Nonlinear Programming and Optimal Control Approach To the study of Social Network

Abstract

This research is a study of social network analysis and we approach it using nonlinear programming, statistics, dynamical systems and differential games theory. The ideas and techniques developed can be adapted to formulate public policy for social intervention, understanding cultural and social groups, marketing strategies by businesses, international relations etc. The study of social networks deals with the mathematical study of the formation and evolution of friendship links between members of a given social group. Each member of a social group has a set of preferred values and attributes and forms links with other members of the social group on the basis of shared values and attributes. This is precisely the basis of the nonlinear programming approach. That is, one seeks to construct an appropriate nonlinear programming on the basis of identified values and attributes of a social group. The solution of the nonlinear programming problem is used to decide whether or not a link is likely to exist between any two members of the social group.

A friendship network can be conveniently presented by using a matrix called a social matrix. A type of social matrix that is commonly used is one where each entry of the matrix is either one or zero corresponding to the presence or absence of friendship respectively.

Each member of a group, in general, acts on the basis of self interest, for example, to get as many links as possible with controlled time varying strategic compromises on personal preferences and attributes resulting in a time evolving social network. To capture the essence of the time evolution of the friendship network a differential games approach is appropriate.

In this dissertation the study of social networks is initially approached using nonlinear programming. Then, dynamic models are considered for time evolving social networks. The solutions of these models are then analyzed for their qualitative and long time behavior. The dynamic models are then used to formulate differential games models for social networks. Illustrative examples, numerical computations, and analyses are presented to illustrate how one uses these twin approaches for the study of social networks.