Expected Emergence of Algorithmic Information from a Lower Bound for Stationary Prevalence

Abrahão, Felipe S.; Wehmuth, Klaus; Ziviani, Artur

doi:10.5753/etc.2018.3149

Cited by 1 publication

(2 citation statements)

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“…Thus, the present work and the investigation on networked computable systems using E-mail address: {fsa,klaus,ziviani}@lncc.br. In [3], some preliminary results of this article are presented as an extended abstract. Authors acknowledge the partial support from CNPq through their individual grants: F. S. Abrahão (313.043/2016-7), K. Wehmuth (312599/2016-1), and A. Ziviani (308.729/2015-3).…”

Section: Introductionmentioning

confidence: 99%

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Emergent Open-Endedness from Contagion of the Fittest

Abrahão¹,

Wehmuth²,

Ziviani³

2018

ComplexSystems

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In this paper, we study emergent irreducible information in populations of randomly generated computable systems that are networked and follow a "Susceptible-Infected-Susceptible" contagion model of imitation of the fittest neighbor. We show that there is a lower bound for the stationary prevalence (or average density of "infected" nodes) that triggers an unlimited increase of the expected local emergent algorithmic complexity (or information) of a node as the population size grows. We call this phenomenon expected (local) emergent open-endedness. In addition, we show that static networks with a power-law degree distribution following the Barabási-Albert model satisfy this lower bound and, thus, display expected (local) emergent open-endedness. Emergent Open-Endedness from Contagion of the Fittest 3The population of our present model plays the Busy Beaver Imitation Game (BBIG), in which each node always imitates the fittest neighbor only. Nevertheless, differently from the model in [4], we present a variation on the information-sharing (or communication) protocol. The major difference in respect to this previous work comes from allowing nodes to become "cured" (with rate δ). Additionally, now nodes also get "infected" with rate ν-which may have a value different from 1. In [4], one has that ν = 1 always holds. Summarizing, although still playing a BBIG, susceptible nodes follow a rule of imitating the neighbor that had output the largest integer (which corresponds to the fittest individual outcome in the population). However, they follow this rule with probability ν, and "infected" nodes come back-become "cured"-to the initial stage with probability δ. Thus, the effective spreading rate λ = ν/δ defined in [33][34][35] assumes a direct interpretation of the rate in which the Imitation-of-the-Fittest Protocol [4] was applied on a node-and this is the reason why we are using the words "infection" and "cure" between quotation marks. Therefore, the diffusion or "infection" scheme of the best output returned by a randomly generated node is ruled by the Susceptible-Infected-Susceptible epidemic model (SIS) in which susceptible nodes have a constant probability ν of being "infected" by a previously "infected" neighbor and "infected" nodes have a constant probability δ of becoming "cured". We also assume, as in [33][34][35], that the prevalence of "infected" nodes (i.e., the average density of "infected" nodes) becomes stationary after sufficient time 1 .Our proofs follow mainly from information theory, computability theory, and graph theory applied on a variation on the information-sharing protocol of the model in [4]. In particular, we have proved results for general dynamic networks and for dynamic networks with a small diameter-O(log(N )) compared to the network size N -in [4]. Further, these results are also directly extended to static networks [4] with the small-diameter property. Therefore, we have shown that there are topological conditions that trigger a phase transition in which eventually the algorithmic networ...

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Section: Introductionmentioning

confidence: 99%

“…Authors also acknowledge the partial support from CAPES, FAPESP, and FAPERJ. algorithmic networks [1,3,4] have shown a way on how to bring these topics into an abstract mathematical theory. Therefore, enabling one to formally define sound and crucial properties and to prove fruitful theorems.…”

Section: Introductionmentioning

confidence: 99%