Abstract

Networks have a rich club organization when highly-connected nodes preferentially connect to other highly-connected nodes. Rich club organization is present in a variety of systems, including transportation networks, scientific collaboration, and the brain. Rich clubs can increase network efficiency by serving as a high-throughput backbone for signal trafficking. For networks with unweighted links, rich clubs are measured with a well-defined metric. For weighted networks, multiple possible metrics exist which each capture different features of the rich club. We describe an integrated framework for measuring unweighted and weighted rich clubs that resolves the many different metrics into a single method. This method focuses on the role of random controls, particularly on selecting what network features to control for. Using this framework, we show how previously reported weighted rich clubs can be biased by inappropriate controls. We also identify that rich club organization of links and weights may be separate, and introduce the method of controlling for link structure and measuring solely the allocation of weight. We use this method to describe the rich club organization of fully connected networks, including the stock market and human brain activity.