# Class 12: Communities Network Science: Communities Dr. Baruch Barzel.

## Presentation on theme: "Class 12: Communities Network Science: Communities Dr. Baruch Barzel."— Presentation transcript:

Class 12: Communities Network Science: Communities Dr. Baruch Barzel

The Modular Structure of Networks

Is a Network Modular Clustering implies modularity Small Worldness tends to wipe out modularity Functionality requires modularity

Is a Network Modular Clustering implies modularity Small Worldness tends to wipe out modularity Functionality requires modularity Hubs tends to wipe out modularity

Is a Network Modular Clustering at the periphery only Low degree nodes typically belong to a single module Hubs bridge between different modules

Is a Network Modular Clustering at the periphery only Low degree nodes typically belong to a single module Hubs bridge between different modules But how do we unveil the modules

The Modular Structure of Networks Functional modularity Natural partition lines

Network Partitioning Optimally dividing the network into a predefined number of partitions Dividing a task into sub-tasks

Network Partitioning Optimally dividing the network into a predefined number of partitions Dividing a task into sub-tasks, while minimizing the transmission between tasks

Network Partitioning Optimally dividing the network into a predefined number of partitions Dividing a task into sub-tasks, while minimizing the transmission between tasks

Network Partitioning Minimizing the Cut: The index vector: The Laplacian Matrix:

The Laplacian Matrix Minimizing the Cut: Consider the Eigenvector: Choose the Eigenvector with the minimal Eigenvalue

The Laplacian Matrix Minimizing the Cut: Consider the Eigenvector: Choose the Eigenvector with the minimal Eigenvalue

The Laplacian Matrix The matrix: The trivial partitioning – put the entire network together: or

The Laplacian Matrix The matrix: The case of isolated components The number of Eigenvectors with λ = 0 equals the number of connected components

The Laplacian Matrix The matrix: The case of almost isolated components The Eigenvectors with λ close to zero capture the partitioning

From Partitioning to Communities The number of communities and their size should be given by the network itself.

Hierarchical Clustering Edges 4 Sides Stable Equal 1. Square + + + + 2. Rectangle + + + -- 3. Circle -- -- -- -- 4. Triangle + -- + +

Hierarchical Clustering Edges 4 Sides Stable Equal 1. Square + + + + 2. Rectangle + + + -- 3. Circle -- -- -- -- 4. Triangle + -- + +

Hierarchical Clustering Edges 4 Sides Stable Equal 1. Square + + + + 2. Rectangle + + + -- 3. Circle -- -- -- -- 4. Triangle + -- + +

Hierarchical Clustering Edges 4 Sides Stable Equal 1. Square + + + + 2. Rectangle + + + -- 3. Circle -- -- -- -- 4. Triangle + -- + +

Dendograms

Topologically Induced Weights

Betweeness Edge Betweeness – the number of paths through an edge

Football and Karate Networks Zacharys Karate Club College Football

Football and Karate Networks Zacharys Karate Club College Football

Ising and Potts Models

Groups of nodes with high link density will tend to have the same polarization Sparseness of connections between groups will allow different communities to have unrelated spins

Ising and Potts Models Groups of nodes with high link density will tend to have the same polarization Sparseness of connections between groups will allow different communities to have unrelated spins Potts Model

Ising and Potts Models

Link Communities Community - A group of densely connected nodes A group of topologically similar links Project Presentations (5 min.) 1.Define your network (nodes, links) 2.How will you get the data 3.Estimated size of network 4.Why