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
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
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
Newman and Watts Newmann, Watts (1999): Renormalization group analysis of the small-world network model , <ul><li>Ring like with Watts and Strogatz's
Pareto distributions - small number of highly connected nodes, most nodes have a small number of connections - Barabasi and Albert called them scale-free networks
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
Klemm and Eguíluz <ul>" 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. "This continues until eventually the node under consideration loses its popularity." <li>From: http://ifisc.uib-csic.es/victor/Nets/trn.html
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
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