The Degree Distribution of Human Brain Functional Connectivity is Generalized Pareto: A Multi-Scale Analysis

Abstract

Are degree distributions of human brain functional connectivity networks heavy-tailed? Initial claims based on least-square fitting suggested that brain functional connectivity networks obey power law scaling in their degree distributions. This interpretation has been challenged on methodological grounds. Subsequently, estimators based on maximum-likelihood and non-parametric tests involving surrogate data have been proposed. No clear consensus has emerged as results especially depended on data resolution. To identify the underlying topological distribution of brain functional connectivity calls for a closer examination of the relationship between resolution and statistics of model fitting. In this study, we analyze high-resolution functional magnetic resonance imaging (fMRI) data from the Human Connectome Project to assess its degree distribution across resolutions. We consider resolutions from one thousand to eighty thousand regions of interest (ROIs) and test whether they follow a heavy or short-tailed distribution. We analyze power law, exponential, truncated power law, log-normal, Weibull and generalized Pareto probability distributions. Notably, the Generalized Pareto distribution is of particular interest since it interpolates between heavy-tailed and short-tailed distributions, and it provides a handle on estimating the tails heaviness or shortness directly from the data. Our results show that the statistics support the short-tailed limit of the generalized Pareto distribution, rather than a power law or any other heavy-tailed distribution. Working across resolutions of the data and performing cross-model comparisons, we further establish the overall robustness of the generalized Pareto model in explaining the data. Moreover, we account for earlier ambiguities by showing that down-sampling the data systematically affects statistical results. At lower resolutions models cannot easily be differentiated on statistical grounds while their plausibility consistently increases up to an upper bound. Indeed, more power law distributions are reported at low resolutions (5K) than at higher ones (50K or 80K). However, we show that these positive identifications at low resolutions fail cross-model comparisons and that down-sampling data introduces the risk of detecting spurious heavy-tailed distributions. This dependence of the statistics of degree distributions on sampling resolution has broader implications for neuroinformatic methodology, especially, when several analyses rely on down-sampled data, for instance, due to a choice of anatomical parcellations or measurement technique. Our findings that node degrees of human brain functional networks follow a short-tailed distribution have important implications for claims of brain organization and function. Our findings do not support common simplistic representations of the brain as a generic complex system with optimally efficient architecture and function, modeled with simple growth mechanisms. Instead these findings reflect a more nuanced picture of a biological system that has been shaped by longstanding and pervasive developmental and architectural constraints, including wiring-cost constraints on the centrality architecture of individual nodes.