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Summary Deciphering the associations between network connectivity and nodal attributes is one of the core problems in network science. The dependency structure and high dimensionality of networks pose unique challenges to traditional dependency tests in terms of theoretical guarantees and empirical performance. We propose an approach to test network dependence via diffusion maps and distance-based correlations. We prove that the new method yields a consistent test statistic under mild distributional assumptions on the graph structure, and demonstrate that it is able to efficiently identify the most informative graph embedding with respect to the diffusion time. The methodology is illustrated on both simulated and real data. 

If you want to estimate whether height is related to weight in humans, what would you do? You could measure the height and weight of a large number of people, and then run a statistical test. Such ‘independence tests’ can be thought of as a screening procedure: if the two properties (height and weight) are not related, then there is no point in proceeding with further analyses. In the last 100 years different independence tests have been developed. However, classical approaches often fail to accurately discern relationships in the large, complex datasets typical of modern biomedical research. For example, connectomics datasets include tens or hundreds of thousands of connections between neurons that collectively underlie how the brain performs certain tasks. Discovering and deciphering relationships from these data is currently the largest barrier to progress in these fields. Another drawback to currently used methods of independence testing is that they act as a ‘black box’, giving an answer without making it clear how it was calculated. This can make it difficult for researchers to reproduce their findings – a key part of confirming a scientific discovery. Vogelstein et al. therefore sought to develop a method of performing independence tests on large datasets that can easily be both applied and interpreted by practicing scientists. The method developed by Vogelstein et al., called Multiscale Graph Correlation (MGC, pronounced ‘magic’), combines recent developments in hypothesis testing, machine learning, and data science. The result is that MGC typically requires between one half to one third as big a sample size as previously proposed methods for analyzing large, complex datasets. Moreover, MGC also indicates the nature of the relationship between different properties; for example, whether it is a linear relationship or not. Testing MGC on real biological data, including a cancer dataset and a human brain imaging dataset, revealed that it is more effective at finding possible relationships than other commonly used independence methods. MGC was also the only method that explained how it found those relationships. MGC will enable relationships to be found in data across many fields of inquiry – and not only in biology. Scientists, policy analysts, data journalists, and corporate data scientists could all use MGC to learn about the relationships present in their data. To that extent, Vogelstein et al. have made the code open source in MATLAB, R, and Python.