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

Abstract This article investigates emergence of algorithmic complexity in computable systems that can share information on a network. To this end, we use a theoretical approach from information theory, computability theory, and complex networks theory. One key studied question is how much emergent complexity 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 increase 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 make these algorithmic networks 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.