Applied Computational Mathematics in Social Sciences

In this chapter, we offer the essential information to our readers regarding the two major fields approached by the book: Applied Computational Mathematics and Social Sciences. The first field emphasizes algorithms, numerical methods, and symbolic methods. The development of this field created new scientific disciplines like: Computational Biology, Computational Chemistry, Computational Economics, Computational Electrodynamics, Computational Finance, Computational Fluid Dynamics, Computational Geophysics, Computational Mechanics, Computational Physics, and Computational Statistics. On the other hand, Social Sciences comprise academic disciplines concerned with the study of the social life of human groups and individuals, and are dealing with complex systems that imply an interdisciplinary research approach, that combines knowledge from Computational Mathematics and Social Sciences. The problems connected with Social Sciences can be solved by using agent-based modeling (ABM). This technique uses a computational model for simulating the actions and interactions of autonomous individuals in a network, with a view to study the effects on the system as a whole. In order to create and implement a multi-agent computational model, you need a software platform. We are going to use NetLogo software platform. NetLogo uses three types of agents A: turtles Ti, patches Pxy and observer O. The main construction blocks of any agent-based computational model are the next: the set of agents (A), the initializations (I), and the simulation specifications (R).

The status of linearity of social sciences was attacked by two major approaches developed inside Mathematics: the catastrophe theory and chaos theory. Under these conditions, the social sciences are in natural continual evolution distinguished in 5 stages: (i) The verbal description of the subject and the logic of problem, (ii) The formal identification of problem and quantification of mathematical relations, (iii) Taking into account the dynamic aspects of mathematical model under the form of linear dynamic models, (iv) The reconsideration of the basic scientific principles, including some nonlinear aspects in dynamic models, (v) The development of new complete nonlinear dynamic models able to explain all the possible phenomena verbally described in the first stage. A theory of nonlinear dynamics of fifth stage does not exist now. Generally speaking, an Agent-based Artificial Society (AAS) is a grouping of individual artificial entities called agents (A) characterized by common interests that may have distinctive culture and institutions, and that are the part of an economic, social, and industrial infrastructure. This type of society is the result of interdisciplinary scientific efforts made by people from fields, such as Applied Computational Mathematics and Social Sciences. The basic elements of the AAS are agents A that encapsulates parameters, variables and algorithms. The other basic element of the AAS is the Artificial Social Group (ASG) defined as a collection of agents, who share certain characteristics, interact with one another, accept expectations and obligations as members of the same social group, and share a common identity.

An agent based model implementation is the result of three different types of scientists: the thematician, the modeler and the computer scientist. The implementation of Artificial Society using NetLogo is the process of transforming the algorithms in the procedures recognized by NetLogo. Any agent-based computational model implementation must be validated and verified. A model is valid to the extent that it provides a satisfactory range of accuracy consistent with the intended application of the model. Verification (sometimes called "internal validation") is the process of ensuring that the model performs in the manner intended by its designers and implementers. NetLogo is a software platform designed by Uri Wilensky it in the year 1999, NetLogo is used for modeling complex systems developing over time. NetLogo has a Models Library with a large collection of agent-based models that can be used and/or modified. Netlogo uses three types of agents: turtles, patches and observer. NetLogo has the next defining characteristics: (i) simplicity, (ii) transparency, (iii) gradual modeling, (iv) cross-platform, (v) extensive options of running, (vi) environment, (vii) BehaviorSpace tool, (viii) System Dynamics Modeler, (ix) speed slider, (x) powerful and flexible plotting system, (xi) HubNet module and (xii) models can be saved as applets to be embedded in web pages.

A computational experiment uses a computer model to make inferences about some underlying system. This type of experiment can be seen as a branch of applied statistics, because the user must take into account three sources of uncertainty. First, the models often contain parameters whose values are not certain; second, the models themselves are imperfect representations of the underlying system; and third, data collected from the system that might be used to calibrate the models are imperfectly measured. Study of multiple experiments involves two aspects: experimental planning, and analysis and modeling of experimental results. In this chapter, we will do 6 computer experiments containing 55 runs with our multi-agent based artificial society.

In this chapter, we draw the conclusions regarding the evolution of Agent-based Artificial Society described in Chapter 2, implemented in Chapter 3 using NetLogo, and used for a set of computational experiments in Chapter 4.

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