Emergent Open-Endedness from Contagion of the Fittest

Abstract

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...

Emergence and Algorithmic Information Dynamics of Systems and Observers

“…In particular, the open-endedness proved in this theorem strictly refers to the unbounded increase of complexity over time, as the evolution goes by. For this reason, it is called evolutionary open-endedness [6,7]. In this sense, we can adopt the convention of classifying the phenomenon in Theorem 4.1 as asymptotically observer-independent diachronic open-endedness.…”

“…The pervasiveness of non-homogeneous network topological properties has fostered the recent field of network science and showed its important role in complex systems science [11]. In this regard, motivated by the pursuit of a unified theory for complexity in network science and complex systems science [10,33], the theory of algorithmic networks [3,6,7] allows the investigation of how network topological properties can trigger emergent behavior that is capable of irreducibly increasing the computational power of the whole network. An algorithmic (complex) network N is a population of computable systems (or TMs) whose members can share information with each other according to a complex network topology.…”

“…In [7], it is shown that there are network topological properties, such as the small diameter, associated with a computationally cheap and simple communication protocol of plain diffusion that asymptotically trigger an unlimited increase of expected emergent algorithmic complexity of a networked node's final output as the number of nodes increases indefinitely. This unlimited increase of expected emergent algorithmic complexity is called expected emergent open-endedness (EEOE) [6,7] and-by simplifying the notation from [6,7] to meet our present purposes-it is mathematically defined by (9) lim…”

“…There is an optimum balance between these two quantities in which, if a large enough average density of these nodes is achieved in a sufficiently small number of communication rounds, then EEOE is triggered. Instead of the communication protocol of plain diffusion, Abrahão et al [6] shows that a susceptible-infected-susceptible (SIS) contagion scheme [38,39] in algorithmic networks with a power-law degree distribution is also sufficient for triggering EEOE. In [4], it is shown that a slight modification in the communication protocol of plain diffusion from [7] is sufficient for enabling the whole algorithmic network to synergistically solve problems at a higher computational class than the computational class of its individual nodes.…”

“…As we mentioned in Section 4.2.1, it was shown in [1,2] that the evolutionary open-endedness from [20,21] also applies to the resource-bounded case. Regarding the EEOE from [6,7], future research is still needed for establishing how a resourcebounded version of the EEOE in algorithmic networks mathematically unfolds.…”