informatics tools in network science
DESCRIPTION
seminar 3 Measurements. Informatics tools in network science. Network Topology. Simple examples. What else? Network Skeleton Visualization (largescale?) Fractal properties Etc. Degree centrality. 6. 1. 3. 4. 5. 7. 2. Network degree centrality. n*: node with highest degree. - PowerPoint PPT PresentationTRANSCRIPT
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