Shadows of the SIS 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 $\lambda$ reaches one. If $\lambda < 1$, the probability that an outbreak will survive by an observation time $t$ tends to zero as $t \rightarrow \infty$; if $\lambda = 1$, this probability is one. We show that treating $\lambda = 1$ as a critical point predicts the $\lambda$-dependence of the survival probability also for more moderate $\lambda$-values. The exponent, however, depends on the underlying network. This fact could, by measuring how a vertex' 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).