1. Models of networks (synthetic networks or generative models): Random, Small-world, Scale-free, Configuration model and Random geometric model
By: Ralucca Gera, NPS
Excellence Through Knowledge

2. The world around us as a network
What do social networks look like?
Watch this video
What categories do we have for networks?
Random networks (normal degree distribution)
Scale free (power-law degree distribution)

3. Erdős-Rényi Random Graphs

4. Random graphs (Erdős-Rényi , 1959)
ER is a model in which some specific set of parameter takes fixed values, and the network is created at random using these values.
Two main examples:
G(n,p): fix n and probability p of the edges between vertices. The number of edges is not fixed. This is the default construction.
G(n, m): fix n and m
The mean value of edges: 𝑚= 𝑛 2 · 𝑝= 𝑛 𝑛−1 𝑝 2

5. .
Generating Erdős-Rényi random networks
ER graphs are models of a network in which some specific set of parameters take fixed values, but the construction of the network is random (see below in Gephi)

6. Generating Erdős-Rényi

7. Generating Erdős-Rényi random networks
Reference for python:

8. Watts-Strogatz Small World Graphs (1998)

9. Small worlds, between perfect order and chaos
the first graph is completely ordered (probability p =0), the graph in the middle is a "small world" graph (0 < p < 1), the graph at the right is complete random (p=1).
Source:

10. Generating Watts-Strogatz

11. Generating Watts-Strogatz networks

12. preferential attachment model (we will consider the Barabasi-Albert example)

13. Network growth & resulting structure
random attachment: new node picks any existing node to attach to
preferential attachment: new node picks from existing nodes according to their degrees (high preference for high degree)

14. Scale Free networks
One example is the one introduced by Barabasi-Albert based on preferential attachment:
Start with a small set of nodes ( 𝑚 0 ) and no edges
Attach new nodes one at the time;
each with the same fixed number 𝑙 of new edges, attaching to the existing ones in the network, with preference for high degrees (once the high degrees appear)
This is not the only way to get scale–free networks!

15. Generating Barabasi-Albert

16. Generating Barabasi-Albert

17. Generating Barabasi-Albert networks

18. Many modifications of this model exists, based on:
Modified BA
Many modifications of this model exists, based on:
Nodes “retiring” and losing their status
Nodes disappearing (such as website going down)
Links appearing or disappearing between the existing nodes (called internal links)
Fitness of nodes (modeling newcomers like Google)
Most researchers still use the standard BA model when studying new phenomena and metrics. Why? It is a simple model, and it was the first model that brought in growth (as well as preferential attachment)

19. The Malloy Reed Configuration model

20. The configuration model
A random graph model created based on Degree sequence of choice (can be scale free)
Maybe more than degree sequence is needed to be controlled in order to create realistic models

21. A zoo of complex networks

22. Random, Small-World, Scale-Free
Scale Free networks:
High degree heterogeneity
Various levels of modularity
Various levels of randomness
Man made, “large world”:

23. Python
References to the classes that exist in python: