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In certain two-dimensional time-dependent flows, the braiding of periodic orbits provides a way to analyze chaos in the system through application of the Thurston-Nielsen classification theorem (TNCT). We expand upon earlier work that introduced the application of the TNCT to braiding of almost-cyclic sets, which are individual components of almost-invariant sets [Stremler et al., "Topological chaos and periodic braiding of almost-cyclic sets," Phys. Rev. Lett. 106, 114101 (2011)]. In this context, almost-cyclic sets are periodic regions in the flow with high local residence time that act as stirrers or "ghost rods" around which the surrounding fluid appears to be stretched and folded. In the present work, we discuss the bifurcation of the almost-cyclic sets as a system parameter is varied, which results in a sequence of topologically distinct braids. We show that, for Stokes' flow in a lid-driven cavity, these various braids give good lower bounds on the topological entropy over the respective parameter regimes in which they exist. We make the case that a topological analysis based on spatiotemporal braiding of almost-cyclic sets can be used for analyzing chaos in fluid flows. Hence, we further develop a connection between set-oriented statistical methods and topological methods, which promises to be an important analysis tool in the study of complex systems. When a body of fluid moves, whether it be in the atmosphere, an ocean, or a kitchen sink, there are often regions of fluid that move together for an extended period of time. As these coherent sets of fluid trace out trajectories in space and time, they can be thought of as "stirring" the surrounding fluid. The entanglement of these trajectories as they "braid" around each other is connected to the level of chaos present in the fluid system. When the sets return periodically to their initial positions in a twodimensional, time-dependent system, the entanglement of their trajectories can be used to predict a lower bound on the rate of stretching in the surrounding domain. We examine the trajectories of such "almost-cyclic sets" in a lid-driven cavity flow and demonstrate that this combination of topological analysis with set-oriented methods can be an effective means of predicting chaos. The characterization of the entanglement and associated prediction of stretching are achieved through application of the Thurston-Nielsen Classification Theorem, which in general classifies the topological complexity of homeomorphisms of punctured surfaces. While a rigorous lower bound on topological entropy is not available in the absence of exactly periodic braiding structures, our approach finds the "topological skeleton" that can be used to get an approximate rate of stretching.

In certain (2 þ 1)-dimensional dynamical systems, the braiding of periodic orbits provides a framework for analyzing chaos in the system through application of the Thurston-Nielsen classification theorem. Periodic orbits generated by the dynamics can behave as physical obstructions that ''stir'' the surrounding domain and serve as the basis for this topological analysis. We provide evidence that, even in the absence of periodic orbits, almost-cyclic regions identified using a transfer operator approach can reveal an underlying structure that enables topological analysis of chaos in the domain.

titioning the variance in firm (business segment) profitability has identified industry, corporate parent, business segment, and time as key sources. However, this variance decomposition research stream has treated firms as atomistic, autonomous entities. We employ a fast-unfolding community-detection algorithm to detect firms' network memberships and use the Shapley Value method to isolate the effect of the firm's alliance network, in addition to industry, corporate parent, business segment, and year effects, on the variance in business unit performance.Our findings demonstrate that the effect of the firm's alliance network explains 11% of the variance in firm ROA among 16,381 business segments from 1979 through 1996. We also extend the time period through 2018 and find that our results broadly hold. Managerial Summary: In the search for superior firm performance, managers typically focus their attention externally on profitable industries in which to operate, as well as internally on their firms' idiosyncratic and valuable resources and capabilities. In addition to these profitability sources, our work suggests another important, but heretofore overlooked, factor in the managerial quest for competitive advantage: the value-creating potential of alliance networks. We employ a machine-learning

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