Degree Distribution, and Scale-free networks
Prof. Ralucca Gera, Applied Mathematics Dept. Naval Postgraduate School Monterey, California [email protected]
Excellence Through Knowledge Learning Outcomes
Understand the degree distributions of observed networks Understand Scale Free networks Power law degree distribution Versus exponential, log-normal, …
2 Degree Distribution
Excellence Through Knowledge Degree distributions
• The degree distribution is a frequency distribution of the degree sequence
• We define pk to be the fraction of vertices in a network that have degree k
• That is the same as saying: pk is the probability that a randomly selected node of the network will have degree k
1 2 4 2 1 p , p , p , p , p , p 0k 5 0 10 1 10 2 10 3 10 4 10 k
• A well connected vertex is called a hub 4 Plot of the degree distribution
The Internet Commonly seen plots of real networks
Right-skewed
• Many nodes with small degrees, few with extremely high – Largest degree is 2407 (not shown). Since 19956 this node is adjacent to 12% of the network 5 Degree distribution in directed networks
• For directed networks we have both in- and out-degree distributions.
The Web
6 Observed networks are SF (power law)
• Power laws appear frequently in sciences: • Pareto : income distribution, 1897 • Zipf-Auerbach: city sizes, 1913/1940’s • Zipf-Estouf: word frequency, 1916/1940’s • Lotka: bibliometrics, 1926 • Yule: species and genera, 1924. • Mandelbrot: economics/information theory, 1950’s+ • Research claims that observed real networks are scale free, not consistent defn., [ref. 1–7], i.e. – The fraction of nodes of degree 𝑘 follows a power law distribution 𝑘 , where 𝛼 1, – Generally being observed 𝛼∈ 2, 3 , see [ref. 8, 9] – Or at least the upper tail of the distribution is power law) [ref. 10] – Or it is more plausible than other thin-tailed distributions [ref. 11]
Broido and Clauset “Scale-free networks are rare”, Jan 2018, https://arxiv.org/abs/1801.03400 [ref. 13] https://www.eecs.harvard.edu/~michaelm/TALKS/Radcliffe.ppt Power law distributions