Northeastern_PHYS7335

Collection of projects and assignments from Northeastern University's Dynamical Processes on Complex Networks course.

Table of Contents

• Random Walks
  • Visualization example
  • Features
  • Installation
  • Usage
    • Basic Usage
    • Custom Parameters
    • Command-Line Options
    • Output
    • Notes
• SIR Epidemic Models
  • Visualization example
  • 1. Deterministic vs Stochastic Analysis (SIR_Deterministic.ipynb)
  • 2. Network Visualization (SIRVisual.py)
    • Features
    • Usage
    • Command-Line Options
  • 3. SLIR Model Exploration (SLIR_Slim.ipynb)
  • 4. Competing Epidemics (SI_Comp_SII.ipynb)
  • 5. Second Epidemic Wave (SIR_Second_Epidemic.ipynb)
  • 6. Growth Rate Analysis (Growth_rate.ipynb)
  • 7. Agent-Based Network Modeling (Agent_based.ipynb)
• Rumor Model
  • The Model
  • Network Topology Comparisons: Watts-Strogatz
  • Running the Simulation

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Random Walks

Simulation and visualization of random walks on various graph topologies (Erdős-Rényi, Watts-Strogatz, Barabási-Albert, and Configuration Model).

Visualization example

Features

• Stochastic Visualization: Generates animated GIFs of random walkers on small graphs.
• Stochastic Analysis: Analyzes walker distribution and mixing times on small graphs.
• Deterministic Analysis: Performs "Degree Block" analysis on larger graphs (1000 nodes).
• Graph Types: Supports ER, WS, BA, and Configuration Model graphs.

Installation

1. Clone the repo or download the directory.
2. Install dependencies:

   pip install -r requirements.txt

Usage

Basic Usage

Run with default parameters (Walkers: 4, Rate: 0.80, Nodes: 30):

python RandomWalks.py

Custom Parameters

Customize the visualization parameters:

python RandomWalks.py --walkers 6 --rate 0.75 --nodes 50

Command-Line Options

Flag	Short	Default	Description
--walkers	-w	4	Number of walkers (Stochastic Visualization)
--rate	-r	0.80	Rate of escape (0-1) for all analyses
--nodes	-n	30	Number of nodes (Stochastic Visualization)
--help	-h	-	Display help message

Output

The script creates an output/ directory containing:

• GIFs: Animations of the random walks (e.g., ER_Graph_random_walks_animation.gif).
• Plots:
  • *_walkers_by_degree.png: Walkers vs Degree.
  • *_cumulative_walkers_by_time.png: Cumulative walkers over time.
  • mixing_time_combined.png: Mixing time analysis.

Notes

• The Stochastic Visualization uses the parameters provided via command line.
• The Deterministic Analysis runs on hardcoded larger graphs (1000 nodes, 10000 walkers) to ensure statistical significance.

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SIR Epidemic Models

Exploration of the Susceptible-Infectious-Recovered (SIR) model through mathematical comparison and network-based simulation.

Visualization example

1. Deterministic vs Stochastic Analysis (SIR_Deterministic.ipynb)

A Jupyter Notebook comparing the classic deterministic approach against stochastic simulations.

• Mathematical Model: Solves the system of discrete-time difference equations for $S(t)$, $I(t)$, and $R(t)$.
• Comparison: Contrasts the "average behavior" of infinite populations (Deterministic) with the variability of finite populations (Stochastic).
• Parameter Sweeps: Analyzes the effect of varying infection rates ($\beta$) on the epidemic curve.

2. Network Visualization (SIRVisual.py)

A Python module for simulating and visualizing SIR epidemics on complex networks. It visualizes the spread of infection node-by-node and compiles the frames into an animated GIF. It is important to note that within SIRVisual.py we are incorrectly using beta, we should be using beta = degree * infect_rate but instead are not factoring in degree into beta. Instead we are rng'ing on each edge for probability beta for each timestep.

