Informatics tools in network science

seminar 3

Measurements

Network Topology

Simple examples

What else?• Network Skeleton• Visualization (largescale?)• Fractal properties• Etc.

Degree centrality

1 3 4

2

5

6

7

Node  ScoreStandardized 

Score

1  1  1/6

2  1  1/6

3  3  3/6 = 1/2

4  2 2/6 = 1/3

5  3  3/6 = 1/2

6  2  2/6  = 1/3

7  2  2/6 = 1/3

Network degree centrality

The higher the value of the measure the higher the difference of the node with the highest Degree Centrality to all other nodes in the network is.

n*: node with highest degree

Infinite:

Minimal:

Betweenness Centrality

The Betweenness Centrality is the normalized number of shortest paths going through a node in a network.

Closeness centrality

The Closeness Centrality is the normalized number of steps required to access every other node from a given node in a network.

Length of the shortest path

Eigenvector centrality, PageRank

PageRank

Eigenvector centrality

Clustering coefficientGlobal clustering coefficient

The local clustering coefficient of a vertex in a graph quantifies how close its neighbors are to being a clique (complete graph).

C = 1/3

Clustering coefficient

degree distribution

Random graph

Scale-free network

(the clustering coefficient behaves the same)

Network motifs

Topological Overlap

the ratio of shared nodes over the number of nodes reachable from a particular pair of nodes

Minimal (0) Maximal (1)

Diameter and Density

The Diameter considers the largest geodesic distance between any pair of nodes in a network.

The measure Density is the proportion of possible edges that are actually present in the network.

Module measurementsEffective number of modules

Overlap value of elements (e.g. effective number of module belongs)

Bridgeness value of elements:

The bridgeness measure of an element or link as the overlap of the given element or link between two or more modules relative to the overlap of the other elements or links.

T is the area-overlap, or common area of element between modules

.

The total bridgeness of element i describes the bridgeness of that element between all modules:

Module similarity of elements:

The similarity of the elements i and j is based on their module membership vectors, di and dj :

Network capacity

e.g. Maximal flow (minimal cut) problem

Robustness

Structural cohesion: how many node needs to be removed to disconnect the graph

2 1 5

Network connectivity

Average geodesic length (the characteristicPath length): normalized average length of all shortest path in the network

infinite in case of disconnected graph

Inverse geodesic length

Effective number

î ii ppe

)log(

j j

ii v

vp

30

60 2 1 1 1100

average sum

165

165

27.5

27.5

eff. num

5.975

1.570

30 2025 20 2530

Take home messages

• separate the giant component (if exists)

• compare measurements (test graph families, controll networks)

• use effective numbers

• check distributions

Programs• R• ModuLand• Pajek• Cytoscape with plugins:

– NetworkAnalyzer: distributions– NetMatch: Motif search– GraMoFoNe: Graph Motif For Networks– CentisCaPe: centrality values

• (lényegiDB)• + python modules

Python