Computational analysis of complex systems: Applications to population dynamics and networks

In most complex evolving systems, we can often find a critical subset of the constituents that can initiate a global change in the entire system. For example, in complex networks, a critical subset of nodes can efficiently spread information, influence, or control dynamical processes over the entire network. Similarly, in nonlinear dynamics, we can locate key variables, or find the necessary parameters, to reach the attraction basin of a desired global state. In both cases, a fundamental goal is finding the ability to efficiently control these systems.;We study two distinct complex systems in this dissertation, exploring these topics. First, we analyze a population dynamics model describing interactions of sex-structured population groups. Specifically, we analyze how a sex-linked genetic trait's ecological consequence (population survival or extinction) can be influenced by the presence of sex-specific cultural mortality traits, motivated by the desire to expand the theoretical understanding of the role of biased sex ratios in organisms.;We analyze dynamics within a single population group, as well as between competing groups. We find that there is a finite range of sex ratio bias that can be maintained in stable equilibrium by sex-specific mortalities. We also find that the outcome of an invasion and the ensuing between-group competition depends not on larger equilibrium group densities, but on the higher allocation of sex-ratio genes.;When we extend the model with diffusive dispersal, we find that a critical patch size for achieving positive growth only exists if the population expands into an empty environment. If a resident population is already present that can be exploited by the invading group, then any small seed of invader can advance from rarity, in the mean-field approximation, as long as the local competition dynamics favors the invader's survival.;Most spatial models assume initial populations with a uniform distribution inside a finite patch; a simple, but not a cost-efficient approach. We show, using a novel application of simulated annealing, that a specific, non-trivial shape of spatial distribution can minimize the total cost of successful invasion, i.e., the cost of ecological restoration. Further, our approach can be generalized to essentially any reaction-diffusion model with diffusive spreading.;In the second part of the dissertation we conduct an extensive study of minimum dominating sets (MDS) in complex networks; particularly, in scale-free networks. MDS is the smallest subset of nodes in a network that can reach every other node as nearest neighbors, thus it provides a key subset of nodes that play critical role in controllability and observability of social, biological, and technological networks. Continued interest in network control, monitoring and influencing of complex networks motivates our research of understanding the properties and practical application-related issues of the MDS.;Our study of the scaling behavior reveals that the size of MDS always scales linearly with network size, as long as the power-law degree exponent gamma of the degree distribution is larger than 2. However, when gamma<2, a domination transition occurs, allowing the MDS size to become O(1), leading to easily dominated networks, under certain structural conditions.;Motivated by practical applicability in large networks, we develop a new dominating set selection method, derived from probabilistic node selection techniques, which can select small dominating sets without complete network topology information. We also show that the effectiveness of our method, as well as the effectiveness of other heuristics of dominating set selection, strongly depends on the assortativity of networks.;Finally, we conduct a numerical study to analyze the fraction of nodes that remain dominated, after the network is damaged, and some nodes are removed. We find that dominating sets optimized for small size are particularly vulnerable to damage; a significant amount of "domination coverage" may be lost if key dominator nodes are deleted. However, we also find that increasing the redundancy of dominating sets by adding a few well-picked nodes can successfully increase the post-damage dominated fraction of the network. Based on this idea, we develop two algorithms to build dominating sets with flexible balance between size and damage resilience.