This paper defines and explores the direct citations between citing publications (DCCPs) of a publication. We construct an ego-centered citation network for each paper that contains all of its citing papers and itself, as well as the citation relationships among them. By utilizing a large-scale scholarly dataset from the computer science field in the Microsoft Academic Graph (MAG-CS) dataset, we find that DCCPs exist universally in medium and highly cited papers. For those papers that have DCCPs, DCCPs do occur frequently; highly cited papers tend to contain more DCCPs than others. Meanwhile, the number of DCCPs of papers published in different years does not vary dramatically. The current paper also discusses the relationship between DCCPs and some indirect citation relationships (e.g., co-citation and bibliographic coupling).