# Mathematical Modeling (MAT352)

## Basic course information

- Course Title: Mathematical Modeling
- Instructor: Kostas Alexandridis, Ph.D.
- Frequency: every odd-year Fall academic semester
- Credits: 3
- Max Students: 25 (UVI St. Thomas Campus)
- Prerequisites: algebra and/or statistics
- Course Description: the course is open to upper-level UVI students of the College of Science and Mathematics. It provides a comprehensive review of mathematical modeling applications, from working with random variables and data structures, probability calculus, distributions and probabilistic modeling, game-theoretical modeling, graph-theoretic and network (including social networks and Bayesian networks) modeling, and introduction to agent-based modeling techniques. The students obtain skills in multiple modeling software and modeling tools.

## Course introduction and rationale

The course develops a fundamental understanding of the role of mathematical modeling in bridging theoretical, methodological, and empirical knowledge in the fields of mathematics, modeling, and simulation. By using examples and case studies from various field in mathematics, engineering, physics, geography, biology, ecology, computational sciences, and other fields will enable a deeper understanding of applications of mathematical modeling in many scientific fields and in interpreting and explaining phenomena and the emergence of complex interactions in real-world settings.

Finally, the course provides the students with a useful and unique skill set, knowledge and understanding of mathematical modeling and simulation in the context and across multiple scientific domains. It is aimed to be an enjoyable yet pedagogical tool for science-technology integration.

## Course aims and objectives

The overall goal of the course is to expose students to the basic principles of mathematical modeling along with a set of rudimental skills and competencies on performing mathematical modeling and simulation. It especially emphasizes the role of mathematical modeling and simulation in advancing fundamental human scientific knowledge, aiding social and technological evolution, and addressing critical global, regional, national and local problems and challenges. The specific aims and objectives of the course are summarized below:

- Develop a principal and comprehensive understanding of the value of mathematical modeling and its applications, emphasizing the principles and basic motivations, modeling principles and methodologies as well goals driving them, and the role that mathematical modeling applications play in advancing science and its relationship with practical, empirical and theoretical aspects of problems and challenges. Specifically, the course will enhance the understanding of:
- Basic foundations of mathematical modeling and systems
- Exposure and appreciation of computational mathematics and its principles
- The role of data collection, numerical recipes, analyses and information in mathematical modeling and simulation
- The contributions of mathematical modeling in multiple scientific domains and knowledge areas.
- Understanding and evaluating the role of mathematical modeling applications in advancing the boundaries of scientific investigation, and providing the bases for methodological, theoretical and empirical advances in knowledge.

- Enhance the critical thinking ability of the students in conceptualizing constructing and simulating computational mathematical modeling systems. Specifically:
- Provide students with examples and case studies showcasing different applications and uses of mathematical modeling.
- Provide both the rationale and the scientific quest for knowledge behind major mathematical modeling applications and advances in computational and simulation methods that explain and interpret every-day phenomena and problems.
- Make students aware of the multidimensional character of the scientific method and how it can be served through mathematical modeling implementations.

- Improve the students’ skill-set in relation to the use of computers and mathematics as tools for scientific investigation. Specifically:
- Understanding the role of data types and imputation methods and relationships.
- Monitoring, evaluating and developing principles for mathematical modeling applications.
- Acquiring and using resources for mathematical modeling and simulation.

- Synthesizing the student’s collective and individual learning by promoting an academic in-class culture of dialogue, communication, expressiveness and synergistic attitudes. The objective will be achieved by:
- Encouraging students to participate in active dialogue/debate processes in classroom, and to combine their own experiences and knowledge in informing the discussions.
- Assigning students critical thinking challenges and evaluations of selected case studies and mathematical modeling examples.
- Enhancing student’s communication skills by in-classroom participation and computational-visually aided implementations of projects and assignments.
- Providing students with computational and technology-advanced demonstrations and interactive examples of mathematical modeling applications.

