class 12: communities network science: communities dr. baruch barzel

Class 12: Communities

Network Science: Communities

Dr. Baruch Barzel

The Modular Structure of Networks

)(kP

k

C

ijD

Is a Network Modular

Clustering implies modularity

Small Worldness tends to wipe out modularity

Functionality requires modularity


Clustering implies modularity

Small Worldness tends to wipe out modularity

Functionality requires modularity

Hubs tends to wipe out modularity


Clustering at the periphery only Low degree nodes typically belong to a single module

Hubs bridge between different modules

C

k


Clustering at the periphery only Low degree nodes typically belong to a single module

Hubs bridge between different modules

C

k

But how do we unveil the modules

The Modular Structure of Networks

Functional modularityNatural partition lines

Network Partitioning

Optimally dividing the network into a predefined number of partitions

nx

x

x

2

1

Dividing a task into sub-tasks


Optimally dividing the network into a predefined number of partitions

nx

x

x

2

1

Dividing a task into sub-tasks, while minimizing the transmission between tasks


Minimizing the Cut:

)(2

1

jQiijAR

The index vector:

bluein is if 1

redin is if 1

i

isi

ij

ijji AssR 12

1

2

1

sLssAksRij

Tjijijii

/ 4

1

4

1

ij i

ijijiiij ksskA The Laplacian Matrix:

Otherwise

0 and

0

1

ij

i

ij Aji

jik

L

The Laplacian Matrix

Minimizing the Cut: sLsR T /4

1

Otherwise

0 and

0

1

ij

i

ij Aji

jik

L

Consider the Eigenvector: ssL/

ssR T 4

1

Choose the Eigenvector with the minimal Eigenvalue


Minimizing the Cut: sLsR T /4

1

Consider the Eigenvector: ssL/

ssR T 4

1

Choose the Eigenvector with the minimal Eigenvalue

negative iselement if

positive iselement if

1

1

is


The matrix: ijijiij AkL

0)( iij

ijijij

jij kkAksLsL /

The trivial partitioning – put the entire network together:

ors



The case of isolated components

The number of Eigenvectors with λ = 0 equals the number of connected components

L/

s



The case of almost isolated components

The Eigenvectors with λ close to zero capture the partitioning

L/

s

From Partitioning to Communities

The number of communities and their size should be given by the network itself.

Hierarchical Clustering

4123

1410

2143

3034

ijW

Edges 4 Sides Stable Equal1. Square + + + +

2. Rectangle + + + --3. Circle -- -- -- --4. Triangle + -- + +

4


4123

1410

2143

3034

ijW



4

3


4123

1410

2143

3034

ijW



4

3

2


4123

1410

2143

3034

ijW



4

3

2

1

Dendograms

Topologically Induced Weights

jiWij and between paths ofnumber the toRelated

1

0

)()(

AIAWl

l

pathst independen node ofNumber ijW

jiA ijl and between paths ofNumber

Betweeness

Edge Betweeness – the number of paths through an edge

Football and Karate Networks

Zachary’s Karate Club

College Football

Ising and Potts Models

ji

ssij jiJE

,

0T T

1is


ji

ssij jiJE

,

0T T

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

1is


ji

ssij jiJE

,

0T T

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

mmmsi ,,1, Potts

Model


ji

ssij jiJE

,

0T T

0T

T

mmmsi ,,1,

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

class 12: communities network science: communities dr. baruch barzel

Documents

modularity slide

partitioning slide

edge slide

dendograms slide

minimal eigenvalue slide

baruch barzel slide

network partitioning

network nodes