Polynomial-time Approximation Algorithm for finding Highly Comfortable Team in any given Social Network

TL;DR

This paper introduces a new social network index called 'comfortability', proves finding highly comfortable teams is NP-Complete, and provides a polynomial-time approximation algorithm with performance guarantees.

Contribution

It defines the concept of comfortability in social networks, proves the NP-Completeness of finding highly comfortable teams, and offers a polynomial-time approximation algorithm with performance bounds.

Findings

NP-Completeness of forming highly comfortable teams

Polynomial-time approximation algorithm with ratio O(ln Δ)

Reduced dispersion rate in identified teams

Abstract

There are many indexes (measures or metrics) in Social Network Analysis (SNA), like density, cohesion, etc. In this paper, we define a new SNA index called "comfortability". One among the lack of many factors, which affect the effectiveness of a group, is "comfortability". So, comfortability is one of the important attributes (characteristics) for a successful team work. It is important to find a comfortable and successful team in any given social network. In this paper, comfortable team, better comfortable team and highly comfortable team of a social network are defined based on \textbf{graph theoretic concepts} and some of their structural properties are analyzed. It is proved that forming better comfortable team or highly comfortable team in any connected network are NP-Complete using the concepts of domination in graph theory. Next, we give a polynomial-time approximation algorithm for finding such a highly comfortable team in any given network with performance ratio O(\ln \Delta), where \Delta is the maximum degree of a given network (graph). The time complexity of the algorithm is proved to be O(n^{3}), where n is the number of persons (vertices) in the network (graph). It is also proved that our algorithm has reasonably reduced the dispersion rate.