Thomas Gebhart, Russell J. Funk
The growth of science and technology is a recombinative process, wherein new discoveries and inventions are built from prior knowledge. Yet relatively little is known about the manner in which scientific and technological knowledge develop and coalesce into larger structures that enable or constrain future breakthroughs. Network science has recently emerged as a framework for measuring the structure and dynamics of knowledge. While helpful, existing approaches struggle to capture the global properties of the underlying networks, leading to conflicting observations about the nature of scientific and technological progress. We bridge this methodological gap using tools from algebraic topology to characterize the higher-order structure of knowledge networks in science and technology across scale. We observe rapid growth in the higher-order structure of knowledge in many scientific and technological fields. This growth is not observable using traditional network measures. We further demonstrate that the emergence of higher-order structure coincides with decline in lower-order structure, and has historically far outpaced the corresponding emergence of higher-order structure in scientific and technological collaboration networks. Up to a point, increases in higher-order structure are associated with better outcomes, as measured by the novelty and impact of papers and patents. However, the nature of science and technology produced under higher-order regimes also appears to be qualitatively different from that produced under lower-order ones, with the former exhibiting greater linguistic abstractness and greater tendencies for building upon prior streams of knowledge.
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Living systems exhibit complex yet organized behavior on multiple spatiotemporal scales. To investigate the nature of multiscale coordination in living systems, one needs a meaningful and systematic way to quantify the complex dynamics, a challenge in both theoretical and empirical realms. The present work shows how integrating approaches from computational algebraic topology and dynamical systems may help us meet this challenge. In particular, we focus on the application of multiscale topological analysis to coordinated rhythmic processes. First, theoretical arguments are introduced as to why certain topological features and their scale-dependency are highly relevant to understanding complex collective dynamics. Second, we propose a method to capture such dynamically relevant topological information using persistent homology, which allows us to effectively construct a multiscale topological portrait of rhythmic coordination. Finally, the method is put to test in detecting transitions in real data from an experiment of rhythmic coordination in ensembles of interacting humans. The recurrence plots of topological portraits highlight collective transitions in coordination patterns that were elusive to more traditional methods. This sensitivity to collective transitions would be lost if the behavioral dynamics of individuals were treated as separate degrees of freedom instead of constituents of the topology that they collectively forge. Such multiscale topological portraits highlight collective aspects of coordination patterns that are irreducible to properties of individual parts. The present work demonstrates how the analysis of multiscale coordination dynamics can benefit from topological methods, thereby paving the way for further systematic quantification of complex, high-dimensional dynamics in living systems.
Topological portraits of multiscale coordination dynamics
Mengsen Zhang, William D. Kalies, J. A. Scott Kelso, Emmanuelle Tognoli