Reducing module size bias of participation coefficient

Both natural and engineered networks are often modular. Whether a network node interacts with only nodes from its own module or nodes from multiple modules provides insight into its functional role. The participation coeffcient (PC) is typically used to measure this attribute although its value also depends on the size of the module it belongs to, often leading to non-intuitive identification of highly connected nodes. Here, we develop a normalized PC that overcomes the module size bias associated with the conventional PC. Using brain, C.elegans, airport and simulated networks, we show that our measure of participation alleviates the module size bias, while preserving conceptual and mathematical properties, of the classic formulation of PC. Unlike the conventional PC, we identify London and New York as high participators in the air traffic network and demonstrate stronger associations with working memory in human brain networks, yielding new insights into nodal participation across network modules.