1.
Some random graphs for network models Birgit Plötzeneder

2.
Bell-shaped node degree distributions

3.
Random model Erdös,Renyi (1960s) On random graphs I; On the evolution of random graphs; On the strength of connectedness of a random grap h - start with N disconnected nodes - connect nodes with probability p to each other

4.
Watts and Strogatz Watts, Strogatz (1998), Collective dynamics of "small-world" networks - one-dimensional ring lattice of N nodes connected to its 2 K nearest neighbors - goes through each of the edges in turn and, independently with probability p "rewire" it to a randomly selected (different) node

5.
Watts and Strogatz - average distance grows like O(log(N) and not O(N). - support high levels of clustering „ The small-world effect (short average distance between nodes and high levelsof clustering) has been detected in networks that include a network of actors in Hollywood, the power generator network in the western US...“ Gerardo Chowell and Carlos Castillo-Chavez, Worst-Case Scenarios and Epidemics

6.
Newman and Watts Newmann, Watts (1999): Renormalization group analysis of the small-world network model , <ul><li>Ring like with Watts and Strogatz's

7.
Don't replace edges, instead create shortcuts </li></ul>

8.
Power-law degree distributions = Pareto distributions

9.
Pareto distributions - small number of highly connected nodes, most nodes have a small number of connections - Barabasi and Albert called them scale-free networks

10.
Barabási and Albert Barabàsi, Albert (1999) Emergence of scaling in random networks - starts with a small number of nodes - a new node connects with higher probability to nodes that have already accumulated a higher number of connections

11.
Klemm and Eguíluz <ul>&quot; When a node is created it is linked to nodes that are popular at the time. It then receives links from nodes created subsequently, he said. &quot;This continues until eventually the node under consideration loses its popularity.&quot; <li>From: http://ifisc.uib-csic.es/victor/Nets/trn.html

12.
Klemm, Eguíluz (2002) Growing scale-free networks with small-world behavior </li></ul>

13.
fat-tailed degree distribution

14.
Dorogovtsev, Mendes, Samukhin Dorogovtsev, Mendes, Samukhin : How to generate a random growing network - with each step, the edges of a growing network are transformed into configurations of edges and new vertices according to some probability function