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Discrete dynamical systems serve as useful formal models to study diffusion phenomena in social networks. Several recent articles have studied the algorithmic and complexity aspects of some decision problems on synchronous Boolean networks, which are discrete dynamical systems whose underlying graphs are directed, and may contain directed cycles. Such problems can be regarded as reachability problems in the phase space of the corresponding dynamical system. Previous work has shown that some of these decision problems become efficiently solvable for systems on directed acyclic graphs (DAGs). Motivated by this line of work, we investigate a number of decision problems for dynamical systems whose underlying graphs are DAGs. We show that computational intractability (i.e.,PSPACE-completeness) results for reachability problems hold even for dynamical systems on DAGs. We also identify some restricted versions of dynamical systems on DAGs for which reachability problem can be solved efficiently. In addition, we show that a decision problem (namely, Convergence), which is efficiently solvable for dynamical systems on DAGs, becomesPSPACE-complete for Quasi-DAGs (i.e., graphs that become DAGs by the removal of asingleedge). In the process of establishing the above results, we also develop several structural properties of the phase spaces of dynamical systems on DAGs. 

Discrete dynamical systems are commonly used to model the spread of contagions on real-world networks. Under the PAC framework, existing research has studied the problem of learning the behavior of a system, assuming that the underlying network is known. In this work, we focus on a more challenging setting: to learn both the behavior and the underlying topology of a black-box system. We show that, in general, this learning problem is computationally intractable. On the positive side, we present efficient learning methods under the PAC model when the underlying graph of the dynamical system belongs to certain classes. Further, we examine a relaxed setting where the topology of an unknown system is partially observed. For this case, we develop an efficient PAC learner to infer the system and establish the sample complexity. Lastly, we present a formal analysis of the expressive power of the hypothesis class of dynamical systems where both the topology and behavior are unknown, using the well-known Natarajan dimension formalism. Our results provide a theoretical foundation for learning both the topology and behavior of discrete dynamical systems. 

Evolutionary anti-coordination games on networks capture real-world strategic situations such as traffic routing and market competition. Two key problems concerning evolutionary games are the existence of a pure Nash equilibrium (NE) and the convergence time. In this work, we study these two problems for anti-coordination games under sequential and synchronous update schemes. For each update scheme, we examine two decision modes based on whether an agent considers its own previous action (self essential) or not (self non-essential) in choosing its next action. Using a relationship between games and dynamical systems, we show that for both update schemes, finding an NE can be done efficiently under the self non-essential mode but is computationally intractable under the self essential mode. We then identify special cases for which an NE can be obtained efficiently. For convergence time, we show that the dynamics converges in a polynomial number of steps under the synchronous scheme; for the sequential scheme, the convergence time is polynomial only under the self non-essential mode. Through experiments, we empirically examine the convergence time and the equilibria for both synthetic and real-world networks. 

The ongoing COVID-19 pandemic underscores the importance of developing reliable forecasts that would allow decision makers to devise appropriate response strategies. Despite much recent research on the topic, epidemic forecasting remains poorly understood. Researchers have attributed the difficulty of forecasting contagion dynamics to a multitude of factors, including complex behavioral responses, uncertainty in data, the stochastic nature of the underlying process, and the high sensitivity of the disease parameters to changes in the environment. We offer a rigorous explanation of the difficulty of short-term forecasting on networked populations using ideas from computational complexity. Specifically, we show that several forecasting problems (e.g., the probability that at least a given number of people will get infected at a given time and the probability that the number of infections will reach a peak at a given time) are computationally intractable. For instance, efficient solvability of such problems would imply that the number of satisfying assignments of an arbitrary Boolean formula in conjunctive normal form can be computed efficiently, violating a widely believed hypothesis in computational complexity. This intractability result holds even under the ideal situation, where all the disease parameters are known and are assumed to be insensitive to changes in the environment. From a computational complexity viewpoint, our results, which show that contagion dynamics become unpredictable for both macroscopic and individual properties, bring out some fundamental difficulties of predicting disease parameters. On the positive side, we develop efficient algorithms or approximation algorithms for restricted versions of forecasting problems. 

Using a discrete dynamical system model for a networked social system, we consider the problem of learning a class of local interaction functions in such networks. Our focus is on learning local functions which are based on pairwise disjoint coalitions formed from the neighborhood of each node. Our work considers both active query and PAC learning models. We establish bounds on the number of queries needed to learn the local functions under both models.We also establish a complexity result regarding efficient consistent learners for such functions. Our experimental results on synthetic and real social networks demonstrate how the number of queries depends on the structure of the underlying network and number of coalitions.