NODE INFLUENCE IN NETWORK-BASED DISCRETE DYNAMICAL SYSTEMS

Abstract

Many complex systems can be modeled as network-based discrete dynamical systems, where individual nodes (representing some variable of interest) are connected by edges that describe their interactions. Each node can take on different states relevant to the network under investigation (for example, a gene may be turned ON or OFF in a genetic regulatory network, or a person may be Infected or Susceptible to a disease in an epidemiological network). These networks are frequently studied by analyzing global properties such as fixed points, robustness (sensitivity to perturbation), and functional organization (modularity). A fundamental problem in complex systems science is to understand how interactions between individual components of a system give rise to such properties; in other words, how can the influence of a node, or set of nodes, be measured and how does it affect the large-scale dynamics of the system? Understanding this influence is crucial to characterize, predict, and control complex systems. Traditional lines of inquiry often analyze only the network structure or assume that the entire network configuration is known (i.e., the state value of all nodes in the network). In practice, however, a network’s structure may not be a good predictor of its dynamics and furthermore, it is reasonable to assume that some nodes may not be measureable or controllable. Some recent approaches, such as causal inference methods, do not make such assumptions; still, calculating influence is in general a NP-hard problem and thus there is a need to further develop feasible approximate methods that work well in practice. This dissertation focuses on methods to calculate node influence and uncover dynamical properties of a network (modular organization, attractor control sets, size of perturbation cascades) that depend only on limited (partial) knowledge of the network configuration. I begin by reviewing the literature on node influence and related problems (influence maximization, control, and modularity), with special focus on methods that utilize either causal inference or approximations of dynamics. I then expand upon these methods with my own contributions to the field. First, I utilize the the existing concept of pathway modules on a dynamical map to define complex (synergistic) modules and use these to describe a network’s dynamics by its underlying causal mechanisms (by calculating direct node influence) and measure its dynamical modularity. Next I use a mean-field approximation of a node’s state based on iterative update of the states of its inputs (the IBMFA) to estimate the influence of that node on long-term configurations and attractors of a network, finding that the approximation performs well in comparison to actual simulations of the system. Finally, I define a thresholded representation of the dynamics (a generalized threshold network) to study different structural representations of the node update functions. I use these different representations and the IBMFA to calculate node influence and find that the choice of which method to use depends on the connectivity of the graph and the precision required. These methods are applied to various networks including random Boolean networks and biological signaling and regulatory networks, with examples given of additional use cases (such as linear threshold models and game-theoretic networks). Taken together, they help to elucidate the role of individual components within complex systems, with applications to dynamical modularity, influence maximization, and attractor/target control. Throughout I try to bridge the gap between literature on dynamical networks (e.g., logical models, Boolean networks) and dynamical processes on networks (e.g., epidemic and information spreading).