Scaling Properties Of Networks Research Articles

Scaling Properties of Dimensionality Reduction for Neural Populations and Network Models.

Recent studies have applied dimensionality reduction methods to understand how the multi-dimensional structure of neural population activity gives rise to brain function. It is unclear, however, how the results obtained from dimensionality reduction generalize to recordings with larger numbers of neurons and trials or how these results relate to the underlying network structure. We address these questions by applying factor analysis to recordings in the visual cortex of non-human primates and to spiking network models that self-generate irregular activity through a balance of excitation and inhibition. We compared the scaling trends of two key outputs of dimensionality reduction—shared dimensionality and percent shared variance—with neuron and trial count. We found that the scaling properties of networks with non-clustered and clustered connectivity differed, and that the in vivo recordings were more consistent with the clustered network. Furthermore, recordings from tens of neurons were sufficient to identify the dominant modes of shared variability that generalize to larger portions of the network. These findings can help guide the interpretation of dimensionality reduction outputs in regimes of limited neuron and trial sampling and help relate these outputs to the underlying network structure.

Fractal relation of mainstream length to catchment area in river networks

Mandelbrot's (1982) hypothesis that river length is fractal has been recently substantiated by Hjelmfelt (1988) using eight rivers in Missouri. The fractal dimension of river length, d, is derived here from the Horton's laws of network composition. This results in a simple function of stream length and stream area ratios, that is, d = max (1, 2 log RL/log RA). Three case studies are reported showing this estimate to be coherent with measurements of d obtained from map analysis. The scaling properties of the network as a whole are also investigated, showing the fractal dimension of river network, D, to depend upon bifurcation and stream area ratios according to D = min (2, 2 log RB/log RA). These results provide a linkage between quantitative analysis of drainage network composition and scaling properties of river networks.