1. Network Properties
…and what they mean

2. 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

3. 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

4. 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

5. 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

6. 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.

7. 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

8. 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

9. 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.

10. 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

11. 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

12. 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.

13. 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

14. Further Reading
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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.