A model of $s$ interacting species is considered with two types of dynamical variables. The fast variables are the populations of the species and slow variables the links of a directed graph that defines the catalytic interactions among them. The graph evolves via mutations of the least fit species. Starting from a sparse random graph, we find that an autocatalytic set (ACS) inevitably appears and triggers a cascade of exponentially increasing connectivity until it spans the whole graph. The connectivity subsequently saturates in a statistical steady state. The time scales for the appearance of an ACS in the graph and its growth have a power law dependence on $s$ and the catalytic probability. At the end of the growth period the network is highly non-random, being localized on an exponentially small region of graph space for large $s$.Comment: 13 pages REVTEX (including figures), 4 Postscript figure
Evolution produces complex and structured networks of interacting components in chemical, biological, and social systems. We describe a simple mathematical model for the evolution of an idealized chemical system to study how a network of cooperative molecular species arises and evolves to become more complex and structured. The network is modeled by a directed weighted graph whose positive and negative links represent ''catalytic'' and ''inhibitory'' interactions among the molecular species, and which evolves as the least populated species (typically those that go extinct) are replaced by new ones. A small autocatalytic set, appearing by chance, provides the seed for the spontaneous growth of connectivity and cooperation in the graph. A highly structured chemical organization arises inevitably as the autocatalytic set enlarges and percolates through the network in a short analytically determined timescale. This self organization does not require the presence of self-replicating species. The network also exhibits catastrophes over long timescales triggered by the chance elimination of ''keystone'' species, followed by recoveries. S tructured networks of interacting components are a hallmark of several complex systems, for example, the chemical network of molecular species in cells (1), the web of interdependent biological species in ecosystems (2, 3), and social and economic networks of interacting agents in societies (4-7). The structure of these networks is a product of evolution, shaped partly by the environment and physical constraints and partly by the population (or other) dynamics in the system. For example, imagine a pond on the prebiotic earth containing a set of interacting molecular species with some concentrations. The interactions among the species in the pond affect how the populations evolve with time. If a population goes to zero, or if new molecular species enter the pond from the environment (through storms, floods, or tides), the effective chemical network existing in the pond changes. We discuss a mathematical model that attempts to incorporate this interplay between a network, populations, and the environment in a simple and idealized fashion. The model [including an earlier version (8, 9)] was inspired by the ideas and results in refs. 10-18. Related but different models are studied in refs. 19-21.
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