Algorithmic Networks: central time to trigger expected emergent open-endedness

TL;DR

This paper explores how network topology influences the emergence of complex behavior in interconnected computational systems, identifying conditions that lead to unbounded complexity growth as the network size increases.

Contribution

It introduces a model of algorithmic networks and establishes topological conditions under which expected emergent open-endedness occurs in large populations.

Findings

Emergent complexity tends to infinity with increasing population size.

Small diameter networks meet the conditions for emergent open-endedness.

A central time exists to trigger emergent open-endedness in dynamic networks.

Abstract

This article investigates emergence and complexity in complex systems that can share information on a network. To this end, we use a theoretical approach from information theory, computability theory, and complex networks. One key studied question is how much emergent complexity (or information) arises when a population of computable systems is networked compared with when this population is isolated. First, we define a general model for networked theoretical machines, which we call algorithmic networks. Then, we narrow our scope to investigate algorithmic networks that optimize the average fitnesses of nodes in a scenario in which each node imitates the fittest neighbor and the randomly generated population is networked by a time-varying graph. We show that there are graph-topological conditions that cause these algorithmic networks to have the property of expected emergent open-endedness for large enough populations. In other words, the expected emergent algorithmic complexity of a node tends to infinity as the population size tends to infinity. Given a dynamic network, we show that these conditions imply the existence of a central time to trigger expected emergent open-endedness. Moreover, we show that networks with small diameter compared to the network size meet these conditions. We also discuss future research based on how our results are related to some problems in network science, information theory, computability theory, distributed computing, game theory, evolutionary biology, and synergy in complex systems.