Modeling dynamics of adaptive complex systems: From gene regulatory networks to financial markets

This dissertation aims to model the dynamics of two types of adaptive complex systems: gene regulatory networks and financial markets. In modeling gene regulatory networks, a dynamics-driven rewiring mechanism is introduced to Boolean networks and it is found that a critical state emerges spontaneously resulting from the interplay between topology and dynamics during evolution. For biologically realized network sizes, significant finite-size effects are observed. In networks of competing Boolean nodes, we find that in small networks, the evolutionary dynamics selects for input inverting functions rather than canalizing functions in infinitely large networks. It is found that finite sizes can cause symmetry breaking in the evolutionary dynamics. Using the Polya theorem, we show the number of the function classes increases to 46, in contrast to 14 in infinitely large networks, due to the reduced symmetry which matches our simulation results well. In addition, we find that an optimum amount of stochastic noise in the signals exchanged between nodes can result in maximum excess canalization. In modeling financial markets, we simulate a double-auction virtual market by utilizing reaction-diffusion processes to describe the dynamics of limit orders. We find that the log-returns produced have a dynamical scaling exponent of 1/4 and nonstationary, negatively autocorrelated increments. By investigating the microstructure of the virtual market, we find that the mean interarrival time between transactions satisfies an increasing power-law function of time. We propose an inhomogeneous compound Poisson process with a decreasing power-law intensity rate function and demonstrate that this purely jump process captures the essential macroscopic dynamics of the virtual market.