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26 April 2024 |
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On Convergence and Threshold Properties of Discrete Lotka-Volterra Population Protocols | | Jurek Czyzowicz
; Leszek Gasieniec
; Adrian Kosowski
; Evangelos Kranakis
; Paul G. Spirakis
; Przemyslaw Uznanski
; | | Date: |
31 Mar 2015 | | Abstract: | In this work we focus on a natural class of population protocols whose
dynamics are modelled by the discrete version of Lotka-Volterra equations. In
such protocols, when an agent $a$ of type (species) $i$ interacts with an agent
$b$ of type (species) $j$ with $a$ as the initiator, then $b$’s type becomes
$i$ with probability $P\_{ij}$. In such an interaction, we think of $a$ as the
predator, $b$ as the prey, and the type of the prey is either converted to that
of the predator or stays as is. Such protocols capture the dynamics of some
opinion spreading models and generalize the well-known Rock-Paper-Scissors
discrete dynamics. We consider the pairwise interactions among agents that are
scheduled uniformly at random. We start by considering the convergence time and
show that any Lotka-Volterra-type protocol on an $n$-agent population converges
to some absorbing state in time polynomial in $n$, w.h.p., when any pair of
agents is allowed to interact. By contrast, when the interaction graph is a
star, even the Rock-Paper-Scissors protocol requires exponential time to
converge. We then study threshold effects exhibited by Lotka-Volterra-type
protocols with 3 and more species under interactions between any pair of
agents. We start by presenting a simple 4-type protocol in which the
probability difference of reaching the two possible absorbing states is
strongly amplified by the ratio of the initial populations of the two other
types, which are transient, but "control" convergence. We then prove that the
Rock-Paper-Scissors protocol reaches each of its three possible absorbing
states with almost equal probability, starting from any configuration
satisfying some sub-linear lower bound on the initial size of each species.
That is, Rock-Paper-Scissors is a realization of a "coin-flip consensus" in a
distributed system. Some of our techniques may be of independent value. | | Source: | arXiv, 1503.9168 | | Services: | Forum | Review | PDF | Favorites | |
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