Network Properties

…and what they mean

Complex Networks

• A Complex Network is just a network that shows features that you would not expect in a normal network

• Scale-free and small-world are two very common types of network

• Common properties include hierarchical structure and high clustering coefficient.

• Some other features will be covered later

Degree Distribution

• One of the most basic properties of a graph, but central to a lot of network analysis

• Quite simply, how many nodes in the graph have each degree

• Could follow many different distributions, such as Poisson, power-law, lognormal etc

• Random graphs follow the Poisson distribution

Scale-free Property

• Scale free is a very important degree distribution

• Very simply, means the degree distribution follows a power law

• Fraction of nodes having a degree of k is roughly K-γ. Usually 2 < γ < 3

• Many networks are conjectured to have this property. Some wikis seem to, but not all

Some results from Wikis

• ‘Club Penguin’ appears to follow a clear power law.• Note that 0 degree nodes do not ‘fit’• ‘Legopedia’, meanwhile, seems to mostly follow Poisson• If 0 degree nodes are ignored, most wikis

seem to follow power law, as expected

Some results from None-Wikis

• Here, the ‘terror’ network clearly follows a power law• As does the ‘protein’ network• In fact, only the random networks (of the results so far) show a different distribution.• These vary, but ER is Poisson.

Self Similarity

• An interesting property of complex networks• Explains the Scale Free property• Basically, complex networks generally consist

of finitely self-repeating patterns• I have not studied this in much detail yet, but

it is looking very interesting so far

Small World Property

• Small world is another very important property of networks

• Informally, it means that every node can reach every other node via a short number of steps

• Formally, it means that the shortest path length grows proportionately to the log of the number of nodes

• i.e. L log N∝

Small World Continued

• Scale-free networks are even smaller worlds, with the shortest paths scaling as:

L log log N∝• Wikis somewhat follow this property, with

some variation. Some of the variance makes sense, some does not, yet.

Network Motifs

• Motifs are another way to classify networks• Harder to visualise and compare.• A motif is a pattern of edges between a small number of nodes• Five and six node patterns can also be

analysed.• Frequency of motifs may be useful

Clustering Coefficient

• There are two types of clustering coefficient, global and local

• Global is simply the number of connected triangles divided by the total number of triangles in the graph

• Local is the proportion of links that occur between its neighbours to the number of possible links

Global Clustering Coefficient

• This serves as a measure of how clustered the nodes are.

• Seem to be representative of ‘type’ of network• Values align with structures of the wiki• Expected to be useful for ‘decision’ process.

Heterogeneity

• Seems to be the most useful stat so far• Determines how varied the degree

distribution is• Maximised for a star network• Minimised for ER network• Very complicated algorithm• More results will help here

Further Reading• http://www.mathstat.strath.ac.uk/downloads/publications/25report_heterogeneity.pdf - Heterogeneity

and some basics. A nice paper, if silly at times.• http://polymer.bu.edu/hes/articles/shm05nat.pdf - Self similarity. Quite an interesting read.• http://aris.ss.uci.edu/~lin/50.pdf - First introduction of global clustering coefficient. Quite tedious.• http://www.readcube.com/articles/10.1038/ng881?locale=en - Introduces motifs. Originally aimed for use

in biology.• labs.yahoo.com/files/w_s_NATURE_0.pdf? - Introduces 'small-world' networks. Language only vaguely

resembles English. I would recommend Wikipedia for this one.