Network properties
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This document discusses various properties of complex networks. It describes scale-free networks, which have a power law degree distribution where highly connected hubs are common. Many real-world networks exhibit small-world properties, where the average path length between nodes grows proportionally to the log of the number of nodes. Networks also show features like high clustering, hierarchical structure, self-similarity, and specific patterns of connections between small groups of nodes called motifs. Different network analysis techniques like measuring degree distribution, clustering coefficients, and heterogeneity can help characterize network structure and compare real networks.
Network properties
- 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 • • • • • 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.