Random maps with sociological flavor

Abstract

A map of a set to itself admits a representation by a graph with vertices being the elements of the set and an edge between every vertex and its image. Communities defined as the maximal connected components are uni-cyclic. The distributions of the sizes of communities and lengths of cycles for unconstrained random maps is a classical subject. We call experts the images and followers the remaining vertices, and we further define prophets, egocentrics, and introverts. We introduce and analyze classes of random maps with sociological flavor.