1. Degree Distributions

2. Emergence of networks Many networks naturally emerge and adapt
What kind of connectivity do they present
Nonlinear
Complex

3. 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.

4. 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

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

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

7. 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

8. 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.

9. 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〉