Utilizing topology to generate and test theories of change.

Abstract

Statistical and methodological innovations in the study of change are advancing rapidly, and visual tools have become an important component in model building and testing. Graphical representations such as path diagrams are necessary, but may be insufficient in the case of complex theories and models. Topology is a visual tool that connects theory and testable equations believed to capture the theorized patterns of change. Although some prior work has made use of topologies, these representations have often been generated as a result of the tested models. This article argues that utilizing topology a priori, when developing a theory, and applying analogous statistical models is a prudent method to conduct research. This article reviews topology by demonstrating how to build a topological representation of a theory and recover the implied equations, ultimately facilitating the transition from complex theory to testable model. Finally, topologies can guide researchers as they adjust or expand their theories in light of recent model testing.

Modeling Multi-Agent Self-Organization through the Lens of Higher Order Attractor Dynamics

“…This interpretation is directly relevant to several of the examples (although more general definitions are extant; Butner et al, 2015). An Attractor is when the agents are drawn toward a particular coordinate over time or a particular directional heading.…”

“…These repetitious behaviors describe the most probable system states and their ability to remain in these states (while facing perturbations) conveys the inherent stability of those states. These attractor dynamics can then be modeled using differential/difference equations, allowing for exploration of the inferred dynamics and theorizing the manifolds in which the system functions (Differential equations are based on idealized models for when change in time approaches zero while difference equations estimate models using the observed discrete differences; Butner et al, 2015). …”

“…A Saddle occurs when agents are attracted in one dimension and repelled in another. It is analogous to a topographical ridgeline because it can separate different patterns such as two attractors (Abraham and Shaw, 1983; Butner et al, 2015). A Cycle corresponds to a push/pull of two dimensions on one another.…”

“…In this way, we capture the movement of many agents and can characterize them with attractor descriptions. With this information it is possible to examine and identify patterns of change for the overall system using the particular topological features defined above to describe how multiple agents are moving over time (Butner et al, 2015). It is in these terms that we gain an understanding of the emergent, coordination of many agents.…”

“…For clarity, this equation is written in regression form where velocity in heading at each point in time is treated as the criterion and position (current heading) is the predictor. When the slope in Equation 1 is negative, we observe an attractor where the time series are attracted toward a value of − b 0 / b 1 known as the set point (Butner et al, 2015). A repeller occurs when the slope is positive instead of negative.…”