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# Network properties

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### Network properties

1. 1. Network Properties …and what they mean
2. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 14. 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.