Social Network Analysis Made Easy with SEDIs Model Assessment

Assignment Overview

Task 1: Exploring the SEDIs model

[5 pages maximum]
Recommended reading resource: HELM 6: Matrices, pages 25-28.

• Express the system of equations 1a – 1d in matrix form and verify that the sum of any column in the matrix of coefficients M is zero.

• Use linear algebra to demonstrate a mathematical interpretation for this fact and identify at least one assumption of this model that makes it unrealistic for real-life social networks.

• Find non-trivial expressions for the points where ds/dt = de/dt = dd/dt = di/dt = 0 simultaneously.

• Provide an interpretation for what is happening in the network at this steady-state point.

• Show that 1a – 1d have strictly positive solutions s(t), e(t), d(t), i(t) > 0 for all t > 0 and derive expressions for the lower bound of these solutions.

Task 2: The effect of doubters

Your task is to understand how the parameters and initial conditions associated with the doubter state d(t) influence the evolution of the infected state i(t) in time.

• You will need to reflect on the mathematical results obtained in Task 1, as well as use MATLAB to solve the system of equations 1a – 1d algebraically and numerically for a wide range of conditions and parameters.

• Once you have analysed your numerical results, discuss what they imply about the importance of doubters in a social network and explicitly connect your findings to your mathematical work in Task 1.

• Your work will be evaluated based on criteria set out in A and B below.

• We suggest you dedicate up to 2 pages for Task 2.A and up to 4 pages for Task 2.B.

Task 2.A – Clarity and rationale behind the design of a numerical study in MATLAB

This will involve clearly describing your plan of analysis, including an outline of parameter ranges and initial conditions, the methods and MATLAB functions you will use, and a justification for these choices.

Guidance:

• Present a table outlining the range of parameters and initial conditions you chose to study in your model. Write up to two paragraphs discussing your rationale for choosing them.

• Create a flowchart/schematic to represent your methodology. Include the mathematical and computational methods you will use and describe how they are connected to one another.

• Describe your computational implementation in a short paragraph. Your goal is to write transparently so that someone reading your project could replicate it and check for themselves if the results are correct.

Task 2.B – Quality of results and discussion

This will involve displaying your results effectively and concisely in visual and numerical forms and providing concise and accurate written interpretations.

Guidance:

• Include one figure contrasting the solutions of the system across different initial conditions but with fixed transmission rates. Explain how the initial conditions affect the long-term behaviour of the solution.

• Include one figure displaying how i(t) varies according to the rates associated with the doubter state for two sets of initial conditions explored in part B.i. Discuss your results and describe the role of the doubter state in the model.

• Conclude your report with an appraisal in 2 to 3 paragraphs of the validity and limitations of this model. Use this appraisal to design and present the schematic of a model that would be more realistic. Summarise your proposed model in three bullet points.

Brief Summary of Assessment Requirements

The assessment focuses on exploring the SEDIs model to understand the dynamics of social networks using mathematical and computational techniques. It is divided into two main tasks:

Task 1: Exploring the SEDIs model (30 marks, 5 pages max)

• Express the given system of equations (1a–1d) in matrix form and verify that the sum of each column of the coefficient matrix MMM is zero.

• Use linear algebra to interpret this fact and identify at least one assumption that limits the model’s realism in real-life social networks.

• Find non-trivial expressions for the steady-state points (ds/dt=de/dt=dd/dt=di/dt=0ds/dt = de/dt = dd/dt = di/dt = 0ds/dt=de/dt=dd/dt=di/dt=0) and interpret what occurs in the network at this point.

• Demonstrate that the solutions s(t),e(t),d(t),i(t)>0s(t), e(t), d(t), i(t) > 0s(t),e(t),d(t),i(t)>0 for all t>0t > 0t>0 and derive lower bounds for these solutions.

