Title: Localized-endemic state transition in the susceptible-infected-susceptible model on networks,
Abstract: It is a longstanding debate concerning the absence of threshold for the
susceptible-infected-susceptible spreading model on networks with localized
state. The key to resolve this controversy is the dynamical interaction
pattern, which has not been uncovered. Here we show that the interaction
driving the localized-endemic state transition is not the global interaction
between a node and all the other nodes on the network, but exists at the level
of super node composed of highly connected node and its neighbors. The internal
interactions within a super node induce localized state with limited lifetime,
while the interactions between neighboring super nodes via a path of two hops
enable them to avoid trapping in the absorbing state, marking the onset of
endemic state. The hybrid interactions render highly connected nodes
exponentially increasing infection density, which truly account for the null
threshold. These results are crucial for correctly understanding diverse
recurrent contagion phenomena