Overview of Percolation Theory and Clustering Phenomena

Slide Note
Embed
Share

Explore various aspects of percolation theory, clusters, critical probabilities, and structures like the Sierpinski Gasket. Learn about site percolation, bond percolation, torus structures, and the double cover of the Sierpinski Gasket in a detailed visual format.

layanw
layanw Follow

Uploaded on Nov 19, 2024 | 0 Views

Download Presentation

Please find below an Image/Link to download the presentation.

The content on the website is provided AS IS for your information and personal use only. It may not be sold, licensed, or shared on other websites without obtaining consent from the author. Download presentation by click this link. If you encounter any issues during the download, it is possible that the publisher has removed the file from their server.

E N D

Presentation Transcript

  1. Percolation Clusters

  2. Site Percolation Cells are occupied with probability p, and empty with probability 1-p Clusters are groups of nearest neighboring cells

  3. For finite cases, a cluster percolates if it goes from one boundary to another p = 0.45, 0.55, 0.59, 0.65, and 0.75, respectively

  4. Bond Percolation Keep edge with probability p Remove edge with probability 1-p Cluster: network of connected components

  5. Torus Structure We are using the torus to model the lattice, Z x Z Each vertex has 4 neighbors

  6. Critical Probability Critical Probability: pcsuch that when p < pc the probability of having an infinite (or percolating) cluster is 0, and when p > pcthe probability of having an infinite cluster is 1.

  7. The Sierpinski Gasket SG level 2 Probability: 0.5

  8. SG level 4 Probability: 0.5

  9. SG level 7 Probability: 0.5

  10. Max Cluster at Level 7

  11. Double Cover of SG

  12. SG x SG (double cover)

  13. Critical Probability on SG x SG SG level 2 SG level 3

  14. SG3 x SG3and SG4 x SG4 SG level 4 SG level 3

  15. Cluster Variation SG3x SG3

  16. Cluster Variation Level 3:

  17. Number of Neighbors (SG x SG Level 4)

  18. Still To Come... Laplacian on percolation clusters Random walks Recurrent vs. Transient walks on clusters Cell graphs vs. vertex graph 3 neighbors rather than 4

Related


Understanding Percolation Theory: The Problem, Simulation, and Techniques

Understanding Percolation Theory: The Problem, Simulation, and Techniques

dess_46
Investigation of QGP Formation at Fermi Lab CO Collider

Investigation of QGP Formation at Fermi Lab CO Collider

thoren
Efficient Parameter-free Clustering Using First Neighbor Relations

Efficient Parameter-free Clustering Using First Neighbor Relations

gangloff_s
Text Analytics and Machine Learning System Overview

Text Analytics and Machine Learning System Overview

kap_213
Understanding Clustering Algorithms: K-means and Hierarchical Clustering

Understanding Clustering Algorithms: K-means and Hierarchical Clustering

axton
Understanding Clustering Methods for Data Analysis

Understanding Clustering Methods for Data Analysis

vue_dd
Correlation Clustering: Near-Optimal LP Rounding and Approximation Algorithms

Correlation Clustering: Near-Optimal LP Rounding and Approximation Algorithms

tiber
Achieving Demographic Fairness in Clustering: Balancing Impact and Equality

Achieving Demographic Fairness in Clustering: Balancing Impact and Equality

damb278
Understanding Text Vectorization and Clustering in Machine Learning

Understanding Text Vectorization and Clustering in Machine Learning

jaad
Understanding Clustering Algorithms in Data Science

Understanding Clustering Algorithms in Data Science

boon_win
Insight into De-Confinement in High-Energy Nuclear Collisions

Insight into De-Confinement in High-Energy Nuclear Collisions

selina
Understanding Soil Texture and Its Impact on Rangeland Health

Understanding Soil Texture and Its Impact on Rangeland Health

rayd_76
Social Processes of Science: A Percolation Model

Social Processes of Science: A Percolation Model

nordlund_k
Density-Based Clustering Methods Overview

Density-Based Clustering Methods Overview

maon699
Understanding Transitivity and Clustering Coefficient in Social Networks

Understanding Transitivity and Clustering Coefficient in Social Networks

smora
Understanding Corporate Climate Assessment Using NLP Clustering

Understanding Corporate Climate Assessment Using NLP Clustering

tai_dib
Understanding Major Terms, Cluster Labels, and Themes in IN-SPIRE Training

Understanding Major Terms, Cluster Labels, and Themes in IN-SPIRE Training

swayzeem
Exploring Avatar Path Clustering in Networked Virtual Environments

Exploring Avatar Path Clustering in Networked Virtual Environments

trovato_m
Theories of Interest in Microeconomics II

Theories of Interest in Microeconomics II

myon299
Bioinformatics for Genomics Lecture Series 2022 Overview

Bioinformatics for Genomics Lecture Series 2022 Overview

eleni
Understanding Reservoir Connectivity and Its Impact on Recovery Models

Understanding Reservoir Connectivity and Its Impact on Recovery Models

woessner_h
Theories of International Relations Overview

Theories of International Relations Overview

kohen
Understanding Neural Networks: Models and Approaches in AI

Understanding Neural Networks: Models and Approaches in AI

zackary
Machine Learning Techniques: K-Nearest Neighbour, K-fold Cross Validation, and K-Means Clustering

Machine Learning Techniques: K-Nearest Neighbour, K-fold Cross Validation, and K-Means Clustering

yetz582

More Related Content