SIS Dynamics on Complex Networks

Computational study of epidemic phase transitions, finite-size scaling and Griffiths phases in heterogeneous networks.

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Overview

This project investigates the critical behaviour of the Susceptible–Infected–Susceptible (SIS) model on complex networks through a combination of analytical methods and large-scale numerical simulations.

While originally motivated by epidemic spreading, the SIS model is also one of the fundamental models of non-equilibrium statistical physics, exhibiting a continuous phase transition that belongs to the Directed Percolation universality class.

The main objective of this work is to study how network topology and quenched disorder affect the epidemic threshold and the critical properties of the system.

The project combines:

• Analytical Mean-Field approaches.
• Degree-Based Mean-Field approximations.
• Stochastic simulations on complex networks.
• Finite-size scaling analysis.
• Critical exponent estimation.
• Griffiths phase characterization.
• Extensible disorder modelling.

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Scientific Motivation

The SIS model describes diseases in which infected individuals recover without acquiring immunity, allowing reinfection.

For sufficiently small infection rates, the disease eventually disappears. Above a critical threshold, however, the epidemic reaches an active stationary state and persists indefinitely.

This transition constitutes a paradigmatic example of a non-equilibrium phase transition and provides a powerful framework for studying universal critical phenomena on complex networks.

The project explores two complementary questions:

1. How does network topology modify the epidemic threshold and critical behaviour?
2. How does quenched disorder influence the transition and lead to the emergence of Griffiths phases?

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Part I — Topological Effects on SIS Dynamics

The first part of the project focuses on understanding how different network structures affect epidemic spreading.

Network Models

The SIS process is simulated on several classes of complex networks:

Erdős–Rényi (ER)

Random networks with approximately Poisson degree distributions.

Watts–Strogatz (WS)

Small-world networks combining local clustering and short characteristic path lengths.

Barabási–Albert (BA)

Scale-free networks generated through preferential attachment, characterized by heavy-tailed degree distributions.

Analyses Performed

• Numerical estimation of epidemic thresholds.
• Comparison with Mean-Field predictions.
• Comparison with Degree-Based Mean-Field predictions.
• Susceptibility peak analysis.
• Finite-size scaling.
• Critical exponent estimation.
• Time evolution of epidemic prevalence.

Key Observables

Epidemic Prevalence

The fraction of infected nodes:

$$\rho = \frac{N_I}{N}$$

Susceptibility

Used to locate the epidemic transition:

$$\chi = N \left( \langle \rho^2 \rangle - \langle \rho \rangle^2 \right)$$

Critical Behaviour

Near the epidemic threshold:

$$\rho \sim (\beta - \beta_c)^\beta$$

allowing the extraction of critical exponents and finite-size scaling relations.

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Part II — Griffiths Phases and Quenched Disorder

The second part extends the SIS model by introducing heterogeneous infection rates across the network.

Instead of assigning a single infection rate to all nodes, local infection dynamics can vary from node to node, generating quenched disorder.

Such heterogeneity can produce:

• Rare-region effects.
• Extended critical regions.
• Slow relaxation dynamics.
• Non-universal power-law behaviour.

These phenomena are collectively associated with the emergence of Griffiths phases.

Analyses Performed

• Disorder-driven SIS simulations.
• Density decay measurements.
• Effective exponent estimation.
• Rare-region analysis.
• Griffiths phase detection.
• Finite-size scaling under disorder.

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Extensible Disorder Framework

One of the main design goals of the project is the separation between epidemic dynamics and disorder generation.

The framework is built around a dedicated DisorderModel abstraction:

DisorderModel
      ↓
Node infection rates βᵢ
      ↓
SIS Simulator
      ↓
Observables & Analysis

This architecture allows new disorder mechanisms to be incorporated with minimal modifications to the rest of the codebase.

Potential future extensions include:

• Binary disorder.
• Uniform disorder.
• Gaussian disorder.
• Power-law disorder.
• Correlated disorder.
• Custom node-dependent infection dynamics.

In most cases, adding a new disorder model only requires implementing a new DisorderModel subclass and exposing it through the user interface.

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Project Workflow

The repository is organized around two main execution pipelines.

Topological Study

Network Generation
        ↓
SIS Simulation
        ↓
Observable Computation
        ↓
Threshold Detection
        ↓
Finite-Size Analysis
        ↓
Figure Generation

Griffiths Study

Network Generation
        ↓
Disorder Assignment
        ↓
SIS Simulation
        ↓
Density Decay Analysis
        ↓
Griffiths Phase Detection
        ↓
Figure Generation

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Repository Structure

SIS-Complex-Networks/

├── docs/
│   └── report/
│       ├── imagenes/
│       │   ├── BA_phase_trans.png
│       │   ├── bifurcation.png
│       │   ├── griffiths_finite_size.png
│       │   ├── time_evolution_stacked_500k.png
│       │   ├── comparison_phase_trans_500k.png
│       │   ├── comparison_suscept_500k.png
│       │   ├── comparison_critical_500k.png
│       │   ├── spectrum_q_0.30.png
│       │   ├── spectrum_q_0.90.png
│       │   ├── spectrum_q_2-3.png
│       │   └── phase_diagram.png
│       ├── report.bib
│       ├── report.pdf
│       └── report.tex
│
├── src/
│   ├── scripts/
│   │   ├── main.py
│   │   └── main_griffiths.py
│   │
│   ├── simulators/
│   │   ├── sis_simulator.py
│   │   └── griffiths_simulator.py
│   │
│   └── plotting/
│       ├── plot_summary.py
│       ├── plot_finite_size.py
│       ├── plot_phase.py
│       └── plot_bifurcation.py
│
├── .gitignore
├── README.md
└── requirements.txt

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Running Simulations

Topological Analysis

python src/scripts/main.py

This pipeline generates:

• SIS simulations.
• Threshold estimates.
• Susceptibility curves.
• Finite-size analyses.
• Publication-ready figures.

