission events failing to occur).
Department of Energy Science, Sungkyunkwan University, 440-746 Suwon, Republic of Korea; Department of Physics, Umeå University, 90187 Umeå, Sweden; and Department of Sociology, Stockholm University, 10961 Stockholm, Sweden.
Petter
P
Holme
D008962:Models, Theoretical