The k-step h-index in citation networks at the paper, author, and institution levels

•By incorporating higher generations of citation information, the k-step h-index provides a comprehensive assessment of scholarly influence at multiple levels.•The k-step h-indices exhibit strong alignment with measures of novelty and disruption, validating their consistency and potential for capturing innovative contributions.•The k-step h-indices outperform certain benchmarks in accurately ranking expert-selected items, such as milestone papers, distinguished authors, and prestigious institutions.•Higher k-step h-indices demonstrate superior performance compared to lower ones and alternative metrics, showcasing their predictive power in identifying prominent scientific entities.