Analysis choices can produce apparent rich-club differences: an Alzheimer's disease EEG case study

The weighted rich-club coefficient tests whether a network's most strongly connected nodes are preferentially interconnected. Its interpretation depends on two often-implicit choices: how the randomised comparison networks are constructed, and whether the hubs are re-identified in each comparison network or held fixed at those of the observed network. When hubs are defined by connection strength, these choices change what the coefficient tests.

We examined how these choices affect inference using resting-state EEG from 20 people with Alzheimer's disease and 20 controls. Holding the data fixed, we compared four analysis pipelines, crossing two ways of building the comparison networks (global weight reshuffle or strength-conditioned null) with the two hub choices, and ran a calibration study using data constructed to contain no preferential hub coupling beyond nodal strength.

An apparent eyes-open group difference in rich-club organisation appeared only under the pipeline combining a global weight reshuffle with re-identified hubs. Under the empirical resampling ensemble, that pipeline reported a group difference in 43% of pseudo-studies, always higher in the Alzheimer's disease group, while the other three stayed near the expected 5%.

Unlike global weight reshuffling, strength-conditioned nulls approximately preserve nodal strengths; fixing observed hubs also retains the identity of the core being tested.

The same data can support or dissolve an apparent rich-club difference depending on these choices. No single choice is universally correct, but reporting which were made, and why they fit the question being asked, is essential to interpreting the result.