Chase-escape with death on trees

Chase–escape in dynamic device-to-device networks

“…Now, we consider the chase-escape model as an independent space-time process on realizations of the connectivity graph. This also brings us much closer to the existing literature on chase-escape dynamics on fixed and random graphs; see, for example, [4,7,15,[21][22][23]39], as chase-escape has not been studied on a dynamic graph so far. More precisely, we consider g ρ,r (S, X λ ∪ Y λ W ) under the Palm distribution with respect to the X λ described in (1) and let N(X i ) denote the set of direct neighbors of vertex X i .…”

“…Regarding the infection as some malware, there can be special devices in the network that have the property of removing the malware from infected devices, but are otherwise indistinguishable from the infected or susceptible devices. This gives rise to chase-escape processes [4,7,15,21,39] in which malware is transmitted from infected to susceptible devices and infected devices are patched. Again, the main results are concerned with exhibiting regimes of survival and extinction of the malware in the system, and such regimes have also been established in the continuum [22,23].…”

Distance-dependent chase-escape on trees

“…The authors of [2] introduced a variant called chase-escape with death in which red particles die at a given rate, independently of the spread of parasites. After the death of a red particle, its vertex is blocked and cannot be occupied by others.…”

“…They studied this process on infinite d-ary trees and characterised the phase behaviour with the occurrence of death (the phase transitions of chase-escape were previously understood in [13,5]). Much of the analysis in [2] relied on a novel connection to weighted Catalan numbers. We deepen this connection by generalising to the setting in which the host spreading and death rates depend on the distance to the nearest parasite.…”

“…Once a site is dead, it stays in the state †; in particular, it blocks the advancement of blue and red through such site. The chase-escape with death process from [2] is the special case of our model with λ i ≡ λ and ρ i ≡ ρ for all i ≥ 1 with λ, ρ > 0.…”