Task 2: The effect of doubters (60 marks, 6 pages max)

• Analyse how doubters (d(t)) influence the evolution of the infected state i(t)i(t)i(t) using results from Task 1 and MATLAB.

• Task 2.A (30 marks, up to 2 pages) – Design and rationale for a numerical study:

  • Plan of analysis including parameter ranges and initial conditions

  • Flowchart of methodology connecting mathematical and computational steps

  • Description of computational implementation to ensure reproducibility

• Task 2.B (30 marks, up to 4 pages) – Results and discussion:

  • Display results visually and numerically

  • Discuss how initial conditions and doubter parameters affect i(t)i(t)i(t)

  • Provide an appraisal of the model’s validity and limitations, and propose a more realistic schematic in three key points

How the Assessment Was Approached by the Academic Mentor

The Academic Mentor guided the student step by step to ensure clarity, structure, and thorough coverage of learning objectives:

Step 1: Understanding the Requirements

• The mentor began by breaking down each task, highlighting mark allocation, maximum page limits, and key deliverables.

• Emphasis was placed on linking mathematical derivations to social network interpretations, ensuring conceptual understanding alongside calculations.

Step 2: Task 1 – Mathematical Exploration

1. Matrix Formulation

   • The student was instructed to convert 1a–1d into matrix form and verify the sum of each column is zero.

   • The mentor explained that this property arises from conservation-like assumptions in the SEDIs model.

2. Linear Algebra Interpretation

   • The mentor guided the student to explore eigenvalues and dependencies in the coefficient matrix, demonstrating why the zero column sum reflects unrealistic assumptions (e.g., constant population, perfect mixing).

3. Steady-State Analysis

   • Students were shown how to solve ds/dt=de/dt=dd/dt=di/dt=0ds/dt = de/dt = dd/dt = di/dt = 0ds/dt=de/dt=dd/dt=di/dt=0 algebraically and interpret the equilibrium behavior of the network.

4. Positive Solutions and Lower Bounds

   • The mentor explained methods to derive strictly positive solutions and establish minimum values for s(t),e(t),d(t),i(t)s(t), e(t), d(t), i(t)s(t),e(t),d(t),i(t), reinforcing stability concepts in dynamic systems.

Step 3: Task 2 – Effect of Doubters

2.A: Numerical Study Design

• The mentor advised creating a table of parameter ranges and initial conditions relevant to the doubter state.

• Students were guided to construct a flowchart connecting equations, MATLAB functions, and output analysis.

• A short descriptive paragraph ensured that the computational implementation was reproducible.

• This step reinforced critical thinking about experimental design and parameter sensitivity.

2.B: Results Interpretation

• The mentor emphasized producing clear figures and tables to demonstrate:

  • Differences in system evolution across initial conditions

  • Influence of doubter parameters on i(t)i(t)i(t)

• Students were coached to interpret trends and link back to mathematical insights from Task 1, highlighting the importance of doubters in network behavior.

• Finally, the mentor encouraged a critical appraisal, discussing model limitations and proposing improvements summarized in three key points.

Step 4: Linking to Learning Objectives

• Throughout the process, the mentor ensured that the student achieved the following objectives:

  1. Understand the SEDIs model and its assumptions.

  2. Apply linear algebra to social network systems.

  3. Derive and interpret steady-state solutions.

  4. Conduct a numerical study using MATLAB and analyze results.

  5. Critically evaluate model limitations and propose more realistic network representations.

Outcome Achieved

• Task 1: The student successfully expressed the system in matrix form, analyzed column sums, derived steady-state solutions, and confirmed positivity of solutions with lower bounds.

• Task 2: Using MATLAB, the student explored parameter sensitivities, produced clear visualizations, and demonstrated the influence of doubters on infection dynamics.

• The final report included a flowchart of methodology, a table of parameter ranges, figures illustrating key trends, and a critical appraisal of model validity.

• Learning objectives were fully addressed, combining theoretical derivations, computational modeling, and critical analysis.

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