The distribution of popular persons in a group

Abstract

In order to obtain the sociometric choice structure of a group sociologists apparently use two schemes: fixed choice and variable choice. In the former, each person in the group is asked to name a specific number of others in the group with whom they would like to interact. The instructions are often stated in the form, “name your three best friends”. This type of structure has recently been dealt with in papers by Shamir and Upfal (1982) and also by Capobianco (1981). In the variable choice scheme, each person is asked to name all of his friends in the group. We shall be concerned with both type of choice schemes in this paper. Sociologically, one of the important things to study is the phenomena of members of the group who are chosen a lot by the others, i.e., the popular persons. These will undoubtedly have the greatest influence on the group’s opinions and actions. Also, if one or more of them leaves the group, the structure will probably be badly damaged. In this paper, we use digraphs (directed graphs) to represent groups and the sociometric choices made by the members. Therefore, a popular person would be represented by a point which has many arcs going to it, i.e., a point of high indegree. We refer to these as HIPS (High Indegree Points). The next section deals with the specific definition of a HIP, and in the remainder of the paper we present some theorems related to the distribution of HIPS under randomness.