Random Maps with Sociological Flavor

TL;DR

This paper introduces and analyzes classes of random maps inspired by sociological concepts, exploring community structures, cycle lengths, and classifications like prophets and egocentrics.

Contribution

It presents novel models of random maps with sociological features, extending classical theories on community and cycle distributions.

Findings

Community sizes follow specific distributions

Cycle lengths exhibit characteristic patterns

New classifications of vertices based on sociological roles

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 vertexes, and we further define prophets, egocentrics, and introverts. We introduce and analyze classes of random maps with sociological flavor.