2002/10/30 by Sanjay Jain, Sandeep Krishna, Jain, Sanjay +1
Physics and Astronomy · #Adaptation and Self-Organizing Systems (nlin.AO) #FOS: Physical sciences #nlin.AO
paper · pdf · doi:10.48550/arxiv.nlin/0210070
46 pages, 15 figures, to appear in "Handbook of Graphs and Networks", eds, S. Bornholdt and H. G. Schuster, John Wiley and VCH publishers
arxiv created 2002/10/30 · arxiv updated 2009/11/30
We give a self-contained introduction to the theory of directed graphs, leading up to the relationship between the Perron-Frobenius eigenvectors of a graph and its autocatalytic sets. Then we discuss a particular dynamical system on a fixed but arbitrary graph, that describes the population dynamics of species whose interactions are determined by the graph. The attractors of this dynamical system are described as a function of graph topology. Finally we consider a dynamical system in which the graph of interactions of the species coevolves with the populations of the species. We show that this system exhibits complex dynamics including self-organization of the network by autocatalytic sets, growth of complexity and structure, and collapse of the network followed by recoveries. We argue that a graph theoretic classification of perturbations of the network is helpful in predicting the future impact of a perturbation over short and medium time scales.