Degree Distributions

Emergence of networks Many networks naturally emerge and adapt What kind of connectivity do they present Nonlinear Complex https://www.youtube.com/watch?v=vp8v2Udd_PM

Degree Distribution Defined: Frequency distribution of the degree sequence pk is the fraction of degree-k vertices, or the probability that a randomly-selected node will have degree k Given the below example, determine the degree sequence and the degree distribution.

Degree Distribution (con’t) The Internet What kind of network that we talked about this morning may this be? Right-skewed Many nodes with small degrees, few with extremely high Largest degree is 2407 (not shown). Since 𝑛=19956 this node is adjacent to 12% of the network UNCLASSIFIED

Degree Distribution (con’t) Directed networks have both in- and out-degree distributions The World Wide Web

Degree Distribution (con’t) Directed networks have both in- and out-degree distributions The World Wide Web

Scale-sFree Networks whose degree distribution follows a power law are called scale-free networks. If a network is directed, the scale-free property applies separately to the in- and the out-degrees. The main difference between a random and a scale-free network comes in the tail of the degree distribution, representing the high-k region of pk

Poisson vs. Power-law Distr. Notation: 〈k〉=average degree. Poisson vs. Power-law with a power-law function on a linear plot. Both distributions have 〈k〉= 11. The same curves as in (a), but shown on a log-log plot, allowing us to inspect the difference between the two functions in the high-k regime. A random network with 〈k〉= 3 and 50 nodes, illustrating that most nodes have comparable degree ≈〈k〉 A scale-free network with and 〈k〉=3, illustrating that numerous small-degree nodes coexist with a few highly connected hubs. The size of each node is proportional to its degree. http://networksciencebook.com/chapter/4#hubs

Hubs Hubs are nodes of high degree Emerge in time They are not present in a random network COMPLETE GRAPH Hubs in a scale-free network are several orders of magnitude larger than the biggest degree node in a random network with the same N and 〈k〉 http://networksciencebook.com/chapter/4#hubs