Topological Features of Random Networks in High Dimensions

Topological data analysis characterizes the geometric structure of high-dimensional systems and is emerging as a valuable tool for understanding neural and biological data. For example, Betti numbers count loops, voids, and higher-order cavities in networks built from thresholded correlation matrices. Betti-1, Betti-2, and Betti-3 curves describe how these features emerge and vanish as the threshold changes and depend only on the relative order, not the absolute size, of correlation. Interpreting Betti curves in real systems is difficult because geometry, dimensionality, and input-output nonlinearity can alter correlation structure. To explore the dependence of Betti curves on these factors, we simulated linear and nonlinear network models driven by random projections from a small number of inputs. We systematically varied network size, number of inputs, and readout nonlinearity. At small network sizes, Betti-1 exceeded Betti-2 and Betti-3, but at larger sizes this ordering reversed, indicating that larger networks are expected to admit more complex topological structure. This transition occurred at smaller network sizes in the nonlinear model. Linear and nonlinear readouts of inputs with the same parameters generally produce similar Betti curves, except in sparse activity regimes: there, all three Betti curves have higher peaks in the nonlinear model than the linear model. More broadly, our approach can be used to guide interpretation of Betti curves in neural and biological data.