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Griffiths Analysis

python src/scripts/main_griffiths.py

This pipeline generates:

• Disorder realizations.
• Density decay measurements.
• Griffiths phase analyses.
• Scaling results.
• Publication-ready figures.

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Plotting Existing Data

The Griffiths analysis can be computationally expensive.

For this reason, dedicated plotting utilities are provided to work directly with previously generated datasets.

Examples include:

python src/plotting/plot_summary.py

python src/plotting/plot_finite_size.py

These scripts allow rapid experimentation with visualizations without repeating expensive simulations.

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Analytical Diagrams

Two standalone plotting scripts generate publication-ready analytical figures directly from theory, without requiring any simulation data.

Transcritical Bifurcation Diagram

python src/plotting/plot_bifurcation.py

Produces the mean-field transcritical bifurcation diagram for the homogeneous SIS model. The script computes the two stationary branches of the fixed-point equation as a function of the infection rate β:

• Healthy state (ρ* = 0): stable for β < μ, unstable above threshold.
• Endemic state (ρ* = 1 − μ/β): emerges continuously at β_c = μ and becomes the stable attractor for β > μ.

The bifurcation point is highlighted explicitly, illustrating the exchange of stability that defines the epidemic threshold. The plot is styled following APS journal conventions and is saved as Bifurcacion_SIS.png.

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Theoretical Phase Diagram

python src/plotting/plot_phase.py

Produces the theoretical phase diagram for the disordered SIS model on an Erdős–Rényi network, mapping the full parameter space spanned by the active node fraction (1 − q) and the infection rate β. Four distinct dynamical regimes are identified and colour-coded:

• Active phase: the epidemic sustains itself indefinitely above the critical line β_c(q).
• Griffiths phase: algebraic density decay driven by rare-region effects, present in the non-percolating regime above β_c(q_perc).
• Absorbing phase — stretched exponential: intermediate regime where the network percolates locally but the epidemic cannot survive globally.
• Absorbing phase — pure exponential: the network is deeply subcritical and the epidemic decays with a simple exponential envelope.

The critical line β_c(q) = μ ⟨k⟩/⟨k²⟩ · 1/(1 − q) is derived from the Degree-Based Mean-Field approximation. The percolation threshold and the multicritical point are marked explicitly. The script supports five output modes (full, colors_only, legend_text_only, legend_formulas_only, no_legend_no_params), making it straightforward to adapt the figure for presentations, reports, or publications.

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Main Results

The project produces a variety of scientific visualizations, including:

• Epidemic prevalence diagrams.
• Susceptibility curves.
• Threshold estimations.
• Finite-size scaling collapses.
• Critical exponent fits.
• Density decay curves.
• Griffiths phase signatures.

Example figures can be found in the report's images directory.

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Physics Topics Covered

• Complex Networks
• Network Science
• Epidemic Spreading
• SIS Dynamics
• Contact Process
• Directed Percolation
• Critical Phenomena
• Finite-Size Scaling
• Mean-Field Theory
• Degree-Based Mean-Field Theory
• Quenched Disorder
• Griffiths Phases
• Rare Region Effects
• Statistical Physics
• Computational Physics

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Technologies

• Python
• NumPy
• SciPy
• NetworkX
• Matplotlib
• Joblib

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Future Directions

Possible extensions include:

• Additional disorder distributions.
• Correlated disorder models.
• SIS variants with adaptive networks.
• SIR and SEIR epidemic dynamics.
• Continuous-time simulations.
• Parallel parameter sweeps on HPC systems.
• Automated critical exponent extraction pipelines.

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References

• Muñoz, M. A., Juhász, R., Castellano, C. & Ódor, G. — Griffiths Phases on Complex Networks, Phys. Rev. Lett. 105, 128701 (2010).
• Pastor-Satorras, R., Castellano, C., Van Mieghem, P. & Vespignani, A. — Epidemic processes in complex networks, Rev. Mod. Phys. 87, 925 (2015).
• Albert, R. & Barabási, A.-L. — Statistical mechanics of complex networks, Rev. Mod. Phys. 74, 47 (2002).
• Dorogovtsev, S. N., Goltsev, A. V. & Mendes, J. F. F. — Critical phenomena in complex networks, Rev. Mod. Phys. 80, 1275 (2008).
• Ódor, G. — Universality classes in nonequilibrium lattice systems, Rev. Mod. Phys. 76, 663 (2004).
• Noest, A. J. — New universality for spatially disordered cellular automata and directed percolation, Phys. Rev. Lett. 57, 90 (1986).
• Bray, A. J. — Nature of the Griffiths phase, Phys. Rev. Lett. 59, 586 (1987).
• Vojta, T. — Rare region effects at classical, quantum and nonequilibrium phase transitions, J. Phys. A 39, R143 (2006).
• Lee, M. Y. & Vojta, T. — Absorbing-state phase transitions on percolating lattices, Phys. Rev. E 79, 041112 (2009).
• Roig Oliver, A. & Martorell Argemí, C. M. — Proyectos de Sistemas Complejos: Enunciados y Guía de Modelado, Universidad de Granada (2026).

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About

Developed for the course:

Physics of Complex Systems

Physics Degree — Departamento de Electromagnetismo y Física de la Materia, Universidad de Granada (UGR)

Authors: Eugenio Etcheverría Sanz (implementation & analysis) & Javier Campos Vallejo (theory & analysis)