Graduation Year

2017

Document Type

Dissertation

Degree

Ph.D.

Degree Name

Doctor of Philosophy (Ph.D.)

Degree Granting Department

Mathematics and Statistics

Major Professor

Gangaram S. Ladde, Ph.D.

Committee Member

Dmitry Khavinson, Ph.D.

Committee Member

Razvan Teodorescu, Ph.D.

Committee Member

Yuncheng You, Ph.D.

Keywords

Multiagent Network, Cohesiveness, Lyapunov Second Method, Invariant Sets, Social Network

Abstract

Historically human endeavors around the globe are in search of bilateral relationships. Knowledge and commerce has played a very significant role in increasing the ability for humans to connect for the betterment of the human species. As the means of communication improve, mutual benefits to the community as a whole also increase. Moreover, the benefits are filtered down to members of the overall community. Recent advancement in electronic communication technologies and in knowledge, in particular, physical, chemical, engineering and medical sciences and philosophies, have facilitated nearly instantaneous multi-cultural interactions. Local problems and solutions have become global. This has generated a need for cooperation, coordination, and co-management at local and global levels. This instant communication and easy access to almost all members of the global community, with minimal cost and effort, causes an increase in uncertainty and lack of clarity in communication and misunderstanding between the members of the community. This leads to a fuzzy and stochastic environment. In short, the 21st century has seen a significant increase in the need to consider all human endeavors as being subject to random environmental fluctuations.

More precisely, currently the human species is highly mobile. In this work, an attempt is made (1) to balance communities working cooperatively and cohesively with one another while preserving member ability to retain individuality and fostering an environment of cultural state diversity. We develop (2) constructive analytic algorithms that provide tools to study qualitative and quantitative properties of multicultural diverse dynamic social networks. Under network parametric space/set conditions (3) cohesion and co-existence of members of multicultural dynamic network are insured. The parametric conditions (4) are algebraically simple, easy to verify, and robust. Moreover, the presented study of parametric representations of cohesion, co-existence and consensus attributes and robustness of multicultural dynamic networks provides a quantitative tool for planning, policy and performance of human mobility processes for members of a multicultural dynamic network.

We develop and investigate (5) a deterministic dynamic multicultural network. To exhibit the significance of the analysis, ideas, the complexity and limitations, we present a specific prototype model. This serves to establish the framework for finding explicit sufficient conditions in terms of system parameters for studying a complex dynamic network. Further, we decompose the cultural state domain into invariant subsets (6) and consider the behavior of members within each cultural state subset. Moreover, we analyze the relative cultural affinity between individual members relative to the center of the social network. We then (7) outline the general method for investigating a multicultural cultural network. We also demonstrate the degree of conservatism of the estimates using Euler type numerical approximation schemes. We are then able to consider how changes in the various parametric effects are reflected on the dynamics of the network.

The deterministic multicultural dynamic model and analysis is extended (8) to a multicultural dynamic network under random environmental perturbations. We present a detailed prototype model for the purpose of investigation. Introducing the concept of stochastic cohesion and consensus in the context of probabilistic modes of convergence (9), the explicit sufficient conditions in terms of system parameters are given to exhibit the cohesive property of the stochastic network. The effects of random fluctuations are characterized.

We then extend the stochastic model (10) to a multicultural hybrid stochastic dynamic network model. By considering a hybrid dynamic, we can explore the properties of a multicultural dynamic under the influence of both continuous-time and discrete-time cultural dynamic systems. This model captures external influences and internal changes that may have an impact on the members and system parameters of the dynamic network. We extend the ideas of global cohesion and consensus to local cohesion and consensus (11). Again, the detailed study is focused on a prototype hybrid multicultural dynamic network. Using the ideas and tools developed from the stochastic network (12), we are able to establish conditions on the network parameters for which the cultural network is locally cohesive. Using Euler-Maruyama type numerical approximation schemes to model the network, we better understand to what extent the analytically developed estimates are feasible.

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