## Content and examples of mathematical applications that students in MAT 352 are expected to understand

### Random variables and data structures

- Construct basic random variables from mathematical equations and functions using for example embedded excel functions.
- Generate empirical and random data to simulate mathematical equations and forms e.g., generating data and graphing sine, cosine and tan functions of quadratic equations.

### Basic probability calculations

- Understand the notion of frequentist and empirical probabilities, i.e., empirically derived probabilities from data, including the notion of event space, union, intersection, etc.
- Simulate and represent computationally simple probabilistic structures and event spaces, such as coin tosses, dice rolls, e.t.c., by generatic random events in an event space, obtaining X outcomes in N trials, computing expected and empirical data probabilities, as well as perform joint and conditional probability calculations.
- Perform basic and intermediate problem-solving tasks using probability calculations, including constructing Conditional Probability Tables (CPTs), Joint Probability Tables (JPT), understanding conditional independence and applying the Bayes theorem in empirically and theoretically derived problem-solving situations.

### Mathematical modeling using probability distributions

- Understanding basic concepts of probability distribution theory, such as sample space, the concepts of probability density or mass functions (PDF/PMF), cumulative probability functions (CPF), their similarities and differences and their recognizing their graphical representations.
- Exposed to and recognizing (a) the different classes of probability distributions, i.e., the differences between discrete and continuous distributions and their functions; (b) the shapes, assumptions and general applicability of basic probability distributions in each class, e.g., binomial, Poisson, discrete uniform (discrete ones), or triangular, continuous uniform, normal, Pareto or exponential, Beta, Gamma, Weibull (continuous ones); (c) empirical cases to which different distribution functions can be applied to or fitted empirically.
- Using mathematical and statistical functions to generate empirical and/or theoretical distribution data, e.g., using probability distribution functions in excel.
- Using optimization methodologies to derive and explore probabilistic distributional problems and find solutions to probabilistic/probabilogic problems (e.g., using solver optimization algorithms in MS excel to compute target probabilities from distribution data).

### Game theory essentials

- Understanding basic definitions and assumptions in mathematical game theory, including players, action choices and strategies, outcomes and strategic interactions.
- Represent game-theoretic situations and problems using both extensive forms (e.g., decision tree diagrams) and standard forms (e.g., payoff-matrices) of the game-theoretical configurations, and being able to describe and compute both numerically and symbolically strategies, actions, payoffs and outcomes of each strategic interaction.
- Understand a number of standard games in game theory, such as the nickel-and-dime game, the prisoner’s dilemma game, the battle-of-the sexes game, the Big Monkey-Little Monkey game, the Voting game, e.t.c.
- Understand and check basic assumptions of game theory, including but not limited to preferences and utility-maximizing or loss-minimizing choices, formation of strategies, complete vs incomplete information, sequential vs. non-sequential move games, zero-sum vs. non-zero-sum games, 2-player vs. N-player games, pure strategies vs. mixed strategies games, etc. Furthermore, students should be able to construct empirical assessments of such theoretical concepts in modeled mathematical data.
- Ability to calculate dominant states of game-theoretic strategic interactions (i.e., test for dominance), compute evolutionary stable-state (ESS) solutions, and solve basic 2-person game-theoretic decision problems both in their simple form (seeking for minimax or maximin solutions) or their slightly advanced form (seeking for mixed strategy solutions probabilistically).
- Ability to model and solve a game-theoretic problems in terms of strategy sets for calculating Nash equilibria solutions, as well as understand and explain the rationale for their emergence. Includes ability to demonstrate convergence to asymptotically stable Nash equilibrium solutions in dynamic evolutionary systems,