Features

• Graph Generation: Supports Erdős-Rényi (ER), Watts-Strogatz (WS), and Barabási-Albert (BA) topologies.
• Visualization: Coloring nodes based on state (Susceptible, Infectious, Recovered) over time.
• GIF Export: Automatically saves animations of the outbreak.

Usage

Run with default parameters (Nodes: 100, Beta: 0.3, Gamma: 0.1):

python SIRVisual.py

Custom Parameters:

python SIRVisual.py --nodes 200 --beta 0.5 --gamma 0.05 --steps 150

Command-Line Options

Flag	Short	Default	Description
--nodes	-n	100	Number of nodes in the graph
--beta	-b	0.3	Infection probability ($S \to I$)
--gamma	-g	0.1	Recovery probability ($I \to R$)
--steps	-s	100	Number of simulation steps

3. SLIR Model Exploration (SLIR_Slim.ipynb)

• Exploration and implementation of the Susceptible-Latent-Infectious-Recovered (SLIR) model.
• Model Expansion: Adds a "Latent" compartment to the classic SIR model, capturing the delay between exposure and infectiousness.
• Deterministic vs Stochastic: Implements difference equations and simulations contrasting theoretical latent progression against random agent-level variation.

4. Competing Epidemics (SI_Comp_SII.ipynb)

• Investigates multi-strain/competing disease dynamics using pure Susceptible-Infectious models.
• SI/SII Framework: Analyzes systems where multiple variants or pathogens compete and spread through a population simultaneously.
• Evolution of Infection: Observes the competitive advantage of variants and evolution of the pathogen across populations over time.

5. Second Epidemic Wave (SIR_Second_Epidemic.ipynb)

• Examines the mathematical occurrence and behavior of a renewed outbreak.
• Repeat Infection Modeling: Simulates a second wave ("repeat epidemic") of infection within the SIR framework.
• Comparative Analysis: Contrasts deterministic expectations of trailing epidemic waves with stochastic realities and sudden resurgences.

6. Growth Rate Analysis (Growth_rate.ipynb)

• Focuses on the comparative growth metrics between simple and delayed-onset viral trajectories.
• SEIR vs SIR Comparison: Evaluates and contrasts the mathematical growth curves of models with an "Exposed" period against those without.
• Metric Extraction: Directly calculates and plots infected population growth rates mapped against various transmission ($\beta$) coefficients.

7. Agent-Based Network Modeling (Agent_based.ipynb)

• Granular approach explicitly controlling and stepping over non-compartmental simulated agents.
• Explicit Topologies: Models individual-level infection spread relying heavily on adjacency matrices (like Erdős-Rényi configuration graphs).
• Exact Initialization: Rigorously defines non-compartmental initial state conditions ($S_0$, $I_0$, $R_0$) in discrete networked agent populations (e.g., $N=100$).

Rumor Model

This section explores the spread of rumors across different network topologies using a modified infectious disease model.

The Model

The simulation categorizes nodes into three distinct states:

• Ignorant (Blue): Nodes that have not yet heard the rumor (Susceptible).
• Spreader (Red): Nodes actively spreading the rumor to their neighbors (Infected).
• Stifler (Green): Nodes that have heard the rumor but have stopped spreading it (Recovered).

The dynamics of the simulation are governed by two main parameters:

• Spreading Rate (λ): The probability that a Spreader successfully transmits the rumor to an Ignorant neighbor during contact.
• Stifling Rate (α): The probability that a Spreader becomes a Stifler after contacting another Spreader or a Stifler.

Network Topology Comparisons: Watts-Strogatz

Below are visualizations of the rumor spreading on Watts-Strogatz (Small-World) networks. We chose to compare the Clustering of two extremes to demonstrate how the rumor model is influenced by such model parameters.

Running the Simulation

You can generate your own visualizations using the RumorVisual.py script. By default, the simulation evaluates Erdős-Rényi, Watts-Strogatz, Barabási-Albert, and Configuration models on 100 nodes for 100 steps with λ=0.3 and α=0.1.

python3 RumorVisual.py -n 100 -l 0.3 -a 0.1 -s 100