Science: Graph theory and networks

Dr Andy Evans

Graph/Network maths

Graph theory deals with networks as mixes of nodes and vertices (edges).Was limited to relatively simple graphs until more data on links and more processing power.Now huge research and development area.

Network statistics

Distribution/average of node degree (edges connected).

Distances:Eccentricity: distance from a node to the node furthest from it.Average path length: average eccentricity.Radius: minimum eccentricity in the graph.Diameter: maximum eccentricity in the graph.

Global clustering: how many nodes are connected in complete connection triangles (triadic closures) as a proportion of the connected triplets in the graph.

Other key statistics

Centrality: various measures, including degree, but two are:Betweenness centrality: number of shortest paths passing through a node.Closeness centrality: average of shortest paths to all other nodes.

Node degree (or other) correlation: how similar are nodes to their neighbours?

SoftwareMasses of software E.g. Inflow

Network CentralitySmall-World Networks Cluster Analysis Network Density Prestige / Influence Structural Equivalence Network Neighborhood External / Internal Ratio Weighted Average Path Length Shortest Paths & Path Distribution

http://en.wikipedia.org/wiki/Social_network_analysis_software

Pajek - for Large Network Analysishttp://pajek.imfm.si/doku.php?id=pajek

Programming GraphsGUESS (Open Source Java program)http://graphexploration.cond.org/Nicely uses GraphML, XML for representing graphs.

JUNG libraryhttp://jung.sourceforge.net/

R: various packages, including igraph.