### Graph theoretic network modeling and Social Network Analysis

- Constructing graph-network matrices (adjacency matrices, sociomatrices) for graph and network modeling. The skills include using pivot tables for creating adjacency lists from raw data vectors and datasets.
- Constructing basic and advanced graphs from data, using Excel, and Network modeling software (e.g., UCINET). The skills include input data from adjacency matrices, vector datasets, or attributidinal network data into software, and visualize graphs according to different embedding algorithms.
- Understand key graph-theoretic network properties, including: simple graphs, complete graphs, graph size and order, graph density, degree of vertices and networks, degree distributions, subgraphs, paths, reachable and connected paths, graph and network geodesics, cutpoints, bridges, edge and network connectivity, directed vs. undirected graphs, network centrality (Freeman), degree centrality, closeness centrality, betweeness centrality, network centralization.
- Understand advanced graph-theoretic social network properties, including: cyclic vs. acyclic graphs, network distances, weak and strong ties in social networks (the weak ties hypothesis), signed ties and bridges, congruency in strong and weak ties, G-transitivity, small-world networks, n-degrees of separation (e.g., Stanley Milgram experiments, etc).
- Ability to compute basic and advanced network metrics and statistics, including empirical network univariate statistics, centrality and node/network cohesion metrics, cutpoints, isolates, computing geodesic distances, network redundancies, etc. using network analysis software (e.g., UCINET).
- Understanding mathematical network principles in graph-theroretic and network analysis contexts: power-law distributions and differences between power-relationships and log-log distributional graphic properties of network metric visualization. Scale invariance and its role in networks and graph dynamics. Empirical applications of scale-free distributions in the web, social networks, collaborative networks, and other types of real-world network applications. Scale-free distributions and Pareto rule (80/20 rule). Preferential attachment in scale-free networks (including probabilistic interpretations and applications, such as wealth accumulation, income distribution, evolutionary dynamics, google rankings, etc.)
- Understand, visualize and model Zipf’s law, and its relationship to scale-free networks and power-law distributions. Understanding ranges of exponents in Zipf’s law, and ability to prove that a power-law distribution follows Zipf’s law using mathematical and statistical hypothesis testing.
- Skills on modeling preferential attachment in evolving networks using experimental data. Visualizing preferential attachment distributions in graph networks using spring embedding algorithms and network metrics (e.g., degree - rank distributions and log-log graphs).

### Probabilistic network modeling

- Understanding the notion of Bayesian probability, including the differences between frequentist vs. subjective probabilities. Understanding fundamental Bayesian concepts (mathematical formulation of Baye’s rule, conditional independence assuptions, Bayesian probability and likelihood concepts). Using Baye’s rule and probability distributions to perform probabilistic inference and encode causal and/or associative relationships.
- Skills in constructing Bayesian Belief and Decision networks (BBN, BDN), and understanding and performing different types of Bayesian estimation and learning tasks, such as Bayesian parametrics and non-parametrics, graphical models for causal inferences, variant types of Bayesian networks.
- Skills in performing structural learning and probabilistic learning in Bayesian Networks:
- Bayesian estimation methods (ML, BIC)

- Bayesian Monte Carlo algorithms
- Bayesian structural learning methods: Naïve Bayes, Tree-Augmented Naïve Bayes
- Bayesian probabilistic learning methods: Expectation-Maximization algorithm (E-M), Gradient Learning algorithm (GL)

- Using Bayesian network inference software (e.g., Netica) to perform structural and probabilistic learning from data, and entering evidence for conditional and diagnostic inference, including problem solving from data using probabilistic learning. Examples of medical diagnostic systems, weather prediction systems, Naïve Bayes decision-support systems, etc

### Agent-based modeling

- Construct and parameterize simple agent-based models (e.g., using NetLogo), and explore simulation assumptions, hypotheses and parameter spaces.
- Perform simple simulations and query agents, nodes and links for their state. Calculate basic metrics from agent-based simulation runs, and output results into spreadsheets for analysis.
- Explore dynamics of simulation including perform basic sensitivity analysis, exploring temporal (step-based) sequence of simulation run, and formulate alternative scenario/hypotheses based on simulation parameters and simulation run ensembles.
- Limited exploration of coding behavior of agents, and understanding the emergence of complex simulation patterns from simple rules and agent-agent or agent-environment communications and interactivity.