The Matrix-Forest Theorem and Measuring Relations in Small Social Groups

TL;DR

This paper introduces a family of graph indices based on the Matrix-forest theorem to measure relationships and properties within small social groups, with applications in sociometry.

Contribution

It develops new graph structural indices related to the Matrix-forest theorem and explores their properties and sociometric applications.

Findings

Basic index measures mutual connectivity between vertices.

Derivative indices quantify dissociation, solitariness, and provinciality.

A new metric on vertices based on connectivity is introduced.

Abstract

We propose a family of graph structural indices related to the Matrix-forest theorem. The properties of the basic index that expresses the mutual connectivity of two vertices are studied in detail. The derivative indices that measure "dissociation," "solitariness," and "provinciality" of vertices are also considered. A nonstandard metric on the set of vertices is introduced, which is determined by their connectivity. The application of these indices in sociometry is